Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bar-Natan skein lasagna modules and exotic surfaces in 4-manifolds

Ian A. Sullivan

TL;DR

This work extends the Bar-Natan link homology framework to the skein lasagna setting by defining the Bar-Natan skein lasagna module $\mathcal{S}_{0}^{BN}(X;L)$ for 4-manifolds with boundary links and introducing an $H$-action that encodes 1-handle operations. It develops connect-sum gluing and deformation theories, including the BN$_{H=1}$ localization, showing that primitive, homologically diverse fillings yield exotically knotted pairs that persist under a single internal stabilization in extended 4-manifolds. The approach provides criteria to detect primitiveness via a reduction to $\mathrm{KhR}_{2}$ and produces explicit examples extending Hayden’s one-stabilization phenomena beyond $B^{4}$, including a notable case in $\overline{\mathbb{C}P^{2}}$. The results illuminate how Bar-Natan’s algebraic data control geometric exotica through skein-theoretic constructions, offering a robust toolkit for producing and extending exotically knotted surfaces in a broad class of 4-manifolds. Overall, the paper advances the understanding of when exotic surface pairs survive stabilizations and how to propagate such phenomena via connect sums and local gluings using Bar-Natan skein lasagna modules.

Abstract

We construct and study the skein lasagna module obtained by importing the Bar-Natan Khovanov homology package. For 4-manifolds satisfying a non-vanishing condition, we produce pairs of exotic surfaces (with boundary) by using the behavior of skein lasagna gluing maps associated to connect sums of 4-manifolds. We show that one internal stabilization is generally not enough for these exotic knotted surfaces, generalizing results of Hayden to 4-manifolds that contain homologically diverse surfaces admitting primitive fillings.

Bar-Natan skein lasagna modules and exotic surfaces in 4-manifolds

TL;DR

This work extends the Bar-Natan link homology framework to the skein lasagna setting by defining the Bar-Natan skein lasagna module for 4-manifolds with boundary links and introducing an -action that encodes 1-handle operations. It develops connect-sum gluing and deformation theories, including the BN localization, showing that primitive, homologically diverse fillings yield exotically knotted pairs that persist under a single internal stabilization in extended 4-manifolds. The approach provides criteria to detect primitiveness via a reduction to and produces explicit examples extending Hayden’s one-stabilization phenomena beyond , including a notable case in . The results illuminate how Bar-Natan’s algebraic data control geometric exotica through skein-theoretic constructions, offering a robust toolkit for producing and extending exotically knotted surfaces in a broad class of 4-manifolds. Overall, the paper advances the understanding of when exotic surface pairs survive stabilizations and how to propagate such phenomena via connect sums and local gluings using Bar-Natan skein lasagna modules.

Abstract

We construct and study the skein lasagna module obtained by importing the Bar-Natan Khovanov homology package. For 4-manifolds satisfying a non-vanishing condition, we produce pairs of exotic surfaces (with boundary) by using the behavior of skein lasagna gluing maps associated to connect sums of 4-manifolds. We show that one internal stabilization is generally not enough for these exotic knotted surfaces, generalizing results of Hayden to 4-manifolds that contain homologically diverse surfaces admitting primitive fillings.

Paper Structure

This paper contains 9 sections, 20 theorems, 31 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Topological preliminaries
  4. Bar-Natan/equivariant link homology
  5. A skein lasagna module for mod 2 Bar-Natan homology
  6. Lasagna H-action for $(X,L)$
  7. A connect sum gluing map for Bar-Natan skein lasagna
  8. Deformations and decompositions into relative second homology
  9. Extending the exotically knotted pairs of surfaces in the 4-ball to other 4-manifolds

Key Result

Proposition 1.1

(Proposition prop:main) Let $(F_{g},F^{\prime}_{g})$ be the pair of genus $g$ exotic surfaces constructed in hayden2023atomic with boundary $K_{g}$ (see Figure fig:0) and let $(X,L)$ be a 4-manifold and boundary link pair. Suppose there exists a primitive filling $[S]\in \mathcal{S}_{0}^{BN}(X;L)$,

Figures (4)

  • Figure 1: Left: A diagram of the knot $K_{g}=\partial{F_{g}}=\partial{F_{g}^{\prime}}$. Right: The knot $K_{H}$ with disks $D$ and $D^{\prime}$. Both diagrams are found in hayden2023atomic.
  • Figure 2: Top: The sphere, torus, dotted sphere, and $H$-trading relation over $\mathbb{F}_{2}$ coefficients. Bottom: the neck-cutting relation.
  • Figure 3: The effect of neck-cutting along the small red circle.
  • Figure 4: The $H$-action map for $L\neq{\emptyset}$ as in Definition \ref{['def:H-action']}.

Theorems & Definitions (71)

  • Proposition 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Baykur2015, Theorem 1
  • Definition 2.4
  • Conjecture 2.5
  • Theorem 2.6: hayden2023atomic, Theorem A
  • Remark 2.7: hayden2023atomic, Propositions 4.1 and 4.2
  • ...and 61 more