A new basis for the Homflypt skein module of the solid torus

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• 作者：Ioannis Diamantis <sup>diamantis@math.ntua.gr" class="auth_mail" title="E-mail the corresponding authorsup> ; Sofia Lambropoulou<sup> ; sup><sup>sofia@math.ntua.gr" class="auth_mail" title="E-mail the corresponding authorsup>
• 关键词：57M27 ; 57M25 ; 57N10 ; 20F36 ; 20C08
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：220
• 期：2
• 页码：577-605
• 全文大小：677 K

文摘

In this paper we give a new basis, 螞, for the Homflypt skein module of the solid torus, <span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si1.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=8c8dc9593f366f9002fb0f0c85c4ab9f" title="Click to view the MathML source">S(ST)span><span class="mathContainer hidden"><span class="mathCode">scroll">script">Sstretchy="false">(STstretchy="false">)span>span>span>, which topologically is compatible with the handle sliding moves and which was predicted by J.H. Przytycki. The basis 螞 is different from the basis <span id="mmlsi2" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si2.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=b5e12ed4b0e76e7d1c97ae70432c4445" title="Click to view the MathML source">螞<sup>′sup>span><span class="mathContainer hidden"><span class="mathCode">scroll">sup>螞′sup>span>span>span>, discovered independently by Hoste and Kidwell <span id="bbr0010">[1]span> and Turaev <span id="bbr0020">[2]span> with the use of diagrammatic methods, and also different from the basis of Morton and Aiston <span id="bbr0030">[3]span>. For finding the basis 螞 we use the generalized Hecke algebra of type B, <span id="mmlsi3" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si3.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=dcda08e1067464280ec360303a1698d5" title="Click to view the MathML source">H<sub>1,nsub>span><span class="mathContainer hidden"><span class="mathCode">scroll">sub>H1,nsub>span>span>span>, which is generated by looping elements and braiding elements and which is isomorphic to the affine Hecke algebra of type A <span id="bbr0040">[4]span>. More precisely, we start with the well-known basis <span id="mmlsi2" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si2.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=b5e12ed4b0e76e7d1c97ae70432c4445" title="Click to view the MathML source">螞<sup>′sup>span><span class="mathContainer hidden"><span class="mathCode">scroll">sup>螞′sup>span>span>span> of <span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si1.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=8c8dc9593f366f9002fb0f0c85c4ab9f" title="Click to view the MathML source">S(ST)span><span class="mathContainer hidden"><span class="mathCode">scroll">script">Sstretchy="false">(STstretchy="false">)span>span>span> and an appropriate linear basis <span id="mmlsi322" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si322.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=5fae6d1e73b15e623783a87675745e22" title="Click to view the MathML source">危<sub>nsub>span><span class="mathContainer hidden"><span class="mathCode">scroll">sub>危nsub>span>span>span> of the algebra <span id="mmlsi3" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si3.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=dcda08e1067464280ec360303a1698d5" title="Click to view the MathML source">H<sub>1,nsub>span><span class="mathContainer hidden"><span class="mathCode">scroll">sub>H1,nsub>span>span>span>. We then convert elements in <span id="mmlsi2" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si2.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=b5e12ed4b0e76e7d1c97ae70432c4445" title="Click to view the MathML source">螞<sup>′sup>span><span class="mathContainer hidden"><span class="mathCode">scroll">sup>螞′sup>span>span>span> to sums of elements in <span id="mmlsi322" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si322.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=5fae6d1e73b15e623783a87675745e22" title="Click to view the MathML source">危<sub>nsub>span><span class="mathContainer hidden"><span class="mathCode">scroll">sub>危nsub>span>span>span>. Then, using conjugation and the stabilization moves, we convert these elements to sums of elements in 螞 by managing gaps in the indices, by ordering the exponents of the looping elements and by eliminating braiding tails in the words. Further, we define total orderings on the sets <span id="mmlsi2" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si2.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=b5e12ed4b0e76e7d1c97ae70432c4445" title="Click to view the MathML source">螞<sup>′sup>span><span class="mathContainer hidden"><span class="mathCode">scroll">sup>螞′sup>span>span>span> and 螞 and, using these orderings, we relate the two sets via a block diagonal matrix, where each block is an infinite lower triangular matrix with invertible elements in the diagonal. Using this matrix we prove linear independence of the set 螞, thus 螞 is a basis for <span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si1.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=8c8dc9593f366f9002fb0f0c85c4ab9f" title="Click to view the MathML source">S(ST)span><span class="mathContainer hidden"><span class="mathCode">scroll">script">Sstretchy="false">(STstretchy="false">)span>span>span>.

sp0080"><span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si1.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=8c8dc9593f366f9002fb0f0c85c4ab9f" title="Click to view the MathML source">S(ST)span><span class="mathContainer hidden"><span class="mathCode">scroll">script">Sstretchy="false">(STstretchy="false">)span>span>span> plays an important role in the study of Homflypt skein modules of arbitrary c.c.o. 3-manifolds, since every c.c.o. 3-manifold can be obtained by integral surgery along a framed link in <span id="mmlsi5" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si5.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=9ab3e2ac6abefa020c322a6549af48c7" title="Click to view the MathML source">S<sup>3sup>span><span class="mathContainer hidden"><span class="mathCode">scroll">sup>S3sup>span>span>span> with unknotted components. In particular, the new basis of <span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si1.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=8c8dc9593f366f9002fb0f0c85c4ab9f" title="Click to view the MathML source">S(ST)span><span class="mathContainer hidden"><span class="mathCode">scroll">script">Sstretchy="false">(STstretchy="false">)span>span>span> is appropriate for computing the Homflypt skein module of the lens spaces. In this paper we provide some basic algebraic tools for computing skein modules of c.c.o. 3-manifolds via algebraic means.