Knot polynomials via one parameter knot theory

TL;DR

This paper introduces new knot polynomials derived from a one-parameter knot theory framework, using homology classes and algebraic sums over knot diagrams with triple points and autotangencies, satisfying specific algebraic equations.

Contribution

It constructs novel knot invariants via homomorphisms from the first homology group of a knot space, utilizing algebraic sums over singularities in knot diagrams.

Findings

Five distinct non-trivial solutions to the defining equations.

New knot polynomials associated with homology classes in knot spaces.

Framework extends classical invariants using algebraic sums over singularities.

Abstract

We construct new knot polynomials. Let $V$ be the standard solid torus in 3-space and let $pr$ be its standard projection onto an annulus. Let $M$ be the space of all smooth oriented knots in $V$ such that the restriction of $pr$ is an immersion (e.g. regular diagrams of a classical knot in the complement of its meridian). There is a canonical one dimensional homology class for each connected component of $M$. We construct homomorphisms from the first homology group of $M$ into rings of Laurent polynomials. Each such homomorphism applied to the canonical homology class gives a knot invariant. Let $\gamma$ be a generic smooth oriented loop in $M$ (i.e. a one parameter family of knot diagrams in the annulus). For finitely many points in $\gamma$ the corresponding knot diagram has in the projection $pr$ an ordinary triple point or an ordinary auto-tangency. To each such diagram we associate some Laurent polynomial by using extensions of the Kauffman bracket or of the Kauffman state model for the Alexander polynomial. We take then an algebraic sum of these polynomials over all triple points and all autotangencies in $\gamma$. The resulting polynomial depends only on the homology class of $\gamma$ if and only if it verifies two sorts of equations: the tetrahedron equations and the cube equations. We have found five different non trivial solutions.