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Gage Martin

Publications and source records attributed to Gage Martin.

8 recordsLinked to original sources
arXiv2026

4-dimensional Skein modules, Handle attachments, and Tangles

Skein lasagna modules are a recent tool developed for the study of 4-manifolds. We provide general formula for 1-, 2-, and 3-handle attachments for skein modules defined with any functorial link theory in $S^3 \times I$ generalizing existing formula of Chen, Manolescu-Neithalath, Manolescu-Walker-Wedrich, and Ren-Willis. These formula are derived from a complete description of the gluing homomorphism on skein modules. For this description, we introduce a variation of these skein modules in the presence of distinguished 3-manifolds in the boundary. A similar construction was recently introduced independently by Blackwell-Krushkal-Luo.

math.GT↗
arXiv2024

Khovanov homology and Lagrangian cobordisms

We provide a partial answer to a question of Ekholm, Honda, and K\'alm\'an about the relationship between Khovanov homology and decomposable Lagrangian cobordisms. We also utilize previously defined filtered invariants to give obstructions to decomposable Lagrangian cobordisms from Khovanov homology.

math.GT↗
arXiv2021

Annular Khovanov homology and meridional disks

We exhibit infinite families of annular links for which the maximum non-zero annular Khovanov grading grows infinitely large but the maximum non-zero annular Floer-theoretic gradings are bounded. We also show this phenomenon exists at the decategorified level for some of the infinite families. Our computations provide further evidence for the wrapping conjecture of Hoste-Przytycki and its categorified analogue. Additionally, we show that certain satellite operations cannot be used to construct counterexamples to the categorified wrapping conjecture. We also extend the Batson-Seed link splitting spectral sequence to the setting of annular Khovanov homology.

math.GT↗
arXiv2021

Braids, fibered knots, and concordance questions

Given a knot in $S^3$, one can associate to it a surface diffeomorphism in two different ways. First, an arbitrary knot in $S^{3}$ can be represented by braids, which can be thought of as diffeomorphisms of punctured disks. Second, if the knot is fibered -- that is, if its complement fibers over $S^1$ -- one can consider the monodromy of the fibration. One can ask to what extent properties of these surface diffeomorphisms dictate topological properties of the corresponding knot. In this article we collect observations, conjectures, and questions addressing this, from both the braid perspective and the fibered knot perspective. We particularly focus on exploring whether properties of the surface diffeomorphisms relate to four-dimensional topological properties of knots such as the slice genus.

math.GT↗
arXiv2020

Knot Floer homology, link Floer homology and link detection

We give new link detection results for knot and link Floer homology inspired by recent work on Khovanov homology. We show that knot Floer homology detects $T(2,4)$, $T(2,6)$, $T(3,3)$, $L7n1$, and the link $T(2,2n)$ with the orientation of one component reversed. We show link Floer homology detects $T(2,2n)$ and $T(n,n)$, for all $n$. Additionally we identify infinitely many pairs of links such that both links in the pair are each detected by link Floer homology but have the same Khovanov homology and knot Floer homology. Finally, we use some of our knot Floer detection results to give topological applications of annular Khovanov homology.

math.GT↗
arXiv2020

Khovanov homology detects $T(2,6)$

We show if $L$ is any link in $S^3$ whose Khovanov homology is isomorphic to the Khovanov homology of $T(2,6)$ then $L$ is isotopic to $T(2,6)$. We show this for unreduced Khovanov homology with $\mathbb{Z}$ coefficients.

math.GT↗
arXiv2019

Annular Rasmussen invariants: Properties and 3-braid classification

We prove that for a fixed braid index there are only finitely many possible shapes of the annular Rasmussen $d_t$ invariant of braid closures. Applying the same perspective to the knot Floer invariant $Υ_K(t)$, we show that for a fixed concordance genus of $K$ there are only finitely many possibilities for $Υ_K(t)$. Focusing on the case of 3-braids, we compute the Rasmussen $s$ invariant and the annular Rasmussen $d_t$ invariant of all 3-braid closures. As a corollary, we show that the vanishing/non-vanishing of the $ψ$ invariant is entirely determined by the $s$ invariant and the self-linking number.

math.GT↗