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Some Questions on Braids and Knots Related to Homotopy 2-colored knot theory intersecting subgroups knot groups Lie algebra Some questions on braids and knots related to homotopy Jie Wu National University of Singapore Workshop on knot theory and low-manifold topology Dalian, China, 9-11 November 2017 2-colored knot theory intersecting subgroups knot groups Lie algebra Some questions on braids and knots related to homotopy free product of braid groups with amalgamation and 2-colored knot theory Intersecting subgroups of link groups Homomorphisms between knot groups Lie algebra on Brunnian braids 2-colored knot theory intersecting subgroups knot groups Lie algebra Free product of braid groups with amalgamation Let Bn be the n-strand Artin braid group. Let Pn be the pure braid group. Given a subgroup Q ≤ Pn, • Question 1. study the free product with amalgamation Bn ∗Q Bn and Pn ∗Q Pn such as • representations of Bn ∗Q Bn and • Lie algebra of Pn ∗Q Pn; • Question 2. develop 2-colored knot theory derived from Bn ∗Q Bn. 2-colored knot theory intersecting subgroups knot groups Lie algebra Intuitive description of Bn ∗Q Bn 0 Let Bn be a copy of Bn. Consider braids in Bn to be colored by 0 white, and braids in Bn be colored as black. The braids in Q are 0 colored as grey, which belongs to both Bn and Bn as tunnel of two braid worlds. Then a word in Bn ∗Q Bn admits a form of black braid white braid black braid ··· or white braid black braid white braid ··· 2-colored knot theory intersecting subgroups knot groups Lie algebra Potential relation with virtual/welded braids 0 -- -- Bn ∗ Bn = hσi ; σi j · · · i VBn = hσi ; ρi j · · · i WBn = hσi ; αi j · · · i ? Bn ∗Q Bn 2-colored knot theory intersecting subgroups knot groups Lie algebra k Connection with homotopy theory–π∗(S ) for k ≥ 3—Mikhailov-Wu k • We give a combinatorial description of π∗(S ) for any k ≥ 3 by using the free product with amalgamation of pure braid groups. • Given k ≥ 3; n ≥ 2, let Pn be the n-strand Artin pure braid group with the standard generators Ai;j for 1 ≤ i < j ≤ n. We construct certain (free) explicit subgroup Qn;k of Pn (depending on n and k). 2-colored knot theory intersecting subgroups knot groups Lie algebra k π∗(S ) for k ≥ 3—Mikhailov-Wu Now consider the free product with amalgamation Pn ∗Qn;k Pn: Namely this amalgamation is obtained by identifying the elements yj in two copies of Pn. Let Ai;j be the generators for 0 the first copy of Pn and let Ai;j denote the generators Ai;j for the P ∗ P 0 n Qn;k n second copy of Pn. Let Ri;j = hAi;j ; Ai;j i be the normal 0 closure of Ai;j ; Ai;j in Pn ∗Qn;k Pn. Let Y [Ri;j j 1 ≤ i < j ≤ n]S = [[Ri1;j1 ; Ri2;j2 ];:::; Rit ;jt ] f1;2;:::;ng=fi1;j1;:::;it ;jt g be the product of all commutator subgroups such that each integer 1 ≤ j ≤ n appears as one of indices at least once. 2-colored knot theory intersecting subgroups knot groups Lie algebra Mikhailov-Wu’13 k Theorem. Let k ≥ 3. The homotopy group πn(S ) is isomorphic to the center of the group (Pn ∗Qn;k Pn)=[Ri;j j 1 ≤ i < j ≤ n]S for any n if k > 3 and any n 6= 3 if k = 3. • Note. The only exceptional case is k = 3 and n = 3. In this 3 ⊕4 case, π3(S ) = Z while the center of the group is Z . • Reference: Mikhailov, Roman; Wu, Jie Combinatorial group theory and the homotopy groups of finite complexes. Geom. Topol. 17 (2013), no. 1, 235-272. 2-colored knot theory intersecting subgroups knot groups Lie algebra definitions 3 3 Let L = fl1;:::; lng be an n-link in S , and let G(L) = π1(S r L) be the link group of Ln. Let Ri be the normal closure of the i-th meridian αi in G(Ln). Let Y [[R1; R2];:::; Rn]S = [[Rσ(1); Rσ(2)];:::; Rσ(n)] σ2Σn be the symmetric commutator subgroup of R1;:::; Rn. Then [[R1; R2];:::; Rn]S ≤ R1 \ R2 \···\ Rn: An n-link L is called strongly nonsplittable if any nonempty sublink of L is nonsplittable. For instance, Hopf n-link is strongly nonsplittable. 2-colored knot theory intersecting subgroups knot groups Lie algebra definitions Let L = fl1;:::; lng be an n-link. Let di L = fl1;:::; li−1; li+1;:::; lng denote the sublink of L by removing i-th component. Let L be an n-link in S3. A knot K in S3 r L is called almost 3 trivial if K bounds a disk in S r di L for each 1 ≤ i ≤ n. A knot K in S3 r L is called weakly almost trivial if K represents an element in R1 \ R2 \···\ Rn. In other words, a knot K in S3 r L is weakly almost trivial if and only if the curve K 3 is null homotopic in the link complement S r di L for 1 ≤ i ≤ n. 2-colored knot theory intersecting subgroups knot groups Lie algebra definitions A knot K in S3 r L is called commutatorized if K represents an element in the symmetric commutator subgroup [[R1; R2];:::; Rn]S. 2-colored knot theory intersecting subgroups knot groups Lie algebra Question A weakly almost trivial knot in a link complement may not be almost trivial. Question. Let L be an n-link. Let K be a weakly almost trivial knot in the link complement S3 r L. Does there exist a connected sum decomposition K = K 0#K 00 0 00 such that K is a commutaorized knot in S3 r L and K is an almost trivial knot in S3 r L? 2-colored knot theory intersecting subgroups knot groups Lie algebra Motivation of the question Theorem. Let L be a strongly nonsplittable n-link with n ≥ 2. Then there is a short exact sequence of groups 3 f1g −! [[R1; R2];:::; Rn]S −! R1\R2∩· · ·∩Rn −! πn(S ) −! f1g: Reference. Fang, Fuquan; Lei, Fengchun; Wu, Jie The symmetric commutator homology of link towers and homotopy groups of 3-manifolds. Commun. Math. Stat. 3 (2015), no. 4, 497-526. 2-colored knot theory intersecting subgroups knot groups Lie algebra Simon conjecture in knot theory Simon conjecture. For each knot K in S3 , the fundamental group of its complement can maps surjectively onto only finitely many distinct knot groups by group homomorphisms. For a knot K , S3 r K is a K (π; 1)-space. Thus any group 3 3 0 homomorphism π1(S r K ) ! π1(S r K ) is induced by a map 0 f : S3 r K ! S3 r K . 2-colored knot theory intersecting subgroups knot groups Lie algebra Fengchun Lei-Fengling Li-Wu’14 Let (L; X) be a framed link in S3 with X a vector field defined in a neighborhood of L perpendicular to the tangent field of L. We can obtain a sequence of link L = fL0; L1;:::;:::g, where L0 = L and Ln is a naive n-cabling of L along the vector field X. By taking the fundamental groups of the link complements, we 3 obtain a simplicial group G(L; X) = fπ1(S r Ln)gn≥0. Let L =∼ L[1] t L[2] t · · · t L[p] be the splitting decomposition of the framed link L such that (L[i]; XjL[i]) is a nontrivial nonsplittable framed link for 1 ≤ i ≤ k and (L[i]; XjL[i]) is a trivial framed link for k + 1 ≤ i ≤ p. Then • Theorem. the geometric realization jG(L; X)j is homotopy k equivalent to Ω(W S3). 2-colored knot theory intersecting subgroups knot groups Lie algebra Question Let K and K 0 be framed knots with frames X and X 0, respectively. Question. Characterize the simplicial homomorphisms and simplicial epimorphism G(K ; X) ! G(K 0; X 0). Question. Study the knot invariants obtained from Gn(K ; X)=Moore boundaries, namely the link group Gn(K ; X) modulo the simplicial homotopy. Note 3 πn(S ) ≤ Gn(K ; X)=Moore boundaries for nontrivial framed knots. The group Gn(K ; X)=Moore boundaries may be still very complicated and heavily depends on K . Reference. Lei, Fengchun; Li, Fengling; Wu, Jie On simplicial resolutions of framed links. Trans. Amer. Math. Soc. 366 (2014), no. 6, 3075-3093. 2-colored knot theory intersecting subgroups knot groups Lie algebra Braid Commutators and Vassiliev Invariants Ted Stanford’96: Let L and L0 be two links which differ by a braid p 2 Γn(Pk ). Let v be a link invariant of order less than n. Then v(L) = v(L0). Here the meaning for two links to differ by a braid p is as follows: Let x^ denote the closure of a braid x. Let b and p be any two braids with the same number of strands. Then b^ and pb^ differ by p. In brief, lower central series of pure braid groups =) Vassiliev Invariants. For working on knots, we can fix k-strand braid b and let p run over Pk . 2-colored knot theory intersecting subgroups knot groups Lie algebra Relative Lie algebra Let G ≤ Pk be a subgroup of Pk . The elements in G give a set of special type of braids. Then the set G^ = fx^ j x 2 Gg gives a subset of special type of links. Consider G = Γ1(Pk )\G ≥ Γ2(Pk )\G ≥ · · · ≥ Γi (Pk )\G ≥ Γi+1(Pk )\G ≥ :::: The resulting (relative) Lie algebra 1 P L L k (G) = (Γi (Pk ) \ G)=(Γi+1(Pk ) \ G) is a sub Lie algebra of i=1 the Yang-Baxter Lie algebra L(Pk ).
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