An enhanced Euler characteristic of sutured instanton homology

Abstract

For a balanced sutured manifold $(M,\gamma )$, we construct a decomposition of $SHI(M,\gamma )$ with respect to torsions in $H=H_1(M;\mathbb{Z})$, which generalizes the decomposition of $I^{♯}(Y)$ in previous work of the authors. This decomposition can be regarded as a candidate for the counterpart of the torsion spin${}^c$ decompositions in $SFH(M,\gamma )$. Based on this decomposition, we define an enhanced Euler characteristic ${\chi }_{\mathrm{en}}(SHI(M,\gamma ))\in \mathbb{Z}[H]/\pm H$ and prove that ${\chi }_{\mathrm{en}}(SHI(M,\gamma ))=\chi (SFH(M,\gamma ))$. This provides a better lower bound on ${\dim }_{\mathbb{C}}SHI(M,\gamma )$ than the graded Euler characteristic ${\chi }_{\mathrm{gr}}(SHI(M,\gamma ))$. As applications, we prove instanton knot homology detects the unknot in any instanton L-space and show that the conjecture $KHI(Y,K)\cong \widehat{HFK}(Y,K)$ holds for all $(1,1)$-L-space knots and constrained knots in lens spaces, which include all torus knots and many hyperbolic knots in lens spaces.

arXiv comment45 pages, 12 figures; v2, published version

References in corpus (45)

• Floer homology and knot complements
• On the classification of tight contact structures I
• Floer homology and surface decompositions
• Knot Floer homology and rational surgeries
• Holomorphic disks and three-manifold invariants: properties and applications
• Holomorphic discs and sutured manifolds
• HF=HM I : Heegaard Floer homology and Seiberg--Witten Floer homology
• Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology
• Contact structures, sutured Floer homology and TQFT
• Cobordisms of sutured manifolds and the functoriality of link Floer homology
• Khovanov homology detects the trefoils
• Floer Simple Manifolds and L-Space Intervals
• The decategorification of sutured Floer homology
• Heegaard Floer homology and genus one, one boundary component open books
• Moving basepoints and the induced automorphisms of link Floer homology
• The equivalence of Heegaard Floer homology and embedded contact homology III: from hat to plus
• Instanton Floer homology and the Alexander polynomial
• The sutured Floer homology polytope
• Characterizing slopes for torus knots
• A contact invariant in sutured monopole homology
• Instantons and L-space surgeries
• Quasi-stabilization and basepoint moving maps in link Floer homology
• Instanton Floer homology and contact structures
• Graph cobordisms and Heegaard Floer homology
• Sutured Floer homology and invariants of Legendrian and transverse knots
• Naturality in sutured monopole and instanton homology
• Instanton Floer homology, sutures, and Heegaard diagrams
• Instanton Floer homology for sutured manifolds with tangles
• Joining and gluing sutured Floer homology
• Contact handles, duality, and sutured Floer homology
• Gluing tight contact structures
• A note on the knot Floer homology of fibered knots
• (1,1) L-space knots
• Duality and mapping tori in Heegaard Floer homology
• On the equivalence of contact invariants in sutured Floer homology theories
• The cosmetic crossing conjecture for split links
• Khovanov homology and the cinquefoil
• Gluing maps and cobordism maps for sutured monopole Floer homology
• Instanton Floer homology, sutures, and Euler characteristics
• Constrained knots in lens spaces
• On meridian-traceless SU(2)-representations of link groups
• Knot homologies in monopole and instanton theories via sutures
• On nonsimple knots in lens spaces with tunnel number one
• Connected sums and directed systems in knot Floer homologies
• Characterizing slopes for torus knots, II

Cited by in corpus (6)

• Instanton Floer homology, sutures, and Heegaard diagrams
• Instanton Floer homology, sutures, and Euler characteristics
• Small Dehn surgery and SU(2)
• SU(2) representations and a large surgery formula
• Instantons and Khovanov homology in ${\mathbb{RP}}^3$
• Knot surgery formulae for instanton Floer homology I: the main theorem