Adjunction inequality for spatially refined $s$-invariants

TL;DR

The work investigates adjunction-type inequalities for spatial refinements of Rasmussen's $s$-invariant under link cobordisms in $k\overline{\mathbb{CP}^2}$, noting that general spatial refinements can violate the adjunction bound. It introduces θ-refined, particularly $s_+^\theta$, invariants for links via a 2-dimensional subspace of Lee/Khovanov data and proves cobordism monotonicity $s_+^\theta(L_1)-s_+^\theta(L_0)\ge \chi(\Sigma)$ for suitable cobordisms, while showing that some refinements (e.g., odd $Sq^1$, $Sq^2$ variants) fail (as in the knot $9_{42}$). The paper then concentrates on $s_+^{Sq^1}$ (the even $Sq^1$ refinement), proving the adjunction inequality holds in this setting. It provides concrete calculations for torus links: $s_+^{Sq^1}(T(n,n)_{p,q})=s^{\mathbb F_2}(T(n,n)_{p,q})=(p-q)^2-2p+1$, and establishes additivity under disjoint unions with certain “nice” links via precise filtrations, yielding $s_+^{Sq^1}(L\sqcup T)=s(L\sqcup T)+2$ when $s_+^{Sq^1}(L)=s(L)+2$. These results illuminate when refined invariants preserve adjunction-type bounds and clarify the behavior of spatial refinements under basic operations.

Abstract

We note an adjunction inequality in $k\overline{\mathbb{CP}^2}$ for the $s$-version of the $Sq^1$-refinement of Rasmussen's $s$-invariant. This does not hold for general spatial refinements of $s$-invariants.

Figures (1)

• Figure 1: A diagram of the knot $9_{42}$. Untwisting the twist region results in the unknot.

Theorems & Definitions (5)

• Theorem 1
• Proposition 2
• proof
• Proposition 3
• proof