There are two 2 twist-spun trefoils

Transcription

1 There re two twist-spun trefoils Colin Rourke Brin Snderson Mthemtis Institute, Universit of Wrwik Coventr, CV4 7AL, UK Emil: nd URL: nd ~js/ Astrt We give short proof inspired Crter et l [] tht the twist-spun trefoil is not isotopi to its orienttion reverse. The proof uses omputer lultion of the third homolog group of the three olour rk. We lso give new proof using the sme lultion of the well-known ft tht the left nd right trefoil knots re not isotopi. AMS Clssifition 57Q45; 57M5, 57M7 Kewords Knot, knot, twist-spinning, reversiilit, hirlit, rk, qundle Introdution In reent pper [] Crter, Jelsovsk, Kmd, Lngford nd Sito onstrut stte-sum invrints of knots using notion of qundle ohomolog. As n pplition the prove tht the twie twist-spun trefoil is not isotopi to its orienttion reverse. Qundle ohomolog groups re losel relted to the ohomolog groups of the rk spe, onstruted erlier Roger Fenn nd the uthors [3, 4] nd the stte-sum invrint hs nturl interprettion in terms of the nonil lss in the homotop of the rk spe determined the knot digrm. In this pper we reover the theorem of Crter et l short rgument sed diretl on the nonil lss. We need to lulte the third homolog group of the three-olour rk nd for this we rel on Mple worksheet [7]. The sme lultion is used to prove the well-known ft tht the left nd right trefoil knots in R 3 re different. Here is n outline of the pper. In setion we give some kground mteril on rks nd the rk spe nd in setion 3 we prove the min theorem, tht the twist-spun trefoil is not isotopi to its orienttion reverse preserving orienttions. In setion 4 we prove tht the left nd right trefoils re different nd we onlude in setion 5 with some remrks out the proofs.

2 Bkground Frmings nd digrms for knots in 4 spe For lssil knots, the oriented nd frmed theories differ, for emple the writhe of knot is n invrint of frmed knots ut not of oriented knots. For knots (ie knots of S in R 4 ) the distintion disppers: Let K R 4 e n oriented, possil knotted, sphere in R 4. The norml undle of K hs setion sine K ounds 3 mnifold ( generlised Seifert surfe ) nd furthermore the orienttions now give nonil seond setion, in other words frming of K. Thus we n onfuse setions of the norml undle nd frmings. Now two setions of K differ mp K S (sine the norml undle is trivil) nd sine π (S ) = n two setions re isotopi. Thus the orienttions determine nonil frming of K in R 4. Now onsider the nonil projetion R 4 R 3 nd think of R 3 s horiontl nd the remining R s vertil. Suppose tht we re given knot K suh tht the projetion of K on R 3 hs multiple set omprising trnsverse doule lines nd triple points. The imge of the projetion together with the informtion of the vertil order of sheets t the multiple set is lled knot digrm for K. The digrm determines K up to isotop nd lso determines nturl frming of K orresponding to the setion whih is vertill up. Conversel, the Compression Theorem [6] knot K with setion of its norml undle n e isotoped so tht the setion is vertill up nd hene K projets to n immersion in R 3. B mking this immersion self-trnsverse we now hve knot digrm for K. Furthermore, the prmeter version of the Compression Theorem, isotopi frmed emeddings determine isotopi knot digrms. Comining the two fts just outlined, nmel tht orienttion determines frming nd tht frming determines digrm, we hve: Lemm. There is nturl ijetion etween isotop lsses of oriented emeddings of sphere in R 4 nd isotop lsses of knot digrms in R 3. The nonil lss of digrm Full detils of ll the mteril in this setion n e found in [4]. For kground mteril on rks see []. Here we shll summrise the results from [4] whih re needed for the min theorem. Let R e rk. The rk spe BR of R is uil set with the set of n ues in ijetion with R n. We shll onl need to onsider ues of dimension 3. The interprettion of suh ues is given the pitures in figure.

3 = Figure The figures mke the oundr mps ler. Thus the k fe of the 3 ue drwn (,, ) is glued to the ue (,) nd the top fe is glued to (, ). There re similr desriptions of oundr mps from n ues to (n ) ues nd the generl formul is: i (,..., n )=(,..., i, i+,..., n ), i (,..., n )=( i,..., i i, i+,, n ) for i n. The rk spe determines topologil spe (lso denoted BR) gluing rel ues together using these oundr mps. Now suppose tht we re given knot digrm in R 3.Alelling of the digrm in rk R mens lelling of sheets elements of R so tht t doule lines the rule indited in figure holds (the figure shows setion trnsverse to the doule line nd the usul digrm onvention hs een used, nmel the roken r mens the lower r in the vertil sense; the rrow indites the seond frming vetor). Figure A lelling R is preisel the sme s homomorphism of the fundmentl rk of K to R, see [; pge 384]. Now the frming of K omprises one vetor prllel to the vertil nd seond vetor norml to the imge in R 3. Using this seond vetor we n provide 3

4 the digrm with iollr with ollr lines identified with I (so tht the sumnifold orresponds to I nd the vetor points in the positive diretion long I ). We n ssume tht the ollr lines determine little ues t triple points nd norml squre undles long doule lines. The squres nd ues re nonill identified with I nd I 3 using the vertil ordering of sheets t doule lines nd triple points. Now suppose the digrm is lelled in R then using the lelling these opies of I, I nd I 3 n e identified with ues in BR. The digrm now determines mp of R 3 to BR mpping the ollr lines to the ues of BR with whih the re identified, the little squres long doule lines to the pproprite ues nd the ues t triple points to the pproprite 3 ues. Outside the ollr ll is mpped to the sepoint (the unique ue) nd hene we get sed mp S 3 BR mpping infinit (the sepoint of S 3 ) to the sepoint of BR. This mp is lmost nonil, depending onl on the hoie of ollr, nd we ll it the nonil mp determined the digrm nd the orresponding lss in π 3 (BR) the nonil lss. Given n isotop of digrms (or even oordism) lelled in R, then the nonil mp vries homotop ppling similr onstrution to the oordism. Thus using lemm. we hve: Lemm. The nonil lss in π 3 (BR) of lelled knot digrm is determined () the isotop lss of the oriented knot () the lelling, or equivlentl, the homomorphism of the fundmentl rk to R. This is s muh of the generl theor from [4] tht we shll need. However it is worth remrking tht there is onverse onstrution. Given mp S 3 BR we n use trnsverslit to mke it trnsverse to the 3 skeleton of BR nd this mens tht it pulls k digrm in R 3 (for some surfe in R 4 ) lelled in R. Similrl homotop pulls k oordism of digrms, ie, digrm for oordism emdedded in R 3 I. These onsidertions re summrised in the following lssifition theorem: Theorem.3 There is ijetion etween π 3 (BR) nd the set of oordism lsses of digrms in R 3, lelled in R, of surfes emedded in R 4. See [4] for the detiled proof of this result, for the interprettion in terms of Jmes undles nd for more generl results long the sme lines. 4

5 3 The twist spun trefoil The twist spun trefoil is not (oriented) isotopi to its ori- Min theorem enttion reverse. Proof The proof oupies the reminder of this setion. Let K e the twist spun trefoil. We shll ehiit n epliit digrm for K whih is lelled the three-olour rk T := {,, } with := mod 3, ie, = iff,, rellthesmeorlldifferent. -()-() -() ()+() () Figure 3 In figure 3 we give series of slies of the twist spun trefoil. The pitures re to e understood s follows. We re thinking of R 4 s n open-ook deomposition R 3 + S with S identified with for eh R = R 3 +, ie, R is the spine of the deomposition. Similrl we re thinking of R 3 s n open-ook deomposition with spine R,ie,R 3 is R + S with similr identifition for eh R = R +. The projetion R 4 R 3 is then given the stndrd projetion R 3 + R + rossed with the identit on S. The 5

6 figures re drwn s sequene of digrms in opies of R + orresponding to emeddings in the orresponding opies of R 3 + nd desrie digrm in R 3 for n emedding in R 4. The sepoint in R 4 is hosen to lie on the spine R of the deomposition ove the spine R on the deomposition of R 3.Thusthe sepoint lies ove eh of the digrms drwn. The frming (not shown) is given the left-hnd rule in other words the frming vetor is otined from the orienttion vetor on rs turning to the left. Figure 3 shows one twist of the spun trefoil. To get the twist spun trefoil, we repet the sequene. The figure lso shows the lelling in T. Oserve tht the finishing lels oinide with the strting ones with nd interhnged, so to get the lels for the seond twist, repet the lels with nd interhnged. The moves from one digrm to the net should ll e ovious eept perhps for the right-most pir in oth rows. Here ripple spins round in the surfe to eliminte the two oppposite twists (reting no triple points in the projetion). Now let ψ π 3 (BT) e the nonil lss of this lelled digrm for K.We shll ompute h(ψ) the Hurewi imge in H 3 (BT). Inspeting the definition of the nonil lss given in setion ove, we see tht h(ψ) is represented le C given s sum of 3 ues of BT one for eh triple point of the digrm. In figure 3 we hve written the 3 ues (with sign) determined the triple points (whih pper s R3 moves in the sequene). Figure 4 gives more detil of the identifition of these ues. Figure 4 shows the ues (), (), () nd () the other two ( () nd ()) re the sme s () nd () respetivel, with lel hnges. -() -() () () Figure 4 6

7 We n now red off the le whih represents h(ψ). C = () () () + () + () + () () () () + () + () + () We need two lultions for whih detils re to e found in the Mple worksheet [7]. Clultion H 3 (BT) = Z Z 3 Clultion C represents genertor of the Z 3 summnd of H 3 (BT). Lemm 3. T = {,, }. The Z 3 summnd of H 3 (BT) is fied ll permuttions of Proof Let Q denote the Z 3 summnd of H 3 (BT) ndlets 3 denote the group of permuttions of {,, }. C represents genertor of Q nd onstrution is invrint under the interhnge (, ) S 3.ThusQis fied (, ). Now notie tht there re no smmetries of H 3 (BT) of order 3. It follows tht the 3 le (,, ) S 3 lso fies Q. SineS 3 is generted (, ) nd (,, ), Q is fied the whole of S 3. Corollr 3. in T. h(ψ) is independent of the hoie of (non-onstnt) lelling Proof The lelling is lerl determined the initil lelling of the first digrm nd it follows tht two different (non-onstnt) lels re relted permuttion of T. Now let K denote K with opposite orienttion. Notie tht sine we re lelling in n involutor rk T (ie one in whih = for ll, ) the originl lels give lelling fter the orienttion hnge. Let ψ e the nonil lss of K using this lelling. To lulte h(ψ ) we oserve tht hnging orienttion reverses ll rrows nd from figure 5 we see tht this orresponds to repling eh ue (,, ) theue(,,) with opposite orienttion. Figure 5 7

8 Performing this sustitution for the le C given ove we red off the le C representing h(ψ ): C = () + () + () () () () () + () + () () () () Serendipitousl we notie: Oservtion C + C = It follows tht h(ψ )= h(ψ). Now suppose tht K is isotopi to K preserving orienttion. Then sine the fundmentl rks of K, K nd the isotop (thought of n n emedding of S I in S 4 I ) re ll isomorphi, the lelling of K in T indues lelling through the isotop whih is non-trivil on K. B orollr 3. we n use these lels to lulte h(ψ) ndh(ψ ). But lemm. ψ = ψ nd we hve ontrdition, ompleting the proof of the min theorem. See the remrks in setion 5 for eplntions of some (ut not ll) of the onidenes whih oured in the ove proof. 4 Trefoils left nd right We shll need generlistion of the rk spe. Let R e rk. The etended rk spe B R R hs set of n ues in ijetion with R n+ (ie the n + ues of BR) nd the sme formule for fe mps s in BR with n inde shift (ie, ε i in B R R is ε i+ in BR) f [3,.3. nd 3..]. It follows tht H n (B R R) = H n+ (BR) (f [4, 5.4 nd ove]). The interprettion for digrms is lelling of oth rs nd regions. More preisel, suppose tht D is digrm in R for knot K in R 3.Anetended lelling of D rk R is lelling of rs nd regions of D elements of R with the rules illustrted in figure 6 for lels of djent regions nd t rossings: = Figure 6 8

9 Given n etended lelled digrm, there is mp R B R R onstruted similrl to setion : Choose iollrs, mp little squres t rossings to the pproprite ue of B R R (for emple squre t the rossing in figure 6 would e mpped to the ue (,, )) ollr lines ross rs to the pproprite ue (eg, ollr line ross the lower left r in figure 6 would e mpped to the ue (, )) nd regions to the ue given the lel. Thus the lelling determines nonil lss φ π (B R R) (sed t the verte orresponding to the lel of the infinite region). An isotop of K in R 3 orresponds to digrm in R I to whih the lelling n e nonill etended. This in turn gives mp R I B R R, ie homotop of the nonil lss. Thus s in setion, the nonil lss is n invrint of the isotop lss of the lelled digrm. We now ehiit etended lellings in the three olour rk T for digrms of the right-hnd trefoil K nd the left-hnd trefoil K, see figure 7. -() () -() () -() () () () -() -() -() () Figure 7 Let φ nd φ e the nonil lsses determined these digrms nd let h(φ) ndh(φ ) e their Hurewi imges in H (B T T ) = H 3 (BT) = Z Z 3. We n red off representting les B nd B from the digrms: B = () () () + () + () + () B = () + () + () () () () Oservtion B = B nd hene h(φ) = h(φ ). We need one finl Mple lultion: 9

10 Clultion 3 B represents genertor of the Z 3 summnd of H (B T T ). Lemm 4. The lsses h(φ) nd h(φ ) re independent of the lels in T of rs nd regions in the digrms provided tht the knots themselves hve non-onstnt lels. Proof We notie tht oth digrms hve rottionl smmetr of order 3. But n two lellings whih re non-onstnt on the knots re relted this smmetr followed permtution of T. The result follows from lemm 3.. We n now prove tht right nd left-hnd trefoils re different. Theorem 4. There is no isotop of R 3 whih rries the (unoriented) righthnd trefoil to the left-hnd trefoil. Proof It is es to ontrut n isotop rring n oriented trefoil to its orienttion reverse. Thus it suffies to prove tht the oriented left nd right trefoils re not isotopi. The digrms drwn ove oth hve ero writhe nd if there is n isotop etween the oriented left nd right trefoils then there would e frmed isotop etween the digrms. Now the lelling on the right trefoil determines lelling of the isotop (thought of s digrm of S I in R 3 I ) nd hene lelling of the left trefoil whih is non-onstnt on the knot. B lemm 4. we n use these lels to lulte h(φ) ndh(φ ) nd it follows tht h(φ) =h(φ ) ontrditing the oservtion mde ove. 5 Some remrks Fundmentl qundles The involutive fundmentl qundle (ie the fundmentl qundle with the opertor reltions foreh) of the trefoil is the 3 olour rk/qundle T. This eplins wh there is nturl lelling in this rk. Furthermore using Vn Kmpen rgument it n e seen tht the fundmentl qundle of n n twist-spun knot K is the fundmentl qundle of K with the opertor reltions n foreh. It follows tht the fundmentl qundle of the twist-spun trefoil is T. So the lelling of the twist-spun trefoil we used is the onl non-trivil lelling (in qundle).

11 The nonil lss The fundmentl rk Γ of lssil link (ie link of irles in S 3 ) together with its nonil lss in π (BΓ), is omplete invrint up to isotop. Rell tht the fundmentl rk lone is omplete homeomorphism invrint ut just for non-split links [; theorem 5.]. Thus the nonil lss gives the etr informtion tht llows the rk to lssif split links nd up to isotop rther thn just homeomorphism. This new result will e proved in detil in [5]; it follows from the methods of the proof of [4; theorem 5.8]. Thus the nonil lss distinguishes the left nd right trefoils whih hve the isomorphi fundmentl rks. The nonil lss lies in π (BΓ) whih is H of the universl over of BΓ nd therefore determines lsses in ever onneted over of BΓ. To distinguish the left nd right trefoils we used B T T whih is 3 fold over BT nd the nonil lss we used ws the imge of the lss in the 3 fold over of BΓ determined in this w. This is tpil of the w in whih informtion n e etrted from the min lssifitoin theorem in [5]. The Z ftor Notie tht H 3 (BT) = H (B T T)hsZftor whih pled no role in the proofs. In ft ever homolog group of the rk spe BR where R hs se element whih ts trivill on itself (n element in qundle will do) splits Z ftor whih omes from the inlusion nd nturl projetion B BR B where B denotes the rk spe of the one element rk. Now B is model for Ω(S ) nd hs one ell in eh dimension with ll oundr mps trivil nd hene hs homolog Z in eh dimension [4; 3.3]. Now for knot the nonil lss mps to π 3 (ΩS )=π 4 (S 3 )=Z whih in turn mps to ero in H 3 (ΩS )=Z. This eplins wh the Z ftor ws immteril for the twist spun trefoil. For knot the imge in H (ΩS )=Z is redil seen to e the writhe of the digrm. But we hose our digrms for the left nd right trefoils to hve ero writhe, whih eplins wh the Z ftor ws immteril for this se s well. Invrine under permuttions of T Rk spes re simple in other words π ts trivill on π n for eh n [4; proposition 5.]. The proof is digrm mnipultion. Introdue smll sphere lelled R. Pull the sphere over the digrm (relising the hnge of lels orresponding to the tion of ) nd then pull the sphere k under the digrm nd eliminte it. The sme proof shows tht n lelled digrm is oordnt to the digrm with lels ted on n element of

12 the lelling rk. Thus the opertor group of the rk ts trivill on the nonil lss. Now the three olour rk T hs opertor group equl to its utomorphism group, whih is the smmetri group S 3. This eplins wh the nonil lss determined lelling in T is invrint under hoie of lels: n rk with opertor group equl to its utomorphism group would hve the sme propert. Thus the lelling used for the twist-spun trefoil ws immteril. For etended lels the sme proof shows invrine under hnge of lelling with fied lel t infinit. This together with the ovious smmetr of the hoie of lel t infinit in T eplins wh the lelling ws lso immteril for the left nd right trefoils. Referenes [] J Sott Crter, Dniel Jelsovsk, Lurel Lngford, Seihi Kmd, Mshio Sito, Qundle Cohomolog nd Stte-sum Invrints of Knotted Curves nd Surfes, rxiv referene mth.gt/9965 [] Roger Fenn, Colin Rourke, Rks nd links in odimension two, Journl of Knot theor nd its Rmifitions, (99) , ville from: [3] Roger Fenn, Colin Rourke, Brin Snderson, Trunks nd lssifing spes, Applied tegoril strutures, 3 (995) 3 356, ville from: [4] Roger Fenn, Colin Rourke, Brin Snderson, Jmes undles nd pplitions, Wrwik preprint (996), ville from: [5] Roger Fenn, Colin Rourke, Brin Snderson, The lssifition of links, Monogrph in preprtion [6] Colin Rourke, Brin Snderson, The ompression theorem, (now eing revised) urrent version ville from: [7] Brin Snderson, Mple worksheet,