Link homology and equivariant gauge theory

Abstract

The singular instanton Floer homology was defined by Kronheimer and Mrowka in connection with their proof that the Khovanov homology is an unknot detector. We study this theory for knots and two-component links using equivariant gauge theory on their double branched covers. We show that the special generator in the singular instanton Floer homology of a knot is graded by the knot signature mod 4, thereby providing a purely topological way of fixing the absolute grading in the theory. Our approach also results in explicit computations of the generators and gradings of the singular instanton Floer chain complex for several classes of knots with simple double branched covers, such as two-bridge knots, torus knots, and Montesinos knots, as well as for several families of two-components links. IntroductionThis paper studies the Floer homology I * (Σ, L) of two-component links L ⊂ Σ in homology 3-spheres defined by Kronheimer and Mrowka [26] using singular SO (3) instantons. An important special case of this theory is the singular instanton knot Floer homology I ♮ (k) for knots k ⊂ S 3 obtained by applying I * (S 3 , L) to the link L which is a connected sum of k with the Hopf link. The Floer homology I * (Σ, L) has a relative Z/4 grading, which can be upgraded to an absolute Z/4 grading in the special case of I ♮ (k). [26] used I ♮ (k) and its close cousin I ♯ (k) to prove that the reduced Khovanov homology is an unknot-detector. Kronheimer and MrowkaThe definition of groups I * (Σ, L) uses singular gauge theory, which makes them difficult to compute. We propose a new approach to these computations which uses equivariant gauge theory in place of the singular one.Given a two-component link L in an integral homology sphere Σ, we pass 2010 Mathematics Subject Classification. 57M27, 57R58.Both authors were partially supported by NSF Grant 1065905. 1 to the double branched cover M → Σ with branch set L and observe that the singular connections on Σ used in the definition of I * (Σ, L) pull back to equivariant smooth connections on M . The generators of the Floer chain complex IC * (Σ, L), whose homology is I * (Σ, L), are then derived from the equivariant representations π 1 M → SO(3), and their Floer gradings are computed using equivariant rather than singular index theory † .As our first application of this approach, we determine the grading of the special generator in the Floer chain complex IC ♮ (k) of a knot k ⊂ S 3 , see Section 5. This fixes the absolute Z/4 grading on I ♮ (k) and confirms the conjecture of Hedden, Herald and Kirk [20].Theorem. For any knot k ⊂ S 3 , the grading of the special generator in the Floer chain complex IC ♮ (k) equals sign k mod 4.We also achieve significant simplifications in computing the Floer chain complexes IC ♮ (k) and IC * (Σ, L) for knots and links with simple double branched covers, such as torus and Montesinos knots and links, whose double branched covers are Seifert fibered manifolds. Explicit calculations for these knots and links are possible because the gauge theory on Seife...

Equivariant aspects of singular instanton Floer homology

“…Thus the orbifold pS 3 , K p,q q is the quotient of S 3 by the action of the dihedral group of order 2p. As observed in [PS17], all of the classes in CpLpp, qqq are fixed by the action of τ , and so CpLpp, qqq " CpLpp, qqq τ . Furthermore, each ξ i Lpp,qq is reducible, and thus uniquely lifts to a class ξ i P CpKq which is fixed by ι.…”

“…From this it is clear that ιpθq " θ. More generally, the following elementary lemma is observed in [PS17], where this involution on the character variety is studied: Lemma 2.5. An element of the critical set C is fixed by the flip symmetry ι if and only if its corresponding representation class in X pY, Kq has image in SU p2q conjugate to a binary dihedral subgroup.…”

“…The geometrical input involved in the singular instanton Floer complexes r C ˚pKq for a knot K Ă S 3 as defined in Section 3 can be related to the corresponding data on the double branched cover π : Σ Ñ S 3 over K. On the level of critical sets, this is established in [PS17], and the description extends to the cylinder in a straightforward manner.…”

On the traceless SU(2) character variety of the 6-punctured 2-sphere

“…Some of the statements of the following theorem are known to experts, for example see [4,21]. We include its proof in Appendix B for convenience.…”

Equivariant rho-invariants and instanton homology of torus knots

“…Let Y be a closed oriented Riemannian 3-manifold and α : π 1 (Y ) → U(n) homology by Poudel and Saveliev [PS15]. Calculations for non-free involutions are rather sparse in the literature; examples of such calculations can be found in Degeratu [Deg09] and Saveliev [Sav99].…”