Knot polynomials from 1-cocycles

TL;DR

This paper introduces a family of polynomial invariants derived from 1-cocycles on moduli spaces of long framed knots, capable of detecting non-invertibility and distinguishing knot types without relying on the knot group.

Contribution

The authors construct new polynomial 1-cocycle invariants for parallel n-cables of long framed knots, providing a combinatorial method to detect knot properties and distinguish knot types.

Findings

$R_3$ detects the non-invertibility of knot $8_{17}$.

Calculation of $R_n$ is at most quartic in the number of crossings.

The invariants can distinguish non-torus knots via the ratio $R_n(fh(K))/R_n(rot(K))$.

Abstract

Let $M_n$ be the topological moduli space of all parallel n-cables of long framed oriented knots in 3-space. We construct in a combinatorial way for each natural number $n>1$ a 1-cocycle $R_n$ which represents a non trivial class in $H^1(M_n; \mathbb{Z} [x_1,x_2,...,x_1^{-1},x_2^{-1},...])$, where the number of variables $x_m$ depends on $n$. To each generic point in $M_n$ we associate in a canonical way an arc {\em scan} in $M_n$, such that $R_n(scan)$ is already a polynomial knot invariant. We show that $R_3(scan)$ detects the non-invertibility of the knot $8_{17}$ in a very simple way and without using the knot group. There are two well-known canonical loops in $M_n$ for each parallel n-cable of a long framed knot $K$: Gramain's loop {\em rot} and the Fox-Hatcher loop {\em fh}. The calculation of $R_n$ is of at most quartic complexity for these loops with respect to the number of crossings of $K$ for each fixed $n$. It follows from results of Hatcher that $K$ is not a torus knot if the rational function $R_n(fh(K))/R_n(rot(K))$ is not constant for each $n>1$. $ \oplus_n R_n$ is a natural candidate in order to separate all classes in $H_1(M_1;\mathbb{Q}) \cong H_1(M_n;\mathbb{Q})$, and in particular to distinguish all knot types $\pi_0(M_1)$.