Isotopy invariants for closed braids and almost closed braids via loops in stratified spaces

TL;DR

This paper introduces new isotopy invariants for closed and almost closed braids in the solid torus, which refine finite type invariants, are computationally efficient, and can detect non-invertibility of certain links, surpassing quantum invariants.

Contribution

The authors develop novel isotopy invariants for braids in the solid torus that refine finite type invariants and can detect non-invertibility of 2-component links, a capability not shared by quantum invariants.

Findings

Invariants can detect non-invertibility of certain links.

Invariants are computable with polynomial complexity.

Quantum invariants fail to detect non-invertibility.

Abstract

Let $\phi : S^1\times D^2\to S^1$ be the natural projection. An oriented knot $K\hookrightarrow V = S^1\times D^2$ is called an almost closed braid if the restriction of $\phi$ to K has exactly two (non-degenerate) critical points (and K is a closed braid if the restriction of $\phi$ has no critical points at all). We introduce new isotopy invariants for closed braids and almost closed braids in the solid torus V. These invariants refine finite type invariants. They are still calculable with polynomial complexity with respect to the number of crossings of K. Let the solid torus V be standardly embedded in the 3-sphere and let A be the axis of the complementary solid torus $S^3\setminus V$. We give examples which show that our invariants can detect non-invertibility of 2-component links $K\cup A\hookrightarrow S^3$. Notice that all quantum link invariants fail to do so and that it is not known wether there are finite type invariants which can detect non-invertibility of 2-component links.