Instanton Floer homology, sutures, and Euler characteristics

Abstract

This is a companion paper to an earlier work of the authors. In this paper, we provide an axiomatic definition of Floer homology for balanced sutured manifolds and prove that the graded Euler characteristic of this homology is fully determined by the axioms we proposed. As a result, we conclude that χpSHIpM, γqq " χpSF HpM, γqq for any balanced sutured manifold pM, γq. In particular, for any link L in S 3 , the Euler characteristic χpKHIpS 3 , Lqq recovers the multi-variable Alexander polynomial of L, which generalizes the knot case. Combined with the authors' earlier work, we provide more examples of p1, 1q-knots in lens spaces whose KHI and { HF K have the same dimension. Moreover, for a rationally null-homologous knot in a closed oriented 3-manifold Y , we construct a canonical Z 2 on KHIpY, Kq, the decomposition of I 7 pY q discussed in the previous paper, and the minus version of instanton knot homology KHI ´pY, Kq introduced by the first author.

Small Dehn surgery and SU(2)

“…Our proof involves a novel blend of ingredients, including a relationship between instanton Floer homology and the symplectic Floer homology of genus-2 surface diffeomorphisms, due to Smith [Smi12]; ideas of Baldwin-Hu-Sivek from their recent work on cinquefoil detection [BHS21]; and new results of Li-Ye relating the graded Euler characteristics of instanton and Heegaard knot Floer homology [LY21b,LY21a]. Along the way, we partially characterize instanton L-space knots of genus 2 (Theorem 2.1 and Remark 2.6), and prove several other results (Proposition 3.2, and Theorems 4.1 and 5.1) which may be of independent interest.…”

“…In recent work [LY21b], Li and Ye proved an equality between the graded Euler characteristics of instanton knot homology and (Heegaard) knot Floer homology, χ gr ( HFK (L, J)) = ±χ gr (KHI (L, J)).…”