More 1-cocycles for classical knots

Thomas Fiedler

Published 2020-04-09Version 1

Let $n$ be a natural number and let $M$ be the moduli space of all knots in the solid torus $V^3$ which represent the homology class $n \in \mathbb{Z}\cong H_1(V^3)$. Let $K$ be a framed oriented long knot and let $nK$ be its n-cable twisted by a fixed string link $T$ to a knot in the solid torus $V^3$. Let $M_n^{reg}$ be the topological moduli space of all such knots $nK$ up to regular isotopy with respect to a fixed projection into the annulus. We construct two new sorts of 1-cocycles: two integer 1-cocycles $R_{[a,b,c]}$ and $R_{(a,b,c)}$ for $M$, which use linear weights and which depend on {\em three} integer parameters $a,b,c \in \mathbb{Z}$, and two integer 1-cocycles $R_{a\pm}^{(2)}$ for $M_n^{reg}$, which use {\em quadratic} weights and which depend on a natural number $0<a<n$. The Lagrange interpolation polynomials of $R_{a\pm}^{(2)}(\gamma)$ are candidates for new polynomial knot invariants for classical knots $K$, which can be calculated with polynomial complexity for each homology class $[\gamma] \in H_1(M_n^{reg})$. The 1-cocycles $R_{[a,b,c]}$ and $R_{(a,b,c)}$ are candidates to distinguish the orientations of classical knots.