Qualgebras and knotted 3-valent graphs

Transcription

1 Qulgers nd knotted 3-vlent grphs Vitori Leed To ite this version: Vitori Leed. Qulgers nd knotted 3-vlent grphs <hl > HAL Id: hl Sumitted on 25 Fe 2014 HAL is multi-disiplinry open ess rhive for the deposit nd dissemintion of sientifi reserh douments, whether they re pulished or not. The douments my ome from tehing nd reserh institutions in Frne or rod, or from puli or privte reserh enters. L rhive ouverte pluridisiplinire HAL, est destinée u dépôt et à l diffusion de douments sientifiques de niveu reherhe, puliés ou non, émnnt des étlissements d enseignement et de reherhe frnçis ou étrngers, des lortoires pulis ou privés.

2 Qulgers nd knotted 3-vlent grphs Vitori Leed Ferury 25, 2014 Astrt This pper is devoted to qulgers nd squndles, whih re qundles enrihed with omptile inry/unry opertion. Algerilly, they re modeled fter groups with onjugtion nd multiplition/squring opertions. Topologilly, qulgers emerge s n lgeri ounterprt of knotted 3-vlent grphs, just like qundles n e seen s n lgeriztion of knots; squndles in turn simplify the qulger lgeriztion of grphs. Knotted 3-vlent grph invrints re onstruted y ounting qulger/squndle olorings of grph digrms, nd re further enhned using qulger/squndle 2-oyles. Some lgeri properties nd the eginning of ohomology theory re given for oth strutures. A lssifition of size 4 qulgers/squndles is presented, nd their seond ohomology groups re ompletely desried. Keywords: qundles; knotted 3-vlent grphs; qulgers; squndles; olorings; ounting invrints; Boltzmnn weight; oyle invrints; qulger ohomology. 1 Introdution A qundle is set Q endowed with two inry opertions nd stisfying the following xioms: RIII self-distriutivity: ( ) = ( ) ( ), (Q SD ) RII invertiility: ( ) = ( ) =, (Q Inv ) RI idempotene: =. (Q Idem ) Sine opertion n e dedued from using (Q Inv ), we shll often omit it from the definition. Originting from the work of topologists D.Joye nd S.Mtveev [Joy82, Mt82], this struture n e seen s n lgeri ounterprt of knots. Indeed, onsider olorings of the rs of knot digrms y elements of Q, ording to the rule on Figure 1 A. This oloring rule is omptile with Reidemeister moves (Figure 3) if nd only if Axioms (Q SD )-(Q Idem ) re verified, eh xiom orresponding to the Reidemeister move indited in the left olumn ove. Thus the numer of digrm olorings y fixed qundle defines n invrint of underlying knots nd links. This invrint n e strengthened y endowing eh olored rossing nd hene, summing everything together, eh digrm oloring with weight (Figure 4). The weights re lulted using qundle 2-oyle of Q ording to proedure suggested y Crter-Jelsovsky-Kmd-Lngford- Sito ([CJK + 03]). 2 A B C 2 Figure 1: Colorings y qundles, qulgers nd squndles From the lgeri viewpoint, the qundle struture n e regrded s n xiomtiztion of the onjugtion opertion in group. Conretely, group with the onjugtion opertion = 1 is lwys qundle, nd ll the properties of onjugtion tht hold in every group re onsequenes of (Q SD )-(Q Idem ). 1

3 The purpose of this pper is to find n lgeri ounterprt of knotted 3-vlent grphs (further simply lled grphs for revity) whih would develop the qundle ides. To this end, we introdue the qulger struture. It is qundle (Q, ) endowed with n dditionl inry opertion stisfying RIV trnsltion omposility: ( ) = ( ), (QA Comp ) RVI distriutivity: ( ) = ( ) ( ), (QA D ) RV semi-ommuttivity: = ( ). (QA Comm ) Restriting oneself to well-oriented grphs (i.e., hving only zip nd unzip verties, f. Figure 7) ndextendingthe qundleoloringrules1 A to3-vlentvertiessshownonfigure1 B, onegets rules omptile with Reidemeister moves for grphs (Figure 5) if nd only if Axioms (QA Comp )- (QA Comm ) re stisfied, eh xiom orresponding to the Reidemeister move indited on the left. Imitting wht ws done for qundle olorings of knots, one n thus define qulger ounting invrints for grphs. The ltter n e upgrded to weight invrints using the qulger 2- oyles introdued in this work. Qulger 2-oyles onsist of two mps, one of whih is used for putting weights on rossings, nd the other one for putting weights on 3-vlent verties (Figures 4 nd 16); the weight of olored digrm is otined, s usul, y summing everything together. A group with the onjugtion qundle opertion eomes qulger with the group multiplition s dditionl opertion: =. Algerilly, the dditionl qulger xioms enode the reltions etween onjugtion nd multiplition opertions in group (see Tle 1). Note tht, however, our qulger xioms do not imply ny of those used in the stndrd definition of group. In prtiulr, we shll give exmples of 4-element qulgers for whih the opertion is non-nelltive, non-ssoitive, nd hs no neutrl element. Besides defining qulgers nd onstruting ounting nd weight invrints of grphs out of them, in this work we study some si properties of qulgers; give omplete lssifition of 4- element qulgers(showing tht single qundle n e the se of numerous qulger strutures with signifintly different properties); nd suggest the eginning of qulger ohomology theory, lulting in prtiulr the seond ohomology group for 4-element qulgers. Moreover, we ompute ertin qulger ounting nd weight invrints for some pirs of grphs, showing tht these grphs n e distinguished using our methods. In prllel with the qulger struture, we study the losely relted squndle struture. It is defined s qundle (Q, ) endowed with n dditionl unry opertion 2, oeying the following xioms (modeled fter the properties of onjugtion nd squring opertions in group): RIV 2 = ( ), (SQ 1 ) RVI 2 = ( ) 2. (SQ 2 ) A qulger with the squring opertion 2 = is n exmple of squndle. The oloring rule from Figure 1 C llows to onstrut invrints of grphs y ounting squndle olorings of their digrms; weight invrints re otined with the help of squndle 2-oyles. The terms qulger nd squndle oth ome from the nmes of the two opertions prtiipting in the definition of these strutures, zipped together s indited on Figure 2. qundle lger squring qundle qulger squndle Figure 2: The terms qulger nd squndle The pper is orgnized s follows. The lnguge of olorings, used throughout this pper, is developed in Setion 2. It is illustrted with the fmous exmple of qundle olorings of knot digrms, from whih some of our further onstrutions tke inspirtion. We then turn to invrints of grphs whih extend the qundle invrints of knots. In Setion 3, fter rief survey of suh extensions found in the literture, we propose n originl one sed on qulger olorings. Our 2

4 invrints re defined for well-oriented grphs only, ut they re shown to indue invrints of unoriented grphs. We further show tht groups give n importnt soure of qulger exmples. Construtions from [Ish13] nd [Deh86, Drá95, Deh07], lose to ut different from ours, re lso disussed. The notion of squndle is introdued in Setion 4, motivted y the onept of speil olorings (with isoseles qulger olorings s the mjor exmple here). Squndle olorings re then used for distinguishing Kinoshit-Tersk nd stndrd Θ-urves. Setion 5 ontins short study of si properties of qulgers nd squndles, pplied to omplete lssifition of qulgers/squndles with 4 elements. One of the exoti strutures otined is next used for distinguishing two uff grphs. Setion 6 is devoted to the notions of qulger/squndle 2-oyles nd 2-ooundries, s well s to the indued weight invrints of grphs. Qulger/squndle 2-oyles nd seond ohomology groups re lulted for 4-element strutures. The lst setion ontins severl suggestions for further development of the qulger ides presented here. Aknowledgements The uthor is grteful to Seiihi Kmd nd Józef Przytyki for stimulting disussions, nd to Arnud Mortier for his omments on n erlier version of this mnusript. During the writing of this pper, the uthor ws supported y JSPS Postdotrl Fellowship For Foreign Reserhers nd y JSPS KAKENHI Grnt Colorings: generlities nd the qundle exmple One of the most nturl nd effiient methods of onstruting invrints of ertin topologil ojets (suh s knots, rids, tngles, knotted grphs, knotted surfes, et.) onsists in studying olorings of their digrms y ertin sets of olors. If the oloring rules re refully hosen, one n extrt invrints of underlying topologil ojets y studying digrm olorings for instne, onsidering their totl numer, or some more sophistited oloring hrteristis. In this setion we develop generl frmework for suh oloring invrints nd illustrte it with the elerted exmple of qundle olorings for knots. We prefer nrrtive style to list of definitions here for the ske of redility. The rest of the pper is devoted to severl pplitions of these oloring ides to knotted 3-vlent grphs. Topologil olorings, ounting invrints nd qundles Let us now fix lss of 1-dimensionl digrms on surfe (e.g., fmilir knot digrms in R 2 ). For this lss of digrms, hoose severl types of speil points, with the lol piture of digrm round speil point eing determined y the point type(rossing points, points of lol mximum nd grph verties re typil exmples). These lol pitures re lled type ptterns (see Figure 1 for the exmples of oriented rossing point nd 3-vlent vertex ptterns). We wnt to study digrms up to speil-point-preserving isotopy, nd up to set of lol (i.e., relized inside smll ll) invertile moves, lled R-moves (the exmple inspiring the nme is tht of Reidemeister moves for knots, f. Figure 3). Digrms relted y isotopy nd R-moves re lled R-equivlent. This defines n equivlene reltion on the set of digrms, whih orresponds in the ses of interest to the isotopy equivlene for underlying topologil ojets. RI RII RIII Figure 3: Reidemeister moves for knot digrms An r is prt of digrm delimited y speil points. Fix set S (possily with some lgeri struture), whih we think of s the oloring set. An S-oloring of digrm D is mp C : A(D) S 3

5 from the set of its rs to S, stisfying some presried oloring rules for rs round speil points. The set of suh olorings of D is denoted y C S (D). The notion of S-oloring extends from our lss of digrms to tht of su-digrms (for instne, those involved in n R-move) in the ovious wy. In the pitures, n r α is often deorted with its olor C(α). Definition 2.1. S-oloring rules re lled topologil if for ny (su-)digrm D, ny C C S (D) nd ny D otined from D y pplying one R-move, there exists unique oloring C C S (D ) oiniding with C outside the smll ll where the R-move ws effetuted. Suh oloring rules llow one to onstrut invrints under R-equivlene. The most si ones re ounting invrints: Lemm 2.2. Fix lss of digrms, set S nd topologil S-oloring rules. For ny R-equivlent digrms D nd D, there exists (non-nonil) ijetion etween their S-oloring sets: C S (D) ij C S (D ). (1) In prtiulr, the funtion D #C S (D) (where one llows the vlue ) is well-defined on R- equivlene lsses of digrms. Thus, if R-equivlene of digrms orresponds to the isotopy equivlene for underlying topologil ojets, the lemm produes invrints of these topologil ojets. Proof. If D nd D differ y single R-move, one n tke the ijetion from the definition of topologil oloring rules. Composing these ijetions, one gets the result for the se when D nd D differ y severl R-moves. Before giving n exmple of topologil oloring rules, we need onvention onerning orienttions: Convention 2.3. In lss of oriented digrms, using unoriented strnds in R-moves or oloring rules mens imposing these moves or rules for ll possile orienttions. Exmple 2.4. Consider the lss of oriented knot digrms in R 2, rossing points s the only type of speil points, Reidemeister moves from Figure 3 s R-moves, set Q endowed with inry opertion s the oloring set, nd Q-oloring rules from Figure 1 A. From the pioneer ppers [Joy82, Mt82], these rules re known to e topologil if nd only if the struture (Q, ) is qundle, i.e., stisfies Axioms (Q SD )-(Q Idem ) (eh of whih orresponds to one Reidemeister move). AtypilexmpleonsistsofgroupGwith theonjugtionopertion = 1, lled onjugtion qundle. Counting invrints for suh olorings even y simplest finite qundles Q pper to e rih nd effiiently omputle. Note lso tht they re esily generlized to the digrms of links nd tngles, s well s to their virtul versions. Weight invrints nd qundle 2-oyles Let us return to the generl setting of lss of digrms endowed with topologil S-oloring rules. Counting invrints, though lredy very powerful for qundle olorings of knots, do not exploit the full potentil of the ijetion from (1). More informtion n e extrted out of it using the following onept: Definition 2.5. A weight funtion ω is olletion of mps, one for eh type of speil points on our lss of digrms, ssoiting n integer to ny S-olored pttern of the orresponding type. The ω-weight of n S-olored (su-)digrm (D,C), denoted y W ω (D,C), is the sum of the vlues of ω on ll its speil points (we suppose the numer of the ltter finite). If for ny R-move the ω-weights of the two involved su-digrms orrespondingly S-olored (in the sense of Definition 2.1) oinide, then ω is lled Boltzmnn weight funtion. Boltzmnn weight funtions llow to upgrde ounting invrints to wht we ll here weight invrints: 4

6 Lemm 2.6. Fix lss of digrms, set S, topologils-oloringrules nd Boltzmnn weight funtion ω. Then the multi-sets of ω-weights of ny R-equivlent digrms D nd D oinide: {W ω (D,C) C C S (D)} = {W ω (D,C ) C C S (D )}. (2) In prtiulr, restrited to the digrms D for whih the set C S (D) is finite, the funtion D t Wω(D,C) Z[t ±1 ] C C S(D) is well-defined on R-equivlene lsses of digrms. Proof. If D nd D differ y single R-move, then Definition 2.1 desries ijetion etween C S (D) nd C S (D ) suh tht orresponding olorings C nd C differ only in smll lls where the R-move is effetuted; Definition 2.5 then gives W ω (D,C) = W ω (D,C ), implying the desired multi-set equlity. Iterting this rgument, one gets the result for the se when D nd D differ y severl R-moves. Note tht Equlity (2), s well s most further exmples nd results, remin vlid if weight funtions re llowed to tke vlues in ny Aelin group nd not only in the group of integers Z. Exmple 2.7. ContinuingExmple 2.4, tke mp χ : Q Q Znd onsiderweightfuntion, still denoted y χ, tht depends only on two of the olorsroundrossingpoint (whih is the only type of speil points here) s shown on Figure 4. In [CJK + 03] this weight funtion ws shown to e Boltzmnn if nd only if it stisfies the following xioms for ll elements of Q (orresponding, respetively, to moves RIII nd RI, the remining one eing utomti): χ(,)+χ(,) = χ(, )+χ(,), (3) χ(,) = 0. (4) Moreover, these onditions were interpreted s the definition of 2-oyles from the elerted qundle ohomology theory. In this theory, 2-ooundries re defined y χ ϕ (,) = ϕ( ) ϕ() (5) for ny mp ϕ : Q Z, nd they re preisely the 2-oyles suh tht W χ vnishes on ll Q-olored knot digrms. χ(,) χ(,) Figure 4: Qundle 2-oyle weight funtion for knot digrms Weight invrints of knots onstruted out of qundle 2-oyles re known s qundle oyle invrints. They re even more effiient thn qundle ounting invrints, sine the sme smll qundle n dmit vrious 2-oyles. Moreover, they re stritly stronger thn qundle ounting invrints sine, ontrry to the ltter, they n distinguish knot from its mirror imge. See [CJK + 03, Km02, CJKS01, CKS03, HN07, NP09] nd referenes therein for more detils. 3 Qulger oloring invrints of knotted 3-vlent grphs We now turn to our min ojet of study, nmely, to knotted 3-vlent grphs (i.e., emeddings of strt 3-vlent grphs into R 3 ) nd their digrms in R 2 ; see Figures 14 nd 15 for typil exmples. In wht follows, the word grph is often used insted of knotted 3-vlent grph for revity. Two types of speil points re relevnt for grph digrms: rossing points nd grph verties. In 1989, L.H.Kuffmn, S.Ymd nd D.N.Yetter independently [Ku89, Ym89, Yet89] extended the Reidemeister moves for knots (Figure 3) y the three moves presented on Figure 5, 5

7 showing tht the resulting 6 moves preisely desrie grph isotopy in R 3. We therefore hoose them s R-moves here, noting tht R-equivlene lsses of grph digrms now orrespond to isotopy lsses of represented grphs. The nmes of the moves re hosen here to visully resemle the su-digrms involved. RIV RV RVI Figure 5: Additionl Reidemeister moves for knotted 3-vlent grph digrms Sine qundles worked so well for knots, we would like to use qundle (Q, ) s the oloring set in the generlized setting of grphs s well. This setion is thus devoted to the following question: Question 3.1. How n one extend the Q-oloring rule from Figure 1 A to 3-vlent verties so tht the resulting oloring rules for grphs re topologil? After short disussion of existing nswers, we shll propose n originl one. Sine the oloring rule round rossing points will lwys e tht from Figure 1 A in this pper, we shll often omit it, restriting our study to rules round 3-vlent verties. Colorings for grphs: existing pprohes Requiredoloringrulesreesytodefinegeometrillyforonjugtionqundle(G, = 1 ). Choose sepoint p situted over digrm D of n oriented grph Γ. Consider the Wirtinger presenttion of the grph group π 1 (R 3 \Γ;p) with one genertorθ α for eh r α of D, onstruted ording to Figure 6 A. An (evident) reltion is imposed on the genertors round eh speil point. A representtion of π 1 (R 3 \Γ;p) in G is now mp P from {θ α α A(D)} to G respeting these reltions. But for P to respet these reltions is preisely the sme thing s for the mp C : α P(θ α ) to e oloring with respet to oloring rules from Figures 1 A nd 6 B (where in the reltion olor or its inverse should e hosen ording to the r eing direted from or to the grph vertex). The ltter oloring rules re topologil, s n e seen vi this grph group representtion interprettion, or y n esy diret verifition. For ny digrm D of Γ, one thus gets ijetion C G (D) ij Hom(π 1 (R 3 \Γ),G). These onjugtion qundle olorings for grphs n e generlized in severl wys. First, in 2010 M.Nierzydowski [Nie10] extended the rules from Figure 6 B to generl qundles, s shown on Figure 6 C (here nd fterwrds nottion + stnds for, nd stnds for ; the hoie in ± depends, s usul, on orienttions). Another pproh ws proposed y A.Ishii in his reent preprint [Ish13]. He onsidered qundle opertion on disjoint union of groups X = i G i, whih is the onjugtion opertion when restrited to eh G i nd whih stisfies some dditionl onditions. Suh struture is lled multiple onjugtion qundle (MCQ), nd it inludes s prtiulr ses usul onjugtion qundles nd G-fmilies of qundles, defined in 2012 y Ishii- Iwkiri-Jng-Oshiro [IIJO12]. The oloring rule from Figure 6 B, where one demnds, nd to lie in the sme group G i, is topologil for MCQ. θ α α p B x Q, C A ±1 ±1 ±1 = 1 ((x ± ) ± ) ± = x Figure 6: Possile extensions of qundle olorings to grph digrms 6

8 Well-oriented 3-vlent grphs The oloring rule we introdue in this work is nother generliztion of onjugtion qundle olorings of grphs to roder lss of qundles. It is defined for grphs oriented in speil wy: Definition 3.2. An strt or knotted oriented 3-vlent grph is lled well-oriented if it hs only zip nd unzip verties, f. Figure 7. In other words, one forids soure nd sink verties. zip unzip Figure 7: Zip nd unzip verties for 3-vlent grphs For well-oriented grph digrms, some of the R-moves n e disrded using the so lled Turev s trik (see lso [Pol10] for detiled nd reful study of miniml generting sets of Reidemeister moves in the knot se): Lemm 3.3. Reidemeister moves IV-VI with orienttions s in Figure 8, together with ll oriented versions of moves RI-RIII, imply ll remining well-oriented versions of moves RIV-RVI. RIV z RIV u RV z RV u RVI z RVI u Figure 8: Reidemeister moves for well-oriented grph digrms Supersripts z nd u refer to the zip or unzip vertex involved in the move. Proof. Move RIV u for nother orienttion is treted in Figure 9; n lterntive orienttion of RV u is delt with in Figure 10. Other moves nd orienttions n e treted in similr wy. RII RIV u RII Figure 9: Reidemeister move IV u for nother orienttion RIV u RI RV u RII Figure 10: Reidemeister move V u for nother orienttion Although our orienttion restrition prevents one from working with ritrry oriented grphs, unoriented grphs n e delt with thnks to the following oservtion: Proposition 3.4. Any strt or knotted 3-vlent grph n e well-oriented. 7

9 Proof. Tke n strt unoriented grph Γ. Suppose ll its verties to e of odd vleny. We ll pth sequene of its pirwise distint edges e 1,...,e k, the endpoints (s i,t i ) of eh e i eing ordered, suh tht t i nd s i+1 oinide for eh i. Choose mximl pth γ in Γ i.e., pth whih is not su-pth of longer one. Deleting γ from Γ nd forgetting ll the isolted verties possily formed fter tht, one gets grph Γ\γ, whose verties re still of odd vleny. Indeed, the vleny sutrted from internl verties of γ is even (sine we enter nd leve them the sme numer of times); the sme rgument works for the first nd lst verties if they oinide (in whih se we ll them internl s well); if they re distint, then their full vlenies re sutrted otherwise γ ould e prolonged, whih would ontrdit its mximlity nd so they re disrded. Now let Γ e 3-vlent. Iterting the rgument ove, one presents Γ s disjoint union of pths, eh vertex ourring in t most two pths nd eing internl for the first pth it elongs to. Orienting eh edge e i in eh pth from s i to t i, one well-orients Γ. Thus, in order to ompre two unoriented grphs, it is suffiient to ompre the sets of their well-oriented versions. A new oloring pproh vi qulgers Now, for well-oriented grph digrms, onsider oloring rule from Figure 1 B, where is nother inry opertion on the qundle (Q, ). Trying to render these rules topologil, one rrives to the notion of qulger, entrl to this pper. Definition 3.5. A set Q endowed with two inry opertions nd is lled qulger if it stisfies Axioms (Q SD )-(QA Comm ) (see pge 1). The term qulger omes from terms qundle nd lger zipped together, s shown on Figure 2. It underlines the presene of two interting opertions in this struture. Algerilly, this definition n e restted in more struturl wy. Nmely, onsider set Q endowed with two inry opertions nd, nd define n opertor σ : Q Q Q Q, (,) (, ). Then (Q,, ) is qulger if nd only if (Q,σ, ) is rided lger whih is ridedommuttive ut not neessrily ssoitive, nd suh tht the Yng-Bxter opertor σ preserves the digonl of Q. Remrk tht Axiom (Q SD ) ould e omitted from the definition, s it is onsequene of (QA Comp ) nd (QA Comm ): ( ) (QAComp) = ( ) (QAComm) = ( ( )) (QAComp) = ( ) ( ); we will inlude or omit this xiom ording to our needs. For further referene, let us lso note the omptiility reltions etween opertions nd. Lemm 3.6. A qulger (Q,, ) enjoys the following properties: ( ) = ( ), (6) ( ) = ( ) ( ), (7) ( ) =. (8) Proof. Let us show (6), the proof for the remining reltions eing similr. Applying (QA Comp ) to elements ( ), nd, one gets ( ( )) ( ) = (( ( )) ). The left-hnd side equls euse of (Q Inv ). Now, pply the mp x (x ) to oth sides: ( ) = (((( ( )) ) ) ). Using (Q Inv ) for the right-hnd side this time, one otins (6). 8

10 Now, returning to olorings of grphs, one gets Proposition 3.7. Tke set Q endowed with two inry opertions nd. Coloring rules from Figure 1 A & B re topologil if nd only if (Q,, ) is qulger. Proof. The equivlene etween the omptiility of the oloring rule 1 A with Reidemeister moves I-III on the one hnd, nd Axioms (Q SD )-(Q Idem ) on the other hnd, ws disussed in Exmple 2.4. Let us turn to the remining three moves, with orienttions from Lemm 3.3. Anlyzing move RIV z (Figure 11), one noties tht on eh side the three olors on the top ompletely determine ll the remining olors, in prtiulr the olors on the ottom. Then, the oloring ijetion from Definition 2.1 tkes ple if nd only if the indued ottom olors oinide on the two sides, whih is equivlent to Axiom (QA Comp ). An nlogous rgument shows tht for move RIV u, the oloring ijetion is equivlent to Axiom (6), whih, in the presene of (Q Inv ), is the sme s (QA Comp ) (f. the proof of Lemm 3.6). Similrly, one heks tht for oth the zip nd unzip versions of RVI (respetively, RV) the oloring ijetion is equivlent to Axiom (QA D ) (respetively, (QA Comm )). RIV z ( ) ( ) Figure 11: Qulger xioms vi oloring rules for grph digrms Remrk 3.8. Certinly, we ould hve used different opertions z nd u for oloring rules round zip nd unzip verties. However, our simplified hoie lredy produes powerful invrints; moreover, it is nturl if one thinks in terms of generliztions of (multiple) onjugtion qundle olorings of grphs. Lemm 2.2 now llows one to onstrut qulger oloring invrints for grphs: Corollry 3.9. Tke qulger (Q,, ) nd onsider Q-oloring rules from Figure 1 A & B. The (possily infinite) quntity #C Q (D) does not depend on the hoie of digrm D representing well-oriented 3-vlent knotted grph Γ. Proof. Proposition 3.7 gurntees tht the oloring rules in question re topologil. Lemm 2.2 then tells tht the funtion D #C Q (D) is well-defined on R-equivlene lsses of digrms, whih, ording to [Ku89, Ym89, Yet89], orrespond to isotopy lsses of grphs. One thus gets systemti wy of produing invrints of well-oriented (or unoriented, f. Proposition 3.4) grphs. Group qulgers We now show tht groups re n importnt soure of qulgers, plying lso signifint motivtionl role. Exmple A onjugtion qundle together with the group multiplition opertion = is qulger, lled group qulger; diret verifition of ll the xioms is esy. For this qulger, the oloring rule from Figure 1 B repets tht from Figure 6 B. Thus our qulger oloring rules nd resulting grph invrints generlize the group oloring rules nd orresponding invrints. 9

11 While from the topologil perspetive qundle xioms (Q SD )-(Q Idem ) n e viewed s lgeri inrntions of Reidemeister moves for knots, from the lgeri viewpoint they re often interpreted s n xiomtiztion of the onjugtion opertion in group. Conretely, if reltion involving only onjugtion holds in every group, then it n e dedued from the qundle xioms (f. [Joy82, Deh00]). In similr wy, s shown in the (proof of) Proposition 3.7, topologilly dditionl qulger xioms (QA Comp )-(QA Comm ) n e regrded s lgeri inrntions of speifi R-moves for 3-vlent grphs. Algerilly, they enode mjor reltions etween onjugtion nd multiplition opertions in group (f. Tle 1). However, we shll see elow tht not ll the onjugtion/multiplition reltions re ptured y the qulger struture. strt level qundle xioms speifi qulger xioms group level onjugtion onjugtion/multiplition intertion topologil level moves RI-RIII moves RIV-RVI Tle 1: Different viewpoints on qundles nd qulgers A slight vrition of Exmple 3.10 is first due: Exmple New exmples of qulgers n e derived y onsidering su-qulgers of given qulgers. In the se of group qulgers, these re simply susets losed under onjugtion nd multiplition opertions, ut not neessrily under tking inverse. For instne, positive integers N form su-qulger of the group qulger of Z. Note tht su-qulgers of group qulgers do not neessrily ontin the neutrl element or inverses. However, they lerly remin ssoitive: Definition A qulger (Q,, ) is lled ssoitive if the opertion is suh, i.e., if for ll elements of Q one hs ( ) = ( ). (9) Exmples of non-ssoitive qulgers will e given in Setion 5. Rell tht in the qundle setting, the free qundle on set S n e seen s the S-generted su-qundle of the onjugtion qundle of the free group on S. This explins the fundmentl role of onjugtion qundles mong ll qundles. One would expet similr result in the ssoitive qulger setting (the neessity to impose the ssoitivity is explined ove). However, this is flse: Proposition Tke set S with t lest 2 elements. Consider the mp from the free ssoitive qulger FAQA S on S to the group qulger of the free group FG S on S, sending every S to itself. This mp is not injetive. The proof of this result is slightly tehnil nd is therefore presented in Appendix A. Relted onstrutions nd non-qulgerizle qundles Group qulgers nd their su-qulgers re fr from overing ll exmples of qulger struture. We hve just seen mnifesttion of this ft: Reltion (32), even though utomti in group qulgers, fils in some other ssoitive qulgers. Moreover, in Setion 5 we shll show tht even in smll size there re some exoti qulgers exhiiting very non-group-like properties: they re neither nelltive, nor ssoitive, nor unitl. Our hoie of qulger xioms, resulting in the struture s rihness (illustrted in prtiulr y suh exoti exmples), ws ditted y the desired pplitions to grph invrints. Here we mention some relted strutures from the literture, ppering in different frmeworks nd exhiiting dissimilr properties. First, oserve tht the ssoitivity, sent from our topologil piture, does eome relevnt when one works with hndleody-knots (f. [Ish08]). In prtiulr it ppers, together with Axioms (Q SD ), (Q Inv ), (QA Comp ) nd (QA D ), in A.Ishii s definition of multiple onjugtion qundle, the ltter eing tilored for produing hndleody-knot invrints. Remrk tht lgerilly, MCQs inherit mny properties of groups, sine they re formed y gluing severl groups together. 10

12 Besides the topologil nd lgeri settings desried ove, Axioms (QA Comp )-(QA Comm ) lso emerge in ompletely different set-theoretil ontext. Nmely, together with the ssoitivity of nd the existene of neutrl element 1 for stisfying moreover 1 = 1 nd 1 = for ll Q, they define (right-)distriutive monoid (or, in other soures, RD lger). The exmples of elementry emeddings, Lver tles nd extended rids, ll of whih dmit rih distriutive monoid strutures, hve motivted n extensive study of the onept (f. for instne [Deh86, Drá95, Drá97, Deh98], or Chpter XI of [Deh00] for omprehensive exposition). A weker ugmented (right-)distriutive system struture of P.Dehornoy oeys only three xioms: (Q SD ), (QA Comp ), nd (QA D ); the mjor exmple here is tht of prenthesized rids (f. [Deh06, Deh07]). Our qulgers re prtiulr ses of ugmented distriutive systems. We finish with some remrks onerning the reltions etween qundle nd qulger strutures. Any qundle n e emedded (s su-qundle) into qulger (f. [Le14]). Further, some qundles n e upgrded to qulgers using severl different opertions (f. Setion 5 for exmples). Here we give n exmple of fmily of qundles whih n not e turned into qulgers, nd of qundle dmitting extly one omptile opertion. Exmple A dihedrl qundle is the set Z/nZ endowed with the opertion = 2 (modn). Suppose tht Z/nZ n e endowed with n dditionl opertion stisfying (QA Comp ). Then for ll,, Z/nZ, the element ( ) = 2 2+ would oinide with ( ) = 2( ), thus 2 = 2( ) 2+2 would not depend on, whih is impossile if n 2. Exmple Considerthe onjugtion qundle ofthe symmetri group S 3. As usul, opertion = turns it into group qundle. Let us show tht this is the only qulgeriztion of this qundle. Indeed, Axiom (QA Comp ) imposes the vlues of (12) ( ) nd (123) ( ) for ll, S 3 ; it remins to show tht the vlues (12) x nd (123) x uniquely identify n x S 3. This follows y diret omputtions: (12) if x {Id,(12)}, { (123) if x {Id,(123),(132)}, (12) x = (23) if x {(132),(13)}, (123) x = (213) if x {(12),(23),(13)}. (13) if x {(123),(23)}; 4 Isoseles olorings nd squndles In onrete situtions, one sometimes hs to del with pirs of grphs for whih the Q-oloring ounting invrints from Corollry 3.9 oinide for ertin qulgers Q, ut whih n e distinguished if only prtiulr kind of olorings is tken into ount. After short survey of the development of suh speil oloring ides in the literture, we introdue prtiulr kind of qulger olorings, llowing one to distinguish, for instne, the two thet-urves from Figure 14. Speil olorings Strt with group oloring rules for ritrry oriented grphs (Figures 1 A nd 6 B ). The most nturl prtiulr kind of orresponding olorings is the one where the olors of rs djent to the sme vertex oinide, up to orienttions. This mens using the oloringrule from Figure 12 A, where olor should e hosen for rs oriented from the vertex, nd olor 1 for the remining ones. Suh olorings n e tred k to C.Livingston s 1995 study of vertex onstnt grph groups ([Liv95]). These ides were generlized in 2007 y T.Fleming nd B.Mellor ([FM07]) to the se of symmetri qundle. The ltter is qundle Q endowed with good involution, i.e., mp ρ : Q Q stisfying, for ll elements of Q, ρ(ρ()) =, (10) ρ() = ρ( ), (11) ρ() =. (12) Symmetri qundles were defined y S.Kmd in [Km07]. The si exmple is our fvourite onjugtion qundle, with ρ() = 1. Now, for symmetri qundle Q, Fleming-Mellor s oloring 11

13 rule for grphs is presented on Figure 12 B ; nottions +1 =, 1 = ρ() re used here, nd the hoie in ±1 is ontrolled y the sme rule s for group olorings. This rule generlizes tht from Figure 12 A, nd orresponding olorings n e seen s speil mong the qundle olorings in the sense of 6 C. To see tht one gets topologil oloring rules, it suffies to hek tht speil oloring remins suh fter n R-move nd the orresponding oloring hnge, whih is done y n esy diret verifition (f. the proof of Proposition 4.2). M.Nierzydowski further generlized these ides to n ritrry qundle se (see [Nie10]). ±1 ±1 ±1 3 = 1 A ±1 ±1 ±1 x Q, ((x ) ) = x Figure 12: Exmples of speil oloring B C Isoseles olorings We now return to qulger olorings for well-oriented grphs. The lss of speil olorings we propose to study here is the following: Definition 4.1. Tkequlger(Q,, )nd Q-oloredwell-orientedgrphdigrm(D,C). A 3-vlent vertex of D is lled C-isoseles if C ssigns the sme olors to its two djent o-oriented rs. The oloring C itself is lled isoseles if ll verties of D re C-isoseles. In other words, working with isoseles olorings mens onsidering oloring rule 12 C. Proposition 4.2. Given qulger (Q,, ), the oloring rules from Figures 1 A nd 12 C re topologil. Proof. Sine isoseles olorings re prtiulr instnes of those from Proposition 3.7, whih re ontrolled y topologil rules, it suffies to hek tht n isoseles oloring remins suh fter n R-move nd the orresponding oloring hnge. For moves RI-RIII nd RV it is ovious, sine they do not hnge the olors round isoseles trivlent verties. Move RVI u is treted on Figure 13: the top three olors determine ll the remining ones (note tht the ottom olors oinide due to (7)), nd for ny of the two digrms eing isoseles mens stisfying = (sine the mp x x is ijetion on Q). Moves RVI z nd RIV re treted similrly. RVI u ( ) ( ) ( ) Figure 13: Reidemeister move VI u nd indued olorings Corollry 4.3. Tke qulger (Q,, ). An invrint of well-oriented 3-vlent knotted grphs n e onstruted y ssigning to suh grph the numer of isoseles Q-olorings #CQ iso (D) of ny of its digrms D. Exmple 4.4. The Kinoshit-Tersk Θ-urve Θ KT nd the stndrd Θ-urve Θ st (Figure 14) often serve s litmus test for new grph invrints. One of the resons is the following: when ny edge is removed from Θ KT, the remining two ones form the unknot, just like for Θ st ; however, the three edges of Θ KT re knotted, in the sense tht Θ KT is not isotopi to Θ st. These prtil unknottedness phenomen re of the sme nture s those exhiited y the Borromen rings. 12

14 Now, for these two Θ-urves, onsider the isoseles Q-oloringsof their digrms D KT nd D st, depitedonfigure14. DigrmD st (swellsllthe otherwell-orientedversionsofthe underlying unoriented digrm) hs #Q isoseles Q-olorings: the o-oriented rs n e olored y ny olor x, nd the remining r gets the olor x x. As for D KT, the oloring rule 12 C round 3-vlent verties is tken into onsidertion in Figure 14, nd the rule 1 A round rossing points gives reltions = x (y y) = y x, = x y = y (x x), = (y y) x = (x x) y. Thus, #C iso Q (D KT) is the numer of the solutions of the ove system in x nd y. One esily heks tht x = y = q is solution for ny q Q (f. Lemm 5.9). In order to find other isoseles olorings of D KT, let us try the simplest se of group qulger Q nd of its order 3 elements x nd y. The three reltions ove re now equivlent to single one, nmely xyx = yxy. In the symmetri group S 4 for exmple, distint order 3 elements x = (123) nd y = (432) give solution to the ove eqution. One thus otins #C iso S 4 (D KT ) > #S 4 = #C iso S 4 (D st ). Sine, s mentioned ove, #C iso S 4 (D st ) is the sme for ll well-oriented versions of D st, one onludes tht Θ KT nd Θ st re distint s unoriented grphs. x x x x x x x x Θ st y Θ KT y y y Figure 14: Isoseles olorings for digrms of stndrd nd Kinoshit-Tersk Θ-urves A vrition of qulger ides Restriting our ttention to isoseles olorings only, we do not exploit the whole struture of qulger. Indeed, the only vlues of we need re those for =. In other words, we use only the squring prt ς : of the opertion. Pursuing this remrk, let us try to determine for whih unry opertions ς the oloring rule 1 C is topologil. One rrives to the following notion: Definition 4.5. A set Q endowed with inry opertion nd unry opertion ς (whih we often denote y 2 ) is lled squndle if it stisfies Axioms (Q SD )-(Q Idem ) nd (SQ 1 )-(SQ 2 ) (see pge 1). The term squndle (similrly to the term qulger ) omes from terms squre nd qundle zipped together, f. Figure 2. Let us lso note the omptiility reltions etween opertions ς nd : Lemm 4.6. A squndle (Q,, ς) enjoys the following properties: 2 = ( ), (13) 2 = ( ) 2. (14) 13

15 Exmple 4.7. Aqulger(Q,, )lwysgivesrisetosqundle(q,,ς : ). Moreover, the su-squndles of the ltter (whih re not neessrily su-qulgers) n e of interest. In prtiulr, onjugtion nd squring opertion 2 in group form squndle, lled group squndle. Axioms (SQ 1 )-(SQ 2 ) n now e seen s n strtion of the reltions etween onjugtion nd squring opertions in group. Now, onsidering squndle olorings, one gets the following results, with the sttements nd proofs nlogous to the qulger se: Proposition 4.8. Tke set Q endowed with inry opertion nd unry opertion ς. Coloring rules from Figure 1 A & C re topologil if nd only if (Q,,ς) is squndle. Corollry 4.9. Tke squndle (Q,,ς) nd onsider Q-oloring rules 1 A & C. The (possily infinite) quntity #C Q (D) does not depend on the hoie of digrm D representing welloriented 3-vlent knotted grph Γ. Exmple Let us resume Exmple 4.4. In the symmetri group S 4, onsider the suset S 3 4 of yles of length 3. It ontins 8 elements, nd it is losed under onjugtion nd squring. Hene S 3 4, endowed with onjugtion nd squring opertions, is size 8 squndle (ut not qulger, sine it does not ontin Id = (123) 3 ). Clultions from Exmple 4.4 show tht #C S 3 4 (D st ) = #S 3 4 = 8, nd tht #C S 3 4 (D KT) is the numer of solutions of xyx = yxy in S 3 4. Now, for ny x, the pir (x,x) is solution, while (x,x 1 ) is not. Further, we hve seen tht yles (123) nd (432) form solution, nd one heks tht (123) nd (423) do not. A onjugtion rgument llows to onlude tht for fixed x 0, preisely hlf of the pirs (x 0,y) re solutions, whih totls to #C S 3 4 (D KT ) = 8 4 = 32. Thus, lthough this exmple gives nothing new out the grphs Θ KT nd Θ st (the group qulger of S 4 ws suffiient to distinguish them), it does show tht with squndle olorings, tul omputtion of ounting invrints n e muh esier. 5 Qulgers nd squndles with 4 elements In this setion we ompletely desrie qulgers nd squndles with 4 elements. Compred to qundles, these new strutures ome with undnt exmples even in suh smll size. Generl properties Some generl fts out qulgers nd squndles re neessry efore proeeding to lssifition questions. Nottion 5.1. Given qundle (Q, ) (in prtiulr, qulger or squndle) nd n Q, denote y S the right trnsltion mp x x. We write qundle mps on the right of their rguments, e.g., (x)s = x. Most xioms of qundle-like strutures n e expressed in terms of these right trnsltions, llowing one to work with symmetri groups insted of strt strutures. This pproh ws extensively used for qundles in [LR06]. Here we pply similr ides to qulgers nd squndles. Lemm 5.2. Given qulger (Q,, ) or squndle (Q,,ς), the mp S : Q Aut(Q), (15) S is well-defined qulger/squndle morphism from Q to Aut(Q), the ltter eing the group qulger/squndle of qulger/squndle utomorphisms of Q. Proof. We prove the qulger version of the ssertion, the squndle one eing nlogous. One should first show tht ny S is qulger utomorphism. Indeed, it is invertile due to Axiom (Q Inv ), its inverse S 1 eing the mp x x, nd it respets opertions nd due to (Q SD ) nd (QA D ) respetively. 14

16 It remins to prove tht S is qulger morphism. Reltion S = S()S() diretly follows from (QA Comp ). Next, for ny x Q one lultes (using qundle Axioms (Q SD )-(Q Idem )) (x)s = x ( ) = ((x ) ) ( ) = ((x ) ) = (((x)s 1 )S )S = (x)(s S ), sineinthegroupqulgeraut(q)opertion istheonjugtion. Hene, S = S() S(). Lemm 5.3. For finite qulger Q, the imge S(Q) of the mp (15) is sugroup of Aut(Q). Proof. Sine S is qulger morphism (Lemm 5.2), its imge S(Q) is su-qulger of the group qulger Aut(Q), whih is finite sine Q is finite. Let us now show tht, in generl, non-empty finite su-qulger R of group qulger G is in ft sugroup. Indeed, R is stle under produt sine it is su-qulger; it ontins the unit 1 of the group G sine 1 = p, where is ny element of R nd p is its order in G; nd it ontins ll the inverses, sine, with the previous nottion, 1 = p 1. Note tht this lemm is flse for squndles in generl: ounter-exmple will e given elow. In study of qulger or squndle, the understnding of its lol struture n e useful. Nottion 5.4. Tke qulger or squndle Q nd n Q. The su-qulger/su-squndle of Q generted y is denoted y Q. The set of fixed points x of S (i.e., (x)s = x) is denoted y Fix(). The set of elements x of Q fixing (in the sense tht ()S x = ) is denoted y St(). Lemm 5.5. Tke qulger (Q,, ) or squndle (Q,,ς), nd n Q. The sets Fix() Q nd St() Q re oth su-qulgers/su-squndles of Q ontining Q. Proof. The ssertion out Fix() eing su-qulger/su-squndleof Q holds true euse S is qulger/squndle utomorphism of Q. As for St(), note tht the set St() of mps in Aut(Q) stilizing is sugroup of Aut(Q), hene lso su-qulger/su-squndle, so St(), whih is its pre-imge S 1 ( St()) long the qulger/squndle morphism S, is suqulger/su-squndle of Q (f. Lemm 5.2). Further, oth Fix() nd St() ontin due to the idempotene xiom (Q Idem ). Sine they were oth shown to e su-qulgers/su-squndlesof Q, they hve to inlude the whole Q. Lemm 5.6. Consider set Q endowed with trivil qundle opertion 0 =. Then ny unry opertion ς ompletes it into squndle. Further, inry opertion ompletes it into qulger if nd only if is ommuttive. Proof. With the trivil qundle opertion, ll qulger nd squndle xioms utomtilly hold true exept for (QA Comm ), whih is equivlent to the ommuttivity of. Definition 5.7. The qulgers/squndles from the lemm ove re lled trivil. Oserve tht olorings y trivil qulgers/squndles do not distinguish over-rossings from under-rossings, hene the orresponding ounting invrints n pture only the underlying strt grph nd not the wy it is knotted in R 3. However, weight invrints n e sensile to the knotting informtion even for trivil strutures. In size 3, ll qulgers/squndles turn out to e trivil: Proposition 5.8. A non-trivil qulger or squndle hs t lest 4 elements. Proof. Let Q e non-trivil qulger or squndle, nd e its element with non-trivil right trnsltions. Then S 2 = S 2 is different froms, sofix() ontinst lest2distint elements nd 2 (f. Lemm 5.5). Further, sine S Aut(Q) is not the identity, t lest two elements of Q should lie outside F ix(). Altogether, one gets t lest 4 elements. We finish y showing tht every qulger/squndle is lolly trivil : Lemm 5.9. Tke qulger (Q,, ) or squndle (Q,,ς), nd n Q. The suqulger/su-squndle Q of Q generted y is trivil. In the qulger se, the restrition of opertion to Q is ommuttive. 15

17 Proof. Lemm 5.5 shows tht every x Q fixes. Thus, the set Fix(x) ontins ; ut, eing su-qulger/su-squndle of Q (gin due to Lemm 5.5), it should ontin the whole Q. The trivility of restrited to Q follows. The ommuttivity of on Q is now onsequene of Lemm 5.6. Clssifition of qulgers of size 4 Sine trivil qulgers/squndles were ompletely desried in Lemm 5.6, only non-trivil strutures re studied in the reminder of this setion. We strt with full list of 9 non-trivil qulger strutures on 4 element set P = {p,q,r,s} (up to isomorphism). Involution will e used in this desription. (p)τ = q,(q)τ = p,(r)τ = r,(s)τ = s (16) Proposition Any non-trivil qulger with 4 elements is isomorphi to the set P with the following opertions (here x nd y re ritrry elements of P): x r = (x)τ, x y = x if y r; r r = s, r x = x r = r if x r, s s = s, q s = s q {p,q,s}, p s = s p = (q s)τ, p q = q p = s, q q {p,q,s}, p p = (q q)τ. Moreover, for ny hoies of q s nd q q in {p,q,s}, the resulting struture is qulger. In order to etter feel the qulger strutures from the proposition, think of the element r s the rottion (of p or q), nd of s s the squre (of r). Proof. Fix qulger struture on P. Oserve first tht for ny x P, one hs #Fix(x) 2. Indeed, otherwise the su-qulger P x generted y x, whih is ontined in Fix(x) due to Lemm 5.5, would onsist of x itself only, nd so, ording to Lemm 5.2, S({P x }) = {S x } would e 1-element su-qulger of Aut(P) S 4, whih is possile only if S x = Id, giving #Fix(x) = 4. Now, ondition #Fix(x) 2 implies tht S x moves t most 2 elements of P, so it is trnsposition or the identity. But then S(P) is sugroup of S 4 (Lemm 5.3) ontining nothing exept trnspositions nd the identity, hene either S(P) = {Id} (nd thus the the qulger is trivil), or, without loss of generlity, S(P) = {Id,τ}, with, sy, S r = τ. We next show tht S 1 (τ) onsists ofr only. Indeed, S(P r ) is su-qulgerof Aut(P) (Lemm 5.2) ontined in S(Fix(r)) (Lemm 5.5), so S(Fix(r)) = {S(r),S(s)} = {τ,s s } should inlude τ 2 = Id, hene S s = Id, implying s / S 1 (τ). As for p nd q, they re not fixed y τ, so they nnot lie in S 1 (τ). We n thus restrit our nlysis to the se S r = τ nd S y = Id for y r. This hoie of opertion gurntees (Q Inv ) nd (Q Idem ). Axiom (Q SD ) n e heked diretly, ut we prefer relling tht it is onsequene of (QA Comp )-(QA Comm ). Let us now nlyze speifi qulger xioms (QA Comp )-(QA Comm ). First, (QA Comp ) trnsltes s S = S S, whih here mens tht r x = x r = r for ll x r, while ll other produts tke vlue in {p,q,s}. Next, (QA D ) is equivlent to ll mps from S(P) respeting the opertion, whih here trnsltes s ( )τ = ()τ ()τ. This mens tht r r nd s s re oth τ-stle, so, lying in {p,q,s}, they n equl only s; this gives nothing new when one of, is r nd the other one is not; nd it divides the remining ordered ouples into pirs, with the produt for one ouple from the pir determined y tht for the other (e.g., p s = (q s)τ). At lst, (QA Comm ) is utomti when one of the elements nd is r nd the other one is p or q, nd for the other ouples it mens the ommuttivity of. In prtiulr, this ommuttivity gives p q = q p, whih, omined with (p q)τ = (p)τ (q)τ = q p, implies tht p q is τ-stle, so, lying in 16

18 {p,q,s}, it n equl only s. Putting ll these onditions together, one gets the desription of given in the sttement. It remins to hek tht the 9 qulger strutures otined re pirwise non-isomorphi. Let f : P P e ijetion intertwining strutures (, 1 ) nd (, 2 ) from our list. Sine r is the only element of P with S Id, one hs (r)f = r, nd lso (s)f = (r 1 r)f = r 2 r = s. Two options emerge: either (q)f = q nd (p)f = p, in whih se 1 nd 2 utomtilly oinide; or (q)f = p nd (p)f = q, tht is, f = τ, in whih se one hs x 2 y = ((x)f 1 1 (y)f 1 )f = ((x)τ 1 1 (y)τ 1 )τ = x 1 y, sine, eing right trnsltion, τ = S r respets 1. One onludes tht there re no isomorphisms etween different qulger strutures from our list. Properties nd exmples In spite of very lose definitions, the 9 strutures ove exhiit quite different lgeri properties. Some of them re studied elow. Proposition The opertions from Proposition 5.10 re ll ommuttive; never nelltive; unitl if nd only if q s = s q = q nd p s = s p = p; ssoitive if nd only if q s = s q = p s = s p = s nd either q q = p p = s, or q q = q nd p p = p; never unitl ssoitive. Proof. The ommuttivity is red from the expliit definition of. The non-nelltivity follows from the soring property of the element r with respet to. Further, reltions q p = s nd r s = r imply tht s is the only possile neutrl element. Exmining the definition of, one sees tht it is indeed so if nd only if the vlue of q s = s q is hosen to e q (implying p s = s p = (q s)τ = (q)τ = p). Assoitivity is trikier to del with. First, if is ssoitive, then s q hs to equl s: s q = (r r) q = r (r q) = r r = s. Sine (s)τ = s, this implies q s = p s = s p = s. Next, q q n not e p, sine this would give q = (p)τ = (q q)τ = (q)τ (q)τ = p p = p (q q) = (p q) q = s q = s. Thus, either q q = p p = s, or q q = q nd p p = p. It remins to show tht these two opertions re indeed ssoitive. Consider the diret produt Z 3 4 endowed with the term-yterm multiplition, nd define n injetion P Z 3 4 y p (,0,1), r (0,0,3), q (0,,1), s (0,0,1), for some 0. One esily heks tht this injetion intertwines opertions nd, where one tkes = 2 for the hoie q q = p p = s, nd = 1 for the hoie q q = q, p p = p. Thus the ssoitivity of implies tht of. To onlude, notie tht if unitl ssoitive existed, then it would stisfy inomptile onditions q s = q nd q s = s. Thus, 3 non-trivil qulger strutures with 4 elements re unitl, nd 2 re ssoitive. Further, non of these qulgers n e su-qulger of group qulger euse of the non-nelltivity. Exmple Let us now use the 4-element qulgers otined ove for distinguishing the stndrd uff grph C st from the Hopf uff grph C H. Consider their digrms D st nd D H depited on Figure 15, nd hoose the qulger P from Proposition 5.10 with q q = s nd 17

19 q s = q. The multiplition of this qulger n e riefly desried y sying tht it is ommuttive with neutrl element s, tht the element r sors everything ut itself (in the sense tht r x = r), nd tht x y = s for x = y nd for x = (y)τ. With the orienttion on Figure 15, the oloring rules for D st round 3-vlent verties red = nd =. Further, note tht every orienttion of D st is well-orienttion, nd tht n orienttion hnge results only in n rgument inversion in one or ll of the reltions ove; sine is ommuttive, this does not hnge the reltions. Summrizing, for ny orienttion of D st one gets ijetion C P (D st ) ij {(,,) P =, = }. Now, eqution = (nd similrly = ) hs 6 solutions in P: either is the unit s, nd is ritrry; or is p or q, nd = r. Serhing for pirs of solutions with the sme, one gets C Q (D st ) ij {(,s,), Q} {(r,,r) {p,q}}, nd so #C P (D st ) = = 18. Let us now turn to the Hopf uff grph digrm D H, oriented s shown on Figure 15. Coloring rules round rossing points llow one to express nd in terms of other olors: =, =. In our qulger P, ll the trnsltions S x (rell Nottion 5.1) re either the identity or τ, so they re pirwise ommuting involutions, implying = = ()S 1 = ()S = ()(S S ) = ()S =. Further, round 3-vlent verties oloring rules give = nd =. Using the preeding remrks, this gives C P (D H ) ij {(,,) P =, = }. The ltter system dmits no solutions with = r. For = s, the equtions eome = nd =, for whih the solutions re ll pirs (,) exept = r, {p,q} or vie vers. In the remining se {p,q}, the only possiility is = = r. Summrizing, one gets C P (D H ) ij {(,s,), {p,q,s}} {(r,s,r),(r,s,s),(s,s,r)} {(r,,r) {p,q}}, nd so #C P (D H ) = = 14 #C P (D st ). With the orienttion remrks mde for D st, Corollry 3.9 now gurntees tht the two unoriented uff grphs re not mutully isotopi. C st C H Figure 15: Qulger olorings for the digrms of stndrd nd Hopf uff grphs Clssifition of squndles of size 4 Let us now turn to non-trivil 4-element squndle strutures. We shll see tht 3 out of the 4 of them re indued from the qulger strutures from Proposition 5.10 ording to the proedure desried in Exmple 4.7. Proposition Any non-trivil squndle with 4 elements is isomorphi either to the su-squndle S 2 3 of the group squndle of the symmetri group S 3 onsisting of the identity nd the trnspositions (12), (23) nd (13); or to the set P = {p,q,r,s} with the following opertions (here x nd y re ritrry elements of P, nd τ is the involution defined y (16)): x r = (x)τ, x y = x if y r; r 2 = s 2 = s, q 2 {p,q,s}, p 2 = (q 2 )τ. 18