One-cocycle invariants for closed braids

TL;DR

This paper introduces polynomial isotopy invariants for closed braids based on Gauss diagram 1-cocycles, which can detect non-invertibility of certain links and are computationally efficient.

Contribution

The authors develop new polynomial invariants for closed braids using Gauss diagram 1-cocycles, capable of detecting link non-invertibility with polynomial complexity.

Findings

Invariants can detect non-invertibility of certain links.

Invariants are polynomial-time computable.

Derivatives at x=1 are finite type invariants.

Abstract

We introduce new polynomial isotopy invariants for closed braids. They are constructed as polynomial valued {\em Gauss diagram 1-cocycles} evaluated on the full rotation of the closed braid $\hat \beta$ around the core of the corresponding solid torus. They can be calculated with polynomial complexity with respect to the braid length and their derivatives evaluated at $x=1$ are finite type invariants of closed braids. Let the solid torus V be standardly embedded in the 3-sphere and let L be the core of the complementary solid torus $S^3\setminus V$. We give examples which show that a natural refinement of our invariants can detect (even with linear complexity with respect to the braid length if the number of strands is fixed, and with quadratic complexity if it is not fixed) the non-invertibility of the 2-component link $\hat \beta \cup L\hookrightarrow S^3$, what quantum invariants fail to do.