arXiv subjects

Rob McConkey

Publications and source records attributed to Rob McConkey.

3 recordsLinked to original sources

arXiv2026

Deeply Slice Knot Detection via Immersed Curves

On the Kirby list, Akbulut poses the question of whether there exists a homology 3-sphere $Y$, other than $S^3$, with the following property: Any knot $K$, representing $0\in\pi_{1}(Y),$ which is slice in some contractible 4-manifold $X$ which $Y$ bounds, is already slice in $Y\times[0,1]$. In this paper, we make progress on this question by producing a class of deeply slice knots. We construct these knots by first specifying a pair $(X, K)$, where $X$ is a contractible 4-manifold with integral homology 3-sphere boundary and $K$ is slice in $X$. Then, we show the knot is deeply slice using concordance invariants from Heegaard Floer homology. We employ immersed curve techniques to compute these invariants.

math.GT↗

arXiv2025

Satellite Operations and $\theta$

We study the behavior of the knot invariant $\theta$ under satellite operations. First, we prove that $\theta$ is additive under connected sum. We then introduce a computational tool to generate $t$-twisted Whitehead doubles and apply it to explore the case of untwisted Whitehead doubles. We propose a conjecture describing the behavior of $\theta$ on untwisted Whitehead doubles and verify the conjecture for the first 2977 prime knots. The pair of invariants $\Theta = (\Delta,\theta)$ was introduced by Bar-Natan and van der Veen, where $\Delta$ is the Alexander polynomial. The invariant $\theta$ is easily computable and effective at distinguishing knots. Further exploration of satellite operations and $\theta$ is proposed to reveal new patterns among cables and general satellites.

math.GT↗

arXiv2023

Linear Bounds of the Crosscap Number of Knots

Kalfagianni and Lee found two-sided bounds for the crosscap number of an alternating link in terms of certain coefficients of the Jones polynomial. We show here that we can find similar two-sided bounds for the crosscap number of Conway sums of strongly alternating tangles. Then we find families of links for which these coefficients of the Jones polynomial and the crosscap number grow independently. These families will enable us to show that neither linear bound generalizes for all links.

math.GT↗