The Hilbert Polynomial of Quandles and Colorings of Random Links

Ariel Davis; Tomer M. Schlank

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arXiv · 2304.08314

The Hilbert Polynomial of Quandles and Colorings of Random Links

Ariel Davis·Tomer M. Schlank

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Abstract

Given a finite quandle $Q$, we study the average number of $Q$-colorings of the closure of a random braid in $B_n$ as $n$ varies. In particular we show that this number coincides with some polynomial $P_Q\in \mathbb{Q}[x]$ for $n\gg 0$. The degree of this polynomial is readily computed in terms of $Q$ as a quandle and these invariants are computed for all quandles with $|Q|\le 4$. Additionally we show that the methods in this paper allow to improve on the stability results of arXiv:0912.0325 from "periodic stability" to "stability".

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BibTeX RIS

Ariel Davis, Tomer M. Schlank. 2023-04-17. The Hilbert Polynomial of Quandles and Colorings of Random Links. https://arxiv.org/abs/2304.08314

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