Framed instanton homology and concordance, II

Abstract

We continue our study of the integer-valued knot invariants ${\nu }^{♯}(K)$ and $r_0(K)$, which together determine the dimensions of the framed instanton homologies of all nonzero Dehn surgeries on $K$. We first establish a "conjugation" symmetry for the decomposition of cobordism maps constructed in our earlier work, and use this to prove, among many other things, that ${\nu }^{♯}(K)$ is always either zero or odd. We then apply these technical results to study linear independence in the homology cobordism group, to define an instanton Floer analogue ${\epsilon }^{♯}(K)$ of Hom's $\epsilon$-invariant in Heegaard Floer homology, and to the problem of characterizing a given 3-manifold as Dehn surgery on a knot in $S^3$.

arXiv comment41 pages; v2: accepted version

References in corpus (12)

• Knot Floer homology and rational surgeries
• Knot homology groups from instantons
• The knot Floer complex and the smooth concordance group
• Bordered Heegaard Floer homology and the tau-invariant of cable knots
• Instantons and odd Khovanov homology
• Framed instanton homology and concordance
• Instantons and L-space surgeries
• Filtered instanton Floer homology and the homology cobordism group
• Mapping cone formula in Heegaard Floer homology and Dehn surgery on knots in $S^3$
• Two-fold quasi-alternating links, Khovanov homology and instanton homology
• Instanton L-spaces and splicing
• Characterizing slopes for $5_2$