arXiv subjects

Tuomas Kelomäki

Publications and source records attributed to Tuomas Kelomäki.

4 recordsLinked to original sources

arXiv2026

On detection probabilities of link invariants

We prove that, for many standard link invariants, both the proportion of distinct invariant values and the detection probability among prime alternating links with at most n crossings decay exponentially in n, with an explicit universal rate. In fact, almost every such link belongs to an invariant fiber whose size is itself exponential in n. This phenomenon applies broadly, in particular to the Jones and HOMFLYPT polynomials and integral Khovanov homology. The companion website gives a much more detailed view of the data, including complete distributions of fiber sizes, separate alternating and non-alternating data, and topological data analysis.

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arXiv2026

On computational complexity of Khovanov homology

Computing the Jones polynomial of general link diagrams is known to be $\#$P-hard, while restricting the computation to braid closures on fixed number of strands allows for a polynomial time algorithm. We investigate polynomial time algorithms for Khovanov homology of braids and show that for $3$-braids there is one. In contrast, we show that Bar-Natan's scanning algorithm runs in exponential time when restricted to simple classes of $3$-braids. For more general braids, we obtain that a variation of the scanning algorithm computes the Khovanov homology for a bounded set of homological degrees in polynomial time. We also prove upper and lower bounds on the ranks of Khovanov homology groups.

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arXiv2025

Morse matchings and Khovanov homology of 4-strand torus links

Given a link or a tangle diagram, we define algorithmic Morse theoretic simplifications on their Khovanov homology. In contrast to Bar-Natan's scanning algorithm, the cancellations are postponed until the end and performed in one go. Although our novel approach is computationally inferior to Bar-Natan's algorithm, it side-steps the need for a large amount of iterations, making it more fitting for theoretical analysis. Our main application is towards integral Khovanov homology of 4-strand torus links, for which we compute non-trivial Khovanov homology groups in all homological degrees and find an abundance of $4$-torsion. At the limit $T(4,\infty)$, our computations agree with a conjecture of Gorsky, Oblomkov and Rasmussen. For finite $n$, we use the $λ$-invariant of Lewark, Marino and Zibrowius to derive lower bounds on proper rational Gordian distances from $T(4,n)$.

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arXiv2024

Discrete Morse Theory for Khovanov Homology

The standard methods for calculating Khovanov homology rely either on long exact/spectral sequences or on the algorithmic "divide and conquer" approach developed by Bar-Natan. In this paper, we employ an alternative and arguably simpler tool, discrete Morse theory, which is new in the context of knot homologies. The method is applied for 2- and 3-torus braids in Bar-Natan's dotted cobordism category, where Khovanov complexes of tangles live. This grants a recursive description of the complexes of 2- and 3-torus braids yielding an inductive result on integral Khovanov homology of links containing those braids. The result, accompanied with some computer data, advances the recent progress on a conjecture by Przytycki and Sazdanović which claims that closures of 3-braids only have 2-torsion in their Khovanov homology.

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