Legendrian simple knots not detected by Khovanov homology
In this short note, we exhibit a Legendrian simple knot that is not detected by Khovanov homology. This disproves a conjecture of Chernov and Maguire.
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arXiv subjects
David Suchodoll
Publications and source records attributed to David Suchodoll.
On the multiplicity of Knot Floer order under cabling
The knot Floer order $\operatorname{Ord}(K)$ is a knot invariant derived from knot Floer homology that provides bounds on many other invariants, such as the bridge index $\operatorname{br}(K)$ for which $\operatorname{Ord}(K) + 1 \leq \operatorname{br}(K)$. For all $(p,q)$-cables of L-space knots, we show that $\operatorname{Ord}(K) + 1$ is multiplicative in $p$ when $g(K) > 1$, and the same holds for $g(K) = 1$ provided $q > 2p$. We also compute the knot Floer order in the range $q < 2p$, thereby determining $\operatorname{Ord}(K_{p,q})$ in terms of $\operatorname{Ord}(K)$ for all cables of L-space knots. We establish upper bounds under cabling for $\operatorname{Ord}(K)$ and discuss potential applications to a conjecture by Krishna and Morton, proving that the braid index of an L-space cable appears as an exponent in its Alexander polynomial if it does for its companion, provided $\operatorname{Ord}(K)+1$ is multiplicative.
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Algorithms in 4-manifold topology
We show that there exists an algorithm that takes as input two closed, simply connected, topological 4-manifolds and decides whether or not these 4-manifolds are homeomorphic. In particular, we explain in detail how closed, simply connected, topological 4-manifolds can be naturally represented by a Kirby diagram consisting only of 2-handles. This representation is used as input for our algorithm. Along the way, we develop an algorithm to compute the Kirby-Siebenmann invariant of a closed, simply connected, topological 4-manifold from any of its Kirby diagrams and describe an algorithm that decides whether or not two intersection forms are isometric. In a slightly different direction, we discuss the decidability of the stable classification of smooth manifolds with more general fundamental groups. Here we show that there exists an algorithm that takes as input two closed, oriented, smooth 4-manifolds with fundamental groups isomorphic to a finite group with cyclic Sylow 2-subgroup, an infinite cyclic group, or a group of geometric dimension at most 3 (in the latter case we additionally assume that the universal covers of both 4-manifolds are not spin), and decides whether or not these two 4-manifolds are orientation-preserving stably diffeomorphic.
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