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Cobordism maps in Khovanov homology and singular instanton homology II

Hayato Imori, Taketo Sano, Kouki Sato, Masaki Taniguchi

TL;DR

We extend the cobordism-map framework to immersed link cobordisms, defining Kh^{low}, Kh^{bal}, and related immersed maps on Khovanov homology and proving their compatibility with Kronheimer–Mrowka’s immersed instanton maps via a Khovanov–Floer type spectral sequence. The construction uses crossing-change maps and a Hopf-link formalism to realize immersed cobordism maps on Kh_{h,t}, and is complemented by instanton-cube computations and equivariant instanton theory to anchor the E^2-term and grading constraints. Two key applications follow: (i) a concordance injectivity result for negative two-bridge torus knots with a left inverse given by reversal, extended to I^♯ and generalized Kh; (ii) a detection result showing that relatively exotic slice surfaces remain exotic under positive twist moves, as witnessed by their Kh and I^♯ maps. Section 4 ties the Kh-side maps to the instanton cube maps with explicit Hopf-link calculations, while Section 5 leverages equivariant instanton theory to obtain structural and concordance-compatibility results for two-bridge torus knots, including left-inverse phenomena and generalized Kh extensions.

Abstract

This paper is a continuation of our previous work, where we defined an embedded cobordism map on the instanton cube complex that recovers the cobordism maps both in Khovanov homology and singular instanton theory. In this paper, we extend this construction to immersed cobordisms, where we define an immersed cobordism map on Khovanov homology and prove that it is compatible with the immersed cobordism map on singular instanton homology. We give two applications: (i) For any smooth, oriented concordance $C$ from a two-bridge torus knot, the induced map $\mathit{Kh}(C)$ on Khovanov homology is injective, and its left inverse is given by the reversal of $C$. (ii) Any pair of relatively exotic surfaces in $D^4$ that are detected by the embedded cobordism map in $\mathit{Kh}$ remain exotic even after applying any number of positive twist moves.

Cobordism maps in Khovanov homology and singular instanton homology II

TL;DR

We extend the cobordism-map framework to immersed link cobordisms, defining Kh^{low}, Kh^{bal}, and related immersed maps on Khovanov homology and proving their compatibility with Kronheimer–Mrowka’s immersed instanton maps via a Khovanov–Floer type spectral sequence. The construction uses crossing-change maps and a Hopf-link formalism to realize immersed cobordism maps on Kh_{h,t}, and is complemented by instanton-cube computations and equivariant instanton theory to anchor the E^2-term and grading constraints. Two key applications follow: (i) a concordance injectivity result for negative two-bridge torus knots with a left inverse given by reversal, extended to I^♯ and generalized Kh; (ii) a detection result showing that relatively exotic slice surfaces remain exotic under positive twist moves, as witnessed by their Kh and I^♯ maps. Section 4 ties the Kh-side maps to the instanton cube maps with explicit Hopf-link calculations, while Section 5 leverages equivariant instanton theory to obtain structural and concordance-compatibility results for two-bridge torus knots, including left-inverse phenomena and generalized Kh extensions.

Abstract

This paper is a continuation of our previous work, where we defined an embedded cobordism map on the instanton cube complex that recovers the cobordism maps both in Khovanov homology and singular instanton theory. In this paper, we extend this construction to immersed cobordisms, where we define an immersed cobordism map on Khovanov homology and prove that it is compatible with the immersed cobordism map on singular instanton homology. We give two applications: (i) For any smooth, oriented concordance from a two-bridge torus knot, the induced map on Khovanov homology is injective, and its left inverse is given by the reversal of . (ii) Any pair of relatively exotic surfaces in that are detected by the embedded cobordism map in remain exotic even after applying any number of positive twist moves.

Paper Structure

This paper contains 38 sections, 75 theorems, 344 equations, 6 figures.

Table of Contents

  1. Introduction
  2. A Khovanov--Floer type statement for immersed cobordisms
  3. Immersed cobordism maps on Khovanov homology
  4. Applications
  5. Injectivity from concordance
  6. Detection of exotic slice surfaces
  7. Immersed cobordism maps on Khovanov homology
  8. Crossing change maps
  9. Geometric description
  10. Crossing changes and Reidemeister moves
  11. Immersed cobordism maps
  12. Khovanov homology of the negative (2, q)-torus knots
  13. Structure of Kh(mT(2, 2k + 1))
  14. Relating Kh(mT(2, q)) and Kh(mT(2, q+2))
  15. Immersed cobordism maps for instanton cube complexes
  16. ...and 23 more sections

Key Result

Theorem 1.1

For any links $L, L'$ with diagrams $D, D'$, and normally immersed cobordism $S$ between $L$ and $L'$, there exists a doubly filtered chain map between the instanton cube complexes of order whose induced map on $E^2$-term with respect to the $h$-filtration coincides up to sign with and whose induced map on homology coincides with the cobordism map of singular instanton knot Floer homology Here

Figures (6)

  • Figure 1: Diagrams $D$, $D'$ and their resolutions $D_0, D_1$ at $x$.
  • Figure 2:
  • Figure 4:
  • Figure 6:
  • Figure 7:
  • ...and 1 more figures

Theorems & Definitions (147)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Proposition 1.10
  • Theorem 1.11
  • Remark 2.1
  • Proposition 2.2
  • ...and 137 more