The tangle-valued 1-cocycle for knots

TL;DR

This paper introduces the Alexander tree, a new computable knot invariant derived from a topological 1-cocycle, capable of distinguishing knot properties like invertibility with manageable computational complexity.

Contribution

It constructs a novel combinatorial 1-cocycle for the moduli space of tangles, leading to the Alexander tree, a potentially complete and locally computable knot invariant.

Findings

The Alexander tree can distinguish non-invertible knots.

The invariant is computable with polynomial complexity.

It provides new insights into knot invertibility and classification.

Abstract

This paper contains the strongest and at the same time most calculable knot invariant ever. Let $\Theta$ be the topological moduli space of all ordered oriented tangles in 3-space. We construct a non-trivial combinatorial 1-cocycle $\mathbb{L}$ for $\Theta$ that takes its values in $H_0(\Theta;\mathbb{Z})$. The 1-cocycle $\mathbb{L}$ has a very nice property, called the {\em scan-property}: if we slide a tangle $T$ over or under a given crossing $c$ of a fixed tangle $T'$, then the value of $\mathbb{L}$ on this arc $scan(T)$ in $\Theta$ is already an isotopy invariant of $T$. In particular, let $D$ be a framed long knot diagram. We take the product with a fixed long knot diagram $K$ and we consider the 2-cable, with a fixed crossing $c$ in $2K$. $\mathbb{L}(scan(2D))$ gives an element in $H_0(\Theta)$. To this element we associate the {\em set of Alexander vectors}, consisting of the corresponding integer multiples of the one-variable Alexander polynomials of (the standard closures) of all sub-tangles of each of the tangles. We can vary the knots $(K,c)$ and moreover we can iterate our construction by starting now again the $scan$ with the tangles in $\mathbb{L}(scan(2D))$ and so on. The result is the infinite {\em Alexander tree}, which is an isotopy invariant of the knot represented by $D$. {\em As examples we show with just one edge of the Alexander tree that the knot $8_{17}$ and the Conway knot are not invertible!} This makes the Alexander tree a very promising candidate for a complete and "locally" calculable knot invariant, because the tangles in $\mathbb{L}(scan(2D))$ can be drawn with linear complexity and their Alexander polynomials can be calculated with quartic complexity with respect to the number of crossings of $D$.