Low complexity algorithms in knot theory

TL;DR

This paper demonstrates that the genus and equivalence problems for alternating knots can be solved efficiently using low complexity algorithms, with specific results placing these problems in classes like Logspace and TC^0.

Contribution

It establishes linear time and Logspace algorithms for the genus problem and TC^0 algorithms for a subclass of alternating knots, advancing computational knot theory.

Findings

Genus problem for alternating knots is in Logspace(n).

Genus problem for standard alternating knots is in TC^0.

Equivalence problem for these knots has n log(n) time complexity and is in Logspace(n) and TC^0.

Abstract

We show that the genus problem for alternating knots with $n$ crossings has linear time complexity and is in Logspace$(n)$. Almost all alternating knots of given genus possess additional combinatorial structure, we call them standard. We show that the genus problem for these knots belongs to $TC^0$ circuit complexity class. We also show, that the equivalence problem for such knots with $n$ crossings has time complexity $n\log (n)$ and is in Logspace$(n)$ and $TC^{0}$ complexity classes.