Knot polynomials via one parameter knot theory

Thomas Fiedler

Published 2006-12-05, updated 2007-02-06Version 2

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.