Virtual knot theory on a group

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1 Virtul knot theory on group Arnud Mortier To ite this version: Arnud Mortier. Virtul knot theory on group. 35 pges, 29 figures <hl > HAL Id: hl Sumitted on 3 Mr 204 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 Virtul knot theory on group Arnud Mortier Mrh 3, 204 Astrt Given group endowed with Z/2-vlued morphism we ssoite Guss digrm theory, nd show tht for prtiulr hoie of the group these digrms enode fithfully virtul knots on given ritrry surfe. This theory ontins ll of the erlier ttempts to deorte Guss digrms, in wy tht is mde preise vi symmetry-preserving mps. These mps eome ruil when one mkes use of deorted Guss digrms to desrie finite-type invrints. In prtiulr they llow us to generlize Grishnov-Vssiliev s formuls nd to show tht they define invrints of virtul knots. Contents Preliminry: lssil Guss digrms nd their Reidemeister moves 2 2 Knot nd virtul knot digrms on n ritrry surfe 4 2. Thikenings of surfes Digrm isotopies nd detour moves Virtul knot theory on weighted group 6 3. Generl settings nd the min theorem Aout the orits of w-moves Aelin Guss digrms Homologil formuls The energy formul The torsion formul Finite-type invrints 9 4. Generl lgeri settings The Polyk lger The symmetry-preserving injetions Arrow digrms nd homogeneous invrints Bsed nd degenerte digrms Invrine riteri Invrine riterion for w-orits Exmples nd pplitions Grishnov-Vssiliev s plnr hin invrints There is Whitney index for non nullhomotopi virtul knots Guss digrms were introdued in knot theory s mens of representing knots nd their finitetype invrints [20, 9], llowing omptifition nd generliztion of formuls due to J.Lnnes [4]. Sine then, severl generliztions hve een ttempted to dpt them to knot theory in thikened surfes y deorting them with topologil informtion [7,, 7]. Our gol is to onstrut unifying fther frmework, nd to desrie how to get down from there to other versions with less dt. First we define nd study (virtul) knot digrms on n ritrry surfe Σ: these re tetrvlent grphs emedded in Σ, some of whose doule points (the rel ones) re pushed nd desingulrized into rel line undle over Σ. Defining Guss digrms requires glol notion for the

3 rnhes t rel rossing to e one over the other, nd glol notion of writhe of rossing. It is shown tht these notions n e defined simultneously if nd only if Σ is orientle. If it is not, we srifie the glolity of one property, nd tke into ount its monodromy. It is shown tht when the totl spe of the undle is orientle, the writhes re glolly defined nd the monodromy of the over/under dtum is the first Stiefel-Whitney lss of the tngent undle to Σ, w (Σ). In Setion 3 is given definition of Guss digrms deorted y elements of fixed group π, sujet to usul Reidemeister moves, nd to dditionl onjugy moves, depending on fixed group homomorphism w : π F 2. It is shown tht when there is surfe Σ suh tht π = π(σ) nd w = w (Σ), then there is orrespondene etween Guss digrms nd virtul knot digrms, tht indues orrespondene etween the equivlene lsses (virtul knot types) on oth sides. A lighter kind of Guss digrms, lled elin, is defined in Susetion 3.2 following the ide of T.Fiedler s H (Σ)-deorted digrms ([7]) nd shown to e equivlent to the ove when π is elin nd w is trivil. The little drwk of this version is tht it eomes more diffiult to ompute the homologil deortion of n ritrry loop. Two formuls re presented in 3.3 to sort this out, involving quite unexpeted omintoril tools. Finlly, we desrie invrine riteri for the nlog of Goussrov-Polyk-Viro s invrints [0] in this frmework. As n pplition, we otin generliztion of Grishnov-Vssiliev formuls [], nd notion of Whitney index for virtul knots whose underlying immersed urve is non nullhomotopi. Aknowledgements I wish to thnk Christin Blnhet who invited me t the Institut de Mthémtiques de Jussieu where this work ws done. This work hs enefited from disussions with Mihel Polyk, Mih Chrismn, nd Thoms Fiedler. Preliminry: lssil Guss digrms nd their Reidemeister moves Definition.. A lssil Guss digrm is n equivlene lss of n oriented irle in whih finite numer of ouples of points re linked y n strt oriented rrow with sign deortion, up to positive homeomorphism of the irle. A Guss digrm with n rrows is sid to e of degree n. It my hppen tht one regrds Guss digrms s topologil ojets (drwing loops on them, onsidering their first homology). In tht se, one must ewre of the ft tht the rrows do not topologilly interset tht is wht is ment y strt. However, the ft tht two rrows my look like they interset is something omintorilly well-defined, nd interesting for mny purposes. Ft: There is nturl wy to ssoite Guss digrm with knot digrm in the sphere S 2, from whih the knot digrm n e uniquely reovered. Fig. illustrtes this ft. d d d Figure : The writhe onvention, digrm of the figure eight knot, nd its Guss digrm the letters re here only for the ske of lrity. 2

4 However, not every Guss digrm tully omes from knot digrm in tht wy. This oservtion hs led to the development of virtul knot theory [3]: silly virtul knot is Guss digrm whih does not ome from n tul knot. There is knot-digrmmti version of these, using virtul rossings sujet to virtul Reidemeister moves - tht n e thought of s unique detour move. A detour move is nturlly ny move tht leves the underlying Guss digrm unhnged. Of ourse virtul knot digrms re lso sujet to the usul Reidemeister moves, nd these do hnge the fe of the Guss digrm. We ll them R-moves for simpliity - nd to mke it ler whether knot digrms or Guss digrms re onsidered. Here is omintoril desription of R-moves. R -moves An R -move is the irth or deth of n isolted rrow, s shown in Fig.2 (top-left). There is no restrition on the diretion or the sign of the rrow. R 2 -moves An R 2 -move is the irth or deth of pir of rrows with different signs, whose heds re onseutive s well s their tils (Fig.2, top-right). If one restrits oneself to Guss digrms tht ome from lssil knot digrms, then there is n dditionl ondition s for the reting diretion: indeed, two rs in knot digrm n e sujet to Reidemeister II move if nd only if they fe eh other. In the virtul world, there is no suh ondition sine ny two rs n e rought to fe eh other y detour moves. It my e good to know tht this ondition n e red diretly on the Guss digrm: indeed, two rs fe eh other in knot digrm if one n join them y wlking long the digrm nd turning to the left t eh time one meets rossing. Thnks to the deortions of the rrows, it mkes sense for pth in Guss digrm to turn to the left. R R 2 = = = R 3 = = Figure 2: R-moves for Guss digrms (see ove nd elow the rules for the deortions) R 3 -moves Definition.2. In lssil Guss digrm of degree n, the omplementry of the rrows is mde of 2n oriented omponents. These re lled the edges of the digrm. In digrm with no rrow, we still ll the whole irle n edge. Let e e n edge in Guss digrm, etween two onseutive rrow ends tht do not elong to the sme rrow. Put { if the rrows tht ound e ross eh other η(e) = otherwise, 3

5 nd let (e) e the numer of rrowheds t the oundry of e. Then define ε(e) = η(e) ( ) (e). Finlly, define w(e) s the produt of the writhes of the two rrows t the oundry of e. An R 3 -move is the simultneous swith of the endpoints of three rrows s shown on Fig.2 (ottom), with the following onditions:. The vlue of w(e)ε(e) should e the sme for ll three visile edges e. This ensures tht the piee of digrm ontining the three rrows n e represented in knot-digrmmti wy without mking use of virtul rossings. 2. The vlues of (e) should e pirwise different. This ensures tht one of the rs in the knot digrm version tully goes over the others. Remrk.3. From simpliil viewpoint, the sign w(e)ε(e) gives nturl o-orienttion of the -odimensionl strt orresponding to R 3 moves. This is exploited in [5] to onstrut finite-type -oyles. 2 Knot nd virtul knot digrms on n ritrry surfe The gol of this setion is to exmine when nd how one n define ouple of equivlent theories virtul knots Guss digrms tht generlizes knot theory in n ritrry 3-mnifold M. Wht first ppers is tht Guss digrm depends on projetion; so it seems unvoidle to sk for the existene of surfe Σ (mye with oundry, non orientle, or non ompt), nd nie mp p : M Σ. For the over nd under rnhes t rossing to e well-defined t lest lolly, the fiers of p need to e equipped with totl order: this leves only the possility of rel line undle. 2. Thikenings of surfes Let us now split the disussion ording to the two kinds of deortions tht one would expet to find on Guss digrm: signs (lol writhes), nd orienttion of the rrows. Lol writhes For knot in n ritrry rel line undle, there re situtions in whih it is possile to swith over nd under in rossing y mere digrm isotopy. For instne, in the non-trivil line undle over the nnulus S R, full rottion of the losure of the two-strnded elementry rid σ turns it into the losure of σ (Fig. 3). P P P P Figure 3: Non trivil line undle over the nnulus s one reds from left to right, the knot moves towrds the right of the piture. Fig. 3 would e extly the sme (exept for the gluing inditions) if one onsidered the trivil line undle over the Moeius strip. Note tht this digrm would then represent 2-omponent 4

6 link. In ft, it is possile to emed this piture in ny non-orientle totl spe of line undle over surfe. This phenomenon revels the ft tht in these ses, there is no wy to define the lol writhe of rossing. However, ording to [6] (Definition.), there is well-defined writhe s soon s the totl spe of the undle is oientle. Definition 2.. We ll thikened surfe rel line undle over surfe, whose totl spe is orientle. Definition-Lemm 2.2. If M Σ is thikened surfe, then its first Stiefel-Whitney lss oinides with tht of the tngent undle to Σ. This lss indues homomorphism w (Σ) : π (Σ) F 2. The ouple (π (Σ), w (Σ)) is lled the weighted fundmentl group of Σ. Note tht in prtiulr the thikening of Σ is the trivil undle Σ R if nd only if Σ is orientle. Arrow orienttions Note tht the writhe of rossing for knot in M Σ depends only on one hoie, tht of n orienttion for M. The importnt thing is tht this hoie is glol, so tht it mkes sense to ompre the writhes of different rossings (they live in the sme Z/2Z). Similrly, for the orienttion of the rrows in Guss digrm to simultneously mke sense, one needs glol definition of the over/under dtum t the rossings; tht is, the fires of M Σ should e simultneously nd onsistently oriented. In other words, M Σ should e the trivil line undle. Aording to our definition of thikened surfe, this hppens only if the surfe is orientle. So it seems tht one hs hoie to mke, either restriting one s ttention to orientle surfes, or tking into ount the monodromy of whtever is not glolly defined. Additionl onjugy moves will e needed when one defines Guss digrms. The onvention to onsider only fire undles with n orientle totl spe is ritrry, its only use is to redue the numer of monodromy morphisms to insted of 2. Virtul knot digrms on n ritrry surfe Fix n ritrry surfe Σ nd denote its thikening y M Σ. Definition 2.3. A virtul knot digrm on Σ is generi immersion S Σ whose every doule point hs een deorted either with the designtion virtul (whih is nothing ut nme), or with wy to desingulrize it lolly into M, up to lol isotopy. These digrms re sujet to the usul Reidemeister moves, ditted y lol isotopy in M, nd to the virtul detour moves whih re studied in the next setion. As explined efore, if one hooses n orienttion for M, then the rel rossings of suh digrm hve well-defined writhe. 2.2 Digrm isotopies nd detour moves Here y knot digrm we men virtul knot digrm on fixed ritrry surfe Σ, s defined ove. In this se digrm isotopy, usully riefly denoted y H : Id h, is the dtum of diffeomorphism h of Σ together with n isotopy from Id Σ to h. A detour move is oundry-fixing homotopy of n r tht, efore nd fter the homotopy, goes through only virtul rossings (suh n r is lled totlly virtul). Though oth of these proesses seem rther simple, it will e useful to understnd how they intert. Lemm 2.4. A knot digrm otined from nother y sequene of digrm isotopies lternting with detour moves my lwys e otined y single digrm isotopy followed y detour moves. 5

7 Proof. It is enough to show tht detour move d followed y digrm isotopy Id h my e repled with digrm isotopy followed y detour move (without hnging the initil nd finl digrms). The initil digrm is denoted y D. Cll α the totlly virtul r tht is moved y the detour move. By definition, d(α) is oundryfixing homotopi to α, nd is totlly virtul too. Thus, h (d (α)) nd h(α) re totlly virtul nd oundry-fixing homotopi to eh other. Sine h (d (D)) nd h(d) differ only y these two rs, it follows tht there is detour move tking h(d) to h (d (D)). Now n interesting question out digrm isotopies is when two of them led to digrms tht re equivlent under detour moves. Here is quite useful suffiient ondition. Definition 2.5. Let X nd Y e two finite susets of Σ with the sme (positive) rdinlity n. A generlized rid in Σ [0, ] sed on the sets X nd Y is n emedding β of disjoint union of segments, suh tht Im β (Σ {t}) hs rdinlity n for eh t, oinides with X t t = 0 nd with Y t t =. Let D e knot digrm nd H digrm isotopy. Let p P,..., p n P n denote little neighorhoods of the rel rossings of D, nd set P = P i. Then, H(p i, ) defines generlized rid H β in Σ [0, ] with n strnds sed on the sets {p,..., p n } nd {h(p ),..., h(p n )}. The strnd of rid β tht intersets Σ {0} t p i is denoted y β i. Proposition 2.6. Let D nd H e s ove. Then, up to detour moves, h(d) only depends on D nd the oundry fixing homotopy lss of H β. Proof. Let γ e mximl smooth r of D outside P (thus totlly virtul). It egins t some P i nd ends t some P j (of ourse it my hppen tht j = i). Using little rs inside of P i nd P j to join the endpoints of γ with p i nd p j, one otins n oriented pth H β i γ H β j. The ovious retrtion of Σ [0, ] onto Σ {} indues mp π (Σ [0, ], h(p) {}) π (Σ, h(p)) [ ] tht sends the lss H β i γ H β j to [h(γ)]. Sine the former lss is unhnged under oundryfixing homotopy of γ nd H β, so is the ltter, whih proves the result. This proposition sttes tht the only relevnt dtum in digrm isotopy of virtul knot is the pth followed y the rel rossings long the isotopy, up to homotopy: the entnglement of these pths with eh other or themselves does not mtter. It follows tht the rossings my e moved one t time: Corollry 2.7. Let D e knot digrm with its rel rossings numered from to n, nd let H : Id h e digrm isotopy. Then there is sequene of digrm isotopies H,..., H n, suh tht h n... h (D) oinides with h(d) up to detour moves, nd suh tht H i is the identity on neighorhood of eh rel rossing ut the i-th one. Remrk 2.8. It is to e understood tht the i-th rossing of h k... h (D) is h k... h (p i ). Proof. Any generlized rid is (oundry-fixing) homotopi to rid β Σ [0, ] suh tht the i-th strnd is vertil efore the time i n nd vertil gin fter the time i n. Tke suh rid β tht is homotopi to H β. Any digrm isotopy H suh tht β = H β ftorizes into produt H n... H stisfying the lst required ondition. The ft tht h n... h (D) nd h(d) oinide up to detour moves is onsequene of Proposition Virtul knot theory on weighted group In this setion, we define new Guss digrm theory, tht depends on n ritrry group π nd homomorphism w : π F 2 Z/2Z. These two dt together re lled weighted group. When (π, w) is the weighted fundmentl group of surfe (Definition 2.2), this theory enodes, fully nd fithfully, virtul knot digrms on tht surfe (Definition 2.3). 6

8 3. Generl settings nd the min theorem Definition 3.. Let π e n ritrry group nd w homomorphism from π to F 2. A Guss digrm on π is lssil Guss digrm deorted with n element of π on eh edge if the digrm hs t lest one rrow. single element of π up to onjugy if the digrm is empty. Suh digrms re sujet to the usul types of R-moves, plus n dditionl onjugy move, or w-move the dependene on w rises only there. An equivlene lss modulo ll these moves is lled virtul knot type on (π, w). A sudigrm of Guss digrm on π is the result of removing some of its rrows. Removing n rrow involves merging of its (2, 3, or 4) djent edges, nd eh edge resulting from this merging should e mrked with the produt in π of the former mrkings. If ll the rrows hve een removed, this produt is not well-defined, ut its onjugy lss is. The notion of sudigrms is useful to onstrut finite-type invrints (see Setion 4), ut it lredy llows expliit understnding of. The distintion etween empty nd non empty digrms in the definition ove. 2. The merge multiply priniple, whih is omnipresent, in prtiulr in R-moves. An R -move is the lol ddition or removl of n isolted rrow, surrounding n edge mrked with the unit π. The mrkings of the ffeted edges must stisfy the rule indited on Fig.4 (top-left). There re no onditions on the deortions of the rrows. Exeptionl se: If the isolted rrow is the only one in the digrm on the left, then the mrkings nd on the piture tully orrespond to the sme edge, nd the digrm on the right, with no rrow, must e deorted y [], the onjugy lss of. R R 2 ε ε ε ε R 3 d d d Figure 4: The R-moves for Guss digrms on group the exeptionl ses nd the rules for the missing deortions re mde preise in Definition 3.. An R 2 -move is the ddition or removl of two rrows with opposite writhes nd mthing orienttions s shown on Fig.4 (top-right). The surrounded edges must e deorted with, nd the merge multiply rule should e stisfied. Exeptionl se of type : If the mrkings nd d (resp. nd ) orrespond to the sme edge, then the resulting mrking shll e (resp. d). Exeptionl se of type 2: If the middle digrm ontins no rrow t ll, i.e. nd d mth nd so do nd, then the (only) mrking of the middle digrm shll e []. 7

9 g g g g g d g g d g g g g g Figure 5: The generl onjugy move (top-left) nd its two exeptionl ses in every se the orienttion of the rrow swithes if nd only if w(g) =. An R 3 -move my e of the two types shown on Fig.4 (ottom left nd right). The surrounded edges must e deorted y, the vlue of w( )ε( ) must e the sme for ll three of them, nd the vlues of ( ) must e pirwise distint (see Definition.2). A onjugy move depends on n element g π. It hnges the mrkings of the djent edges to n ritrry rrow s indited on Fig.5. Besides, if w(g) = then the orienttion of the rrow is reversed though its sign remins the sme. Remrk 3.2. By omposing R-moves nd w-moves, it is possile to perform generlized moves, whih look like R-moves ut depend on w. Fig.6 shows some of them. ε g ε g ε g ε g g d g d g d gd g g h d e f k h k k d kf h eh Figure 6: Some generlized moves for the R 3 piture, it is ssumed tht ghk =. Wrning: the rules for the rrow orienttions here depend on the vlue of w(g). Theorem 3.3. Let (Σ, x) e n ritrry surfe with se point, nd denote y (π, w) the weighted fundmentl group of (Σ, x) (see Definition 2.2). There is orrespondene Φ etween Guss digrms on π up to R-moves nd w-moves (i.e. virtul knot types on (π, w)), nd virtul knot digrms on Σ up to digrm isotopy, Reidemeister moves nd detour moves (i.e. virtul knot types on Σ). Proof. Fix suset X of Σ homeomorphi to losed 2-dimensionl dis nd ontining the se point x so tht π = π (Σ, X). Also, X eing ontrtile llows one to fix triviliztion of the thikening of Σ over X: this gives mening to the lolly over nd under rnhes when knot 8

10 digrm hs rel rossing in X. Constrution of the ijetion. Pik knot digrm D Σ nd ssume tht every rel rossing of D lies over X. Then D defines Guss digrm on π, denoted y ϕ(d): the signs of the rrows re given y the writhes, their orienttion is defined y the triviliztion of M Σ over X, nd eh edge is deorted y the lss in π of the orresponding r in D. This defines ϕ(d) without miguity if D hs t lest one rel rossing. If it does not, then define ϕ(d) s Guss digrm without rrows, deorted with the onjugy lss orresponding to the free homotopy lss of D. Finlly, put Φ(D) := [ϕ(d)] mod R-moves nd w-moves. Invrine of Φ under digrm isotopy nd detour moves. It is ler from the definitions tht ϕ(d) is stritly unhnged under detour moves on D. Now ssume tht D nd D 2 re equivlent under usul digrm isotopy tht is, digrm isotopy tht my tke rel rossings out of X for some time. By Corollry 2.7, it is enough to understnd wht hppens for digrm isotopy long whih only one rossing goes out of X. In tht se, ϕ(d) is hnged y w-move performed on the rrow orresponding to tht rossing, where the onjugting element g is the loop followed y the rossing long the isotopy. Indeed, sine the first Stiefel-Whitney lss of the thikening of Σ oinides with tht of its tngent undle, it follows tht:. The orienttion of the fire (nd thus the notions of over nd under ) is reversed long g if nd only if w(g) =, whih tully orresponds to the rule for rrow orienttions in w-move. 2. The orienttion of the fire over the rossing is reversed long g if nd only if given lol orienttion of Σ is reversed long g, so tht the writhe of the rossing never hnges. Invrine of Φ under Reidemeister moves. Up to onjugy y digrm isotopy, it n lwys e ssumed tht Reidemeister move hppens inside X. In tht se, t the level of ϕ(d), it lerly orresponds to n R-move s desried in Definition 3.. So fr, Φ is well-defined mp from the set of virtul knot types on Σ to the set of virtul knot types on (π, w). Constrution of n inverse mp Ψ. If G is Guss digrm without rrows, then define ψ(g) s the totlly virtul knot with free homotopy lss equl to the mrking of G it is welldefined up to detour moves. If G hs rrows, then for eh of them drw rossing inside X with the required writhe, nd then join these y totlly virtul rs with the required homotopy lsses. The resulting digrm ψ(g) is well-defined up to digrm isotopy nd detour moves y this onstrution. In oth ses, put Ψ(D) := virtul knot type of ψ(d). Let us prove tht ϕ nd ψ re inverse mps, so tht Ψ will e the inverse of Φ s soon s it is invrint under R-moves nd w-moves. It is ler from the definitions tht ϕ ψ oinides with the identity. It is lso ler tht ψ ϕ is the identity, up to detour moves, for totlly virtul knot digrms. Now fix knot digrm D with t lest one rel rossing (nd ll rel rossings inside X). Rell tht ψ ϕ(d) is defined up to digrm isotopy nd detour moves, so fix digrm D in tht lss. There is nturl orrespondene etween the set of rel rossings of D nd those of D, due to the ft tht oth identify y onstrution with the set of rrows of ϕ(d). Pik digrm isotopy h tht tkes eh rel rossing of D to meet its mth in D, without leving X. Then lerly ϕ(h(d)) = ϕ(d), nd euse ϕ ψ is the identity, one gets ϕ(h(d)) = ϕ(d ). () The hoie of h ensures tht h(d) nd D differ only y totlly virtul rs, nd () implies tht eh of these, in h(d), hs the sme lss in π (Σ, X) s its mth in D, whih mens y definition 9

11 tht h(d) nd D re equivlent up to detour moves. Thus ψ ϕ is the identity up to digrm isotopy nd detour moves. Invrine of Ψ under R-moves. Let us tret only the se of R 2 -moves, whih ontins ll the ides. Let G nd G 2 differ y n R 2 -move, nd ssume tht G is the one with more rrows. By pproprite digrm isotopy nd detour moves inside X, performed on ψ(g ), it is possile to mke the two onerned rossings fe eh other, s in Fig.7 (left). The pths α nd α 2 from this piture re totlly virtul nd trivil in π (Σ, X), thus ψ(g ) is equivlent to the seond digrm of Fig.7 up to detour moves. The ft tht t this point, n R-II move is tully possile is onsequene of (in ft equivlent to) the omintoril onditions defining the R-moves. Denote y D the third digrm of the piture. The merge multiply priniple tht rules R 2 -moves implies tht ϕ(d) = G 2, so tht ψ(g ) D ψ ϕ(d) = ψ(g 2 ), (2) where is the equivlene under digrm isotopy, detour moves nd Reidemeister moves. It follows tht ψ(g ) nd ψ(g 2 ) hve the sme knot type. α X X X α 2 Figure 7: R 2 -moves tully orrespond to Reidemeister moves X X X g Figure 8: Performing w-move the rilwy trik Invrine of Ψ under w-moves. Let G nd G 2 differ y w-move on g π. Cll the orresponding rossing on the digrm ψ(g ). Then, pik two little rs right efore, one on eh rnh, nd mke them follow g y detour move. At the end, one shll see totlly virtul 4-lne 0

12 rilwy s pitured on Fig.8 (middle): the strnds re mde prllel, i.e. ny (virtul) rossing met y either of them is prt of lrger piture s indited y the zoom. This ensures tht, using the mixed version of Reidemeister III moves, one n slide the rel rossing ll long the red prt of the rilwy, ending with the digrm on the right of the piture let us ll it D. The onlusion is identil to tht for R-moves: gin ϕ(d) = G 2 nd (2) holds, whene ψ(g ) nd ψ(g 2 ) hve the sme knot type. 3.. Aout the orits of w-moves It ould feel nturl to try to get rid of w-moves y understnding their orits in syntheti omintoril wy. This is wht is done in Setion 3.2 in the prtiulr se of n elin group π endowed with the trivil homomorphism π F 2. In generl, for Guss digrm on π, G, denote y h (G) the set of free homotopy lsses of loops in the underlying topologil spe of G (it is the set of onjugy lsses in free group on deg(g) genertors). Also, denote y h (π) the set of onjugy lsses in π. Then the π-mrkings of G define mp F G : h (G) h (π). Oserve tht the mp G F G is invrint under w-moves. This rises numer of questions tht mout to tehnil group theoreti prolems, nd whih will not e nswered here (G w denotes the orit of G under w-moves):. Is the mp G w F G injetive? 2. If the nswer to. is yes, then is G w determined y finite numer of vlues of F G, for instne its vlues on the free homotopy lsses of simple loops? 3. Is it possile to detet in simple mnner wht mps h (G) h (π) lie in the imge of G w F G? Remrk 3.4. Guss digrms with deortions in h (Σ) n e met for exmple in [], where they re used to onstrut knot invrints in thikened oriented surfe Σ see lso Setion 4.3. If the nswer to Question. ove is no, then suh invrints, whih ftor through F G, stnd no hne to e omplete. Remrk 3.5. Even for digrms with only one rrow, it still does not seem esy to nswer the simple loop version of Question 2. Given x, y, h, k in finite type free group, is it true tht { hxh hxh kyk = lxl = xy = l, kyk = lyl? Let us end with n exmple tht shows tht the vlues of F G on the (finite) set of simple loops running long t most one rrow is not enough (f. Question 2.). Fig.9 shows Guss digrm with suh deortions {, } is set of genertors for the free group π (Σ) F(, ), where Σ is 2-puntured dis. These prtiulr vlues of F G do not determine the free homotopy lss of the red loop γ, s it is shown in Fig.0. In ft, these two virtul knots re even distinguished y Vssiliev-Grishnov s plnr hin invrints, whih mens they represent different virtul knot types. 3.2 Aelin Guss digrms In this susetion, π is ssumed to e elin, nd w 0 denotes the trivil homomorphism π F 2. We desrie version of Guss digrms tht rries s muh informtion s the previously introdued virtul knot types on (π, w 0 ), with two improvements: The digrms re mde of less dt thn in the generl version. This version is free from onjugy moves.

13 γ [] [] [] [] [] Figure 9: A Guss digrm with h -deortions tht does not define unique virtul knot Figure 0: One red loop is trivil, while the other is ommuttor It is inspired from the deorted digrms introdued y T. Fiedler to study omintoril invrints for knots in thikened surfes (see [7, 8] nd lso [7]). We use the sme nottion G for Guss digrm nd its underlying topologil spe, whih hs -dimensionl omplex struture with edges nd rrows s oriented -ells. H (G) denotes its first integrl homology group. Definition-Lemm 3.6 (fundmentl loops). Let G e lssil Guss digrm of degree n. There re extly n simple loops in G respeting the lol orienttions of edges nd rrows, nd going long t most one rrow. They re lled the fundmentl loops of G nd their homology lsses form sis of H (G). Definition 3.7 (elin Guss digrm). Let π e n elin group. An elin Guss digrm on π is lssil Guss digrm G deorted with group homomorphism µ : H (G) π. It is usully represented y its vlues on the sis of fundmentl loops, tht is, one deortion in π for eh rrow, nd one for the se irle tht lst one is lled the glol mrking of G. A Guss digrm on π determines n elin Guss digrm s follows: The underlying lssil Guss digrm is the sme. Eh fundmentl loop is deorted y the sum of the mrkings of the edges tht it meets (see Fig ). This defines n eliniztion mp. Proposition 3.8. The mp indues nturl orrespondene etween elin Guss digrms on π nd equivlene lsses of Guss digrms on π up to w 0 -moves. Moreover, if 2

14 f e de def d ef Figure : Aelinizing Guss digrm on n elin group π = π (Σ) is the fundmentl group of surfe, then these sets re in orrespondene with the set of virtul knot digrms on Σ up to digrm isotopy nd detour moves. Proof. The proof of the lst sttement is ontined in tht of Theorem 3.3 through the fts tht φ nd ψ re inverse mps up to detour moves nd digrm isotopy, nd tht w-moves t the level of knot digrms n e performed using only detour moves nd digrm isotopies, y the rilwy trik (Fig.8). As for the first sttement, one esily sees tht is invrint under w 0 -moves. We hve to show tht onversely, if (G ) = (G 2 ), then G nd G 2 re equivlent under w 0 -moves. This is ler if G hs no rrows, sine then (G ) = G. Now proeed y indution. Sine G nd G 2 hve the sme eliniztion, they hve in prtiulr the sme underlying lssil Guss digrm, nd there is nturl orrespondene etween their rrows. Cse : No two rrows in G ross eh other. Then t lest one rrow surrounds single isolted edge on one side (s in n R -move). Choose suh n rrow α nd remove it, s well s its mth in G 2. By indution, there is sequene of w 0 -moves on the resulting digrm G tht turns it into G 2. Sine the rrows of G hve nturl mth in G, those w 0 -moves mke sense there, nd tke every mrking of G to e equl to its mth in G 2, exept for those in the neighorhood of α. So we my ssume tht G nd G 2 only differ ner α s in Fig.2. Sine ll the unseen mrkings oinide in G nd G 2, nd sine (G ) nd (G 2 ) hve the sme glol mrking, it follows tht =. Thus w 0 -move on α with onjugting element g = turns G into G 2. G G 2 α α Figure 2: Nottions for se Cse 2: There is t lest one rrow α in G tht intersets nother rrow. By the sme proess s in se, one my ssume tht G nd G 2 only differ ner α see Fig.3, where,, nd d tully orrespond to pirwise distint edges sine α intersets n rrow. Agin, sine ll the unseen mrkings oinide in G nd G 2, one otins d = d, nd =, 3

15 y onsidering the glol mrking, nd the mrking of α, in (G ) nd (G 2 ). Moreover, there is t lest one rrow interseting α: onsidering the mrking of tht rrow gives The lst three equtions my e written s =. = = = d d, so tht, gin, w 0 -move on α with onjugting element g = turns G into G 2. G G 2 α α d d Figure 3: Nottions for se 2 Remrk 3.9. A different proof of this proposition ws given in drft pper, in the speil se π = Z ([6], Proposition 2.2). As n exerise, one n show tht this proof extends to the se of n ritrry elin group. To mke the piture omplete, it only remins to understnd R-moves in this ontext. Definition 3.0 (ostrution loops). Within ny lol Reidemeister piture like those shown on Fig.2 feturing t lest one rrow, there is extly one (unoriented) simple loop. We ll it the ostrution loop. Fig.4 shows typil exmples. Definition 3. (R-moves). A move from Fig.2 is likely to define n R-move only if the ostrution loop lies in the kernel of the deorting mp H (G) π (whih mkes sense even though the loop is unoriented). Under tht ssumption, the R-moves for elin Guss digrms re defined y the usul onditions: i =. No dditionl ondition. i = 2. The rrows hed to the sme edge, nd hve opposite signs. i = 3. The vlue of w(e)ε(e) is the sme for ll three visile edges e, nd the vlues of (e) re pirwise different (see Definition.2). Theorem 3.2. The mp indues nturl orrespondene etween equivlene lsses of elin Guss digrms on π up to R-moves nd virtul knot types on (π, w 0 ). Proof. lerly mps n R-move in the non ommuttive sense to n R-move in the elin sense. Conversely, if (G ) nd (G 2 ) differ from n (elin) R-move, then the vnishing homologil ostrution implies tht G nd G 2 re in position to perform generlized R-move like the exmples pitured on Fig.6. Theorems 3.3 nd 3.2 together imply the following Corollry 3.3. If Σ is n orientle surfe with elin fundmentl group, then there is orrespondene etween elin Guss digrms on π (Σ) up to R-moves, nd virtul knot types on Σ. 4

16 R R 2 R 3 Figure 4: Homologil ostrution to R-moves 3.3 Homologil formuls It my seem not esy to ompute n ritrry vlue of the liner mp deorting n elin Guss digrm, given only its vlues on the fundmentl loops. To end this setion, we give two formuls to fill this gp, y understnding the oordintes of n ritrry loop in the sis of fundmentl loops The energy formul Fix n elin Guss digrm G. Oserve tht s ellulr omplex, G hs no 2-ells, thus every -homology lss hs unique set of oordintes long the fmily of edges nd rrows. For eh -ell (whih my e n rrow or n edge), we denote y, : H (G) Z the oordinte funtion long. It is group homomorphism. Let us denote y [A] H (G) the lss of the fundmentl loop ssoited with n rrow A (Fig.5 left). Definition-Lemm 3.4 (Energy of loop). Fix n edge e in G, nd lss γ H (G). The vlue of E e (γ) = γ, e γ, A (3) [A],e = is independent of e. This defines group homomorphism E : H (G) Z. Proof. Let us ompre the vlues of E (γ) for n edge e nd the edge e right fter it. e nd e re seprted y vertex P, whih is the endpoint of n rrow A. There re two possile situtions (Fig.5):. P is the til of A. Then [A], e = nd [A], e = 0, so tht E e (γ) E e (γ) = γ, e γ, A γ, e. 2. P is the hed of A. Then [A], e = 0 nd [A], e =, so tht E e (γ) E e (γ) = γ, e γ, A γ, e. In oth ses, E e (γ) E e (γ) is equl to γ, P, whih is 0 sine γ is yle. Theorem 3.5. For ny γ H (G), one hs the deomposition γ = A γ, A [A] E(γ) [K]. (4) 5

17 [A] A e [A] A e e A e [A] Figure 5: The fundmentl loop of n rrow nd the two ses in the proof of Lemm 3.4 Proof. This formul is n identity etween two group homomorphisms, so it suffies to hek it on the sis of fundmentl loops, whih is immedite. Remrk 3.6. The existene of mp E suh tht Theorem 3.5 holds ws ler, sine for eh rrow A onsidered s -ell, [A] is the only fundmentl loop tht involves A. With tht in mind, one my red into (3) s follows: E(γ) ounts the (lgeri) numer of times tht γ goes through n edge, minus the numer of those times tht re lredy tken re of y the fundmentl loops of the rrows. This numer hs to e the sme for ll edges, so tht one reovers multiple of [K] The torsion formul Looking t (4) nd Fig.5, one my feel tht it would e more nturl to hve [K] [A] involved in the formul, insted of [A], for ll rrows A suh tht γ, A is negtive tht is, when γ runs long A with the wrong orienttion more often thn not. The formul then eomes γ = γ, A [A] γ, A ([A] [K]) T (γ) [K], (5) where γ,a >0 γ,a <0 T (γ) = E(γ) Definition 3.7. T (γ) is lled the torsion of γ. γ,a <0 γ, A. (6) How is (5) different from (4)? On the negtive side, unlike the energy, T is not group homomorphism. But it tully ehves lmost like one: Lemm 3.8. Let γ nd γ 2 e two homology lsses suh tht Then A, γ, A γ 2, A 0. T (γ γ 2 ) = T (γ ) T (γ 2 ). Proof. It follows from the definition nd the ft tht E(γ) is homomorphism. On the positive side: Lemm 3.9. The torsion of loop in Guss digrm G does not depend on the orienttions of the rrows of G. Proof. By expnding the defining formul, T (γ) = γ, e γ, A < 0 [A], e = 0 γ, A γ, A > 0 [A], e = γ, A, one sees tht reversing n rrow mkes its ontriution (if non zero) swith from one sum to the other, while γ, A lso hnges signs. 6

18 This lemm llows one to expet tht T (γ) should dmit very simple omintoril interprettion. It tully does, ut only for ertin fmily of loops the ERS loops defined elow. Fortuntely enough, this fmily hppens to positively generte H (G), whih llows one to ompute the torsion of ny loop y using Lemm 3.8. Definition The nottion γ is used for loops s well s -homology lsses. A homology lss γ H (G) is sid to e ER (for edge-respeting ), if for every edge e, γ, e 0. simple if it is the lss of simple (injetive) loop, tht is, γ, for every -ell (edge or rrow). ERS if it is ER nd simple. proper if it runs long t lest one rrow. γ () () Figure 6: The lol nd glol look of proper ERS loop Consider permuttion σ S (, n ), nd set ր(σ) := {i, n σ(i) > i}. It is esy to hek tht if σ 0 is the irulr permuttion ( 2... n), then σ S, ր(σ) = ր(σ 0 σσ 0 ). Definition 3.2. The invrine property from ove mens tht T is well-defined for permuttions of set of n points lying in n strt oriented irle. We still denote this funtion y T, nd ll it the torsion of permuttion. Let γ e proper simple loop, then the set of edges e suh tht γ, e 0 n e nturlly ssimilited to finite suset of n oriented irle, nd γ indues permuttion of this set. Let us denote it y σ γ. Theorem For ll proper ERS loops γ, T (γ) = ր(σ γ ). This theorem n e useful in prtie, sine the torsion of permuttion n e omputed t glne on the rid-like presenttion. Oserve tht. Every non proper loop is homologous to multiple of [K], esy to determine. 2. For every proper loop γ, there is n integer n suh tht γ = γ n[k] is proper, ER, nd hs zero oordinte long t lest one edge. Nmely, n = min e γ, e. 3. Every lss γ s ove my e deomposed s sum γ = i γ i suh tht ll the γ i s re proper nd ERS 7

19 i, j, A, γ i, A γ j, A 0 4. By Lemm 3.8, T ( γ) = i T (γ i), nd the T (γ i ) s re given y Theorem This shows tht it is possile to ompute ny homology lss y using the torsion formul. Whether it is more interesting thn the energy formul depends on the ontext. Proof of Theorem One my ssume tht for every rrow A, γ, A =. Indeed, deleting n rrow voided y γ, or reversing the orienttion of n rrow tht γ runs in the wrong diretion, hve no effet on either side of the formul (notly euse of Lemm 3.9). Under this ssumption, hlf of the edges of G re run y γ: ll them the red edges of G, while the other hlf re lled the lue edges. Red nd lue edges lternte long the orienttion of the irle. If e is ny (red or lue) edge, we define: λ(e) := A [A], e. Lemm Under the ssumption tht γ, A = for ll A, the vlue of λ(e) only depends on the olor of the edge e. Moreover, λ(lue) = λ(red) = ր(σ γ ). Let us temporrily dmit this result. By the definition of λ, A [A] = rrows λ(red) (red edges) λ(lue) (lue edges) Lemm 3.23 = rrows (red edges) λ(lue) (red nd lue edges) = γ λ(lue)[k] Lemm 3.23 = γ ր(σ γ )[K]. Sine it ws ssumed tht γ, A = for every rrow, the definition of T (5) reds γ = A [A] T (γ)[k], whih termintes the proof of the theorem, up to Lemm Proof of Lemm In the se of σ 0 = ( 2... n) depited on Fig.7, it is esy to see tht λ(red) = n nd λ(lue) = n, while ր (σ 0 ) = n. The lemm eing true for one digrm, let us show tht it survives elementry hnges tht over ll the digrms.... σ γ σ0... Figure 7: Brid-like representtions of permuttions re to e red from ottom to top Notie tht for every proper ERS loop γ, σ γ is yle, nd onversely permuttion tht is yle uniquely defines n undeorted Guss digrm nd proper ERS loop γ suh tht for every rrow A, γ, A =. Thus, overing ll possile permuttions implies overing ll possile digrms nd proper ERS loops. So ll we hve to hek is tht the formul survives n opertion on σ γ, of the form: (... i j... ) (... j i... ) 8

20 σ (... i j...) σ (... j i...) i j type A σ (i)=j σ (j)=i j i j i type B σ (i)=j σ (j)=i i j i j i j type C σ (i)=j σ (j)=i i j Figure 8: Twist moves on Guss digrms i j i j The orresponding move t the level of Guss digrms my e of six different types, grouped in three pirs of reverse opertions (Fig.8). On eh digrm in Fig.8, the three moving rrows split the se irle into six regions. One omputes the vrition of λ seprtely for eh of these regions, nd sees tht it is the sme for eh of them. The results re gthered in the following tle, proving the lemm. type of move vrition of λ vrition of T (γ) A unhnged unhnged B (from left to right) dereses y dereses y C (from left to right) dereses y dereses y 4 Finite-type invrints One of the min points of using Guss digrms is their ility to desrie finite-type invrints y simple formuls [20, 7, 2, 3]. In the se of lssil long knots in 3-spe, suh formuls tully over ll Vssiliev invrints s ws shown y M.Goussrov [9]. In the virtul se, the two notions tully differ (see [3] nd lso [5, 4]. Finite type invrints for virtul knots tht do dmit Guss digrm formuls shll e lled GPV invrints [0]. In [7], simple set of riteri ws given to detet prtiulr fmily of those formuls, lled virtul rrow digrm formuls. Most of the exmples tht re known elong to this fmily. Tht inludes Chmutov-Khoury-Rossi s formuls for the oeffiients of the Conwy polynomil [2] (nd their generliztion y M. Brndenursky []), s well s the formuls from [7, 8, ] where different kinds of deorted digrms re used. Note however tht the formuls for the invrints extrted from the HOMFLYPT polynomil [3] re rrow digrm formuls only if the vrile is speilized to (whih yields k the result of [2]). In this setion, we extend the results from [7] to n ritrry surfe. Then we show how to pply them to ny other kind of deorted digrms found in the literture, y defining symmetrypreserving mps whih enle one to jump from one theory to nother. 9

21 4. Generl lgeri settings We denote y G n (resp. G n ) the Q-vetor spe freely generted y Guss digrms on π of degree n (resp. n), nd set G = limg n. Unless π is finite group, these spes re not finitely generted, nd we define their ht versions Ĝn (resp. Ĝ n ) s the Q-spes of forml series of Guss digrms of degree n (resp. n). Finlly, set Ĝ = Ĝ n. lim An ritrry element of Ĝ is usully denoted y G nd lled Guss series, of degree n if it is represented in Ĝ n ut not in Ĝ n. The nottion G is sved for single Guss digrms. A Guss digrm G of degree n hs group of symmetries Aut(G), whih is sugroup of Z/2n, mde of the rottions of the irle tht leve unhnged given representtive of G (see Susetion 4..2). G is endowed with the orthonorml slr produt with respet to its nonil sis, denoted y (, ), nd its normlized version,, defined y G, G := Aut(G) (G, G ). (7) There is liner isomorphism I : G n G n, the keystone to the theory, whih mps Guss digrm of degree n to the forml sum of its 2 n sudigrms: I(G) = G (σ), (8) σ {±} n where G (σ) is G deprived from the rrows tht σ mps to (see Definition3. for sudigrms). The inverse mp of I is given y I (G) = sign(σ)g (σ). (9) σ {±} n Definition 4.. A finite-type invrint for virtul knots in the sense of Goussrov-Polyk-Viro is virtul knot invrint given y Guss digrm formul ν G : G G, I(G), (0) where G Ĝ. Suh formul ounts the sudigrms of G, with weights given y the oeffiients of G. Notie tht only one of the two rguments of, needs to e finite sum for the expression to mke sense. We do not mke distintion etween virtul knot invrint nd the liner form indued on G. 4.. The Polyk lger A Guss series G Ĝ defines virtul knot invrint if nd only if the funtion G, I(.) is zero on the suspe spnned y R-moves nd w-moves reltors. Hene one hs to understnd the imge of tht suspe under I with simple fmily of genertors. This is the ide of the onstrution of the Polyk lger ([9, 0]). In the present se, P is defined s the quotient of G y the reltions shown in Fig.9, whih we ll P, P 2, P 3 (or 8T reltion), the W reltion, whih is simply the liner mth of w-moves (i.e. just reple the with = in ll the reltions from Fig.5). Be reful tht unlike R -moves, where n isolted rrow surrounding n edge mrked with simply disppers, in P -move the presene of suh n rrow ompletely kills the digrm. Fig.9 does not feture the π-mrkings for P 3 to lighten the piture, ut they hve to follow the usul merge multiply rule (see Definition 3.). The following proposition extends Theorem 2.D from [0]. Proposition 4.2. The mp I indues n isomorphism G/ R,W G/ P,W =: P. More preisely, I indues n isomorphism etween Spn(R i ) nd Spn(P i ), for i =, 2, 3, nd etween Spn(W) nd itself. It follows tht the mp G I(G) P defines omplete invrint for virtul knots. 20

22 P = 0 P 2 ε ε ε ε ε ε ε ε = 0 = 0 P 3 or "8T" d d d d d d = Figure 9: The three kinds of Polyk reltions only one P 3 reltion is shown, there is seond one otined y reversing ll the rrow orienttions The symmetry-preserving injetions Depending on the ontext, one my hve to onsider simultneously different types of Guss digrms, with more or less deortions. This susetion presents nturl wy to do it, onvenient from the viewpoint of Guss digrm invrints. The onstrution requires one to hoose kind of omintoril ojets tht is the fther of ll other kinds, in the sense of quotienting. We present the onstrution y tking s the fther type tht of Guss digrms on group. In first ple, we do not regrd Guss digrms up to homeomorphisms of the irle: the se irle is ssumed to e the unit irle in C, the endpoints of the rrows re ssumed to e loted t the 2n-th roots of unity, nd the rrows re stright line segments. Suh digrm is lled rigid. By type of rigid Guss digrms we men n equivlene reltion on the set of rigid Guss digrms on π, whih is required to stisfy two properties:. (Degree property) All digrms in given equivlene lss shll hve the sme degree. 2. (Stility property) The tion of Z/2n on the set of Guss digrms of degree n shll indue n tion on the set of degree n equivlene lsses. Sine every onstrution in this susetion is therefore destined to e homogeneous, the degree of ll Guss digrms is one nd for ll set equl to n. A rigid Guss digrm of type is n equivlene lss under the reltion. A Guss digrm (of type ) is the orit of rigid digrm of type under the tion of Z/2n. The orrespondingq-spes re respetively denoted y G rigid nd G. Sine Z/2n is elin, two elements from the sme orit hve the sme stilizer, hene Guss digrm G hs well-defined group of symmetries Aut(G), whih is the stilizer of ny of its rigid representtives under the tion of Z/2n. Consequently, the spe G is endowed with piring, defined y (7). Now onsider two types of rigid Guss digrms, sy nd 2, suh tht reltion is finer thn reltion 2 ( 2 ). 2

23 Definition 4.3 (Forgetful projetions). A -rigid digrm G determines unique 2-rigid digrm whose Z/2n-orit only depends on tht of G. This indues nturl surjetive mp t the level of Guss digrm spes, denoted y T 2 : G () G (2). Note tht this mp my e not well-defined on the spes of forml series of Guss digrms, if some 2-equivlene lss ontins infinitely mny -lsses. Exmple: the eliniztion mp (Definition 3.7) indues y linerity forgetful projetion from Guss digrms on π to elin digrms on π, when π is elin. Definition 4.4 (Symmetry-preserving injetions). In the opposite wy, there is mp G rigid (2) G rigid () tht sends 2-rigid digrm G 2 to the forml sum of ll -lsses tht it ontins. When this sum is pushed in G (), the result is well-defined: 2-rigid digrm nnot ontin infinitely mny rigid representtives of given Guss digrm of type, sine the orits re finite (Z/2n is finite). only depends on the Z/2n-orit of G 2. This indues n injetive symmetry-preserving mp t the level of forml series, S 2 : Ĝ(2) Ĝ(). S 2 is well-defined, omponentwise, sine 2-rigid digrms from different Z/2n-orits ontin - rigid digrms from disjoint sets of Z/2n-orits (the imges of two different Guss digrms do not overlp). It is injetive for the sme reson. The terminology is explined y the following fundmentl formul: Lemm 4.5. With nottions s ove, for ny Guss digrm G 2 of type 2, S 2(G 2 ) = T 2 (G)=G2 Aut(G 2 ) Aut(G ) G. () Informlly, the weight given to preimge of G 2 under T 2 is the mount of symmetry tht it hs lost y the gin of more informtion. Note tht the weights re integers, sine Aut(G ) identifies with sugroup of Aut(G 2 ). Proof. Fix representtive G rigid 2 of G 2. By the stility property, Aut(G 2 ) ts on the set of -lsses ontined in G rigid 2. Moreover, y definition of Aut(G 2 ), two different orits under tht tion still lie in different orits under the tion of Z/2n itself. Therefore there is orrespondene etween the Aut(G 2 )-orits nd the Guss digrms tht hppen in the sum S 2 (G 2). The stilizer of given -lss G rigid is y definition Aut(G ), whene the rdinlity of the orresponding orit, whih is lso the oeffiient of G in S 2 (G 2), is Aut(G2) Aut(G. ) Proposition For ny three reltions suh tht 2 3, the following digrms ommute: T 2 G () G (2) G (3) T 2 3 T 3 S 2 3 Ĝ (3) Ĝ (2) S 2 S 3 Ĝ (). 22

24 2. Injetions nd projetions re pirwise, -djoint, in the sense tht G G (), G 2 Ĝ(2), S 2 (G 2 ), G = G2, T 2 (G ). 3. Im S 2 = Ker T 2. Proof.. The first digrm ommutes diretly from the definition of the mps T j i. As for the mps S j i, sine they re defined omponentwise it is enough to hek it for single digrm G 3. In tht se, it is onsequene of Lemm 4.5 nd the reltion T 3 = T2 3 T In oth sides, it is ler tht only finite numer of terms in G 2 re relevnt, nmely those tht re projetions of some terms of G under T 2. Thus, y ilinerity, it is enough to onsider single digrms G nd G 2. If G 2 T 2 (G ), then oth sides re 0. If G 2 = T 2 (G ), then S 2 (G 2 ), G = Aut(G2) Aut(G G ), G = Aut(G 2 ), while G2, T 2 (G ) = G 2, G 2 = Aut(G 2 ). 3. The inlusion Im S 2 Ker T 2 follows immeditely from 2. For the onverse, pik Guss digrm series G in Ker T 2. For ny two 2-relted Guss digrms of type, G nd G, one hs G, G G = 0. Thus, if G 2 is Guss digrm of type 2, one n define φ(g 2 ) to e the vlue of G, G for ny preimge G of G 2 under T 2, nd set G 2 = φ(g 2 ) Aut(G 2 ) G 2, where the sum runs over ll Guss digrms of type 2. Finlly, φ(t 2 (G)) S 2(G 2 ) = Aut(T 2 (G)) Aut(G ) Aut(T (G)) G 2 = G,G Aut(G G. ) = G In prtie, point 3 is useful in oth diretions: whether one needs hrteriztion of the series tht lie in the imge of some mp S (Lemm 4.4), or of the series tht define invrints under some kind of moves (Propoosition 4.20). Point 2 sttes tht symmetry-preserving mps re the good ditionry to understnd invrints tht were defined vi forgetful projetions. Remrk 4.7. Every onstrution nd result in this susetion n e repeted y repling the set of rigid Guss digrms of degree n with ny set endowed with the tion of n elin finite group Arrow digrms nd homogeneous invrints Definition 4.8 (see [9, 20]). An rrow digrm (on π) is Guss digrm G (on π) of whih the signs deorting the rrows hve een forgotten. As usul, it is onsidered up to homeomorphisms of the irle. Arrow digrm spes A n, A n, A, the ht versions, nd the pirings (, ) nd, re defined similrly to their signed versions (Susetion 4.). We use nottions A for n rrow digrm nd A for n rrow digrm series i.e. n element of Â. 23