There are non homotopic framed homotopies of long knots

Abstract

Let $\mathcal{M}$ be the space of all, including singular, long knots in 3-space and for which a fixed projection into the plane is an immersion. Let $cl({\Sigma }_{iness}^{(1)})$ be the closure of the union of all singular knots in $\mathcal{M}$ with exactly one ordinary double point and such that the two resolutions represent the same (non singular) knot type. We call ${\Sigma }_{iness}^{(1)}$ the {\em inessential walls} and we call ${\mathcal{M}}_{ess}=\mathcal{M}\setminus cl({\Sigma }_{iness}^{(1)})$ the {\em essential diagram space}. We construct a non trivial class in $H^1({\mathcal{M}}_{ess};\mathbb{Z}[A,A^{-1}])$ by an extension of the Kauffman bracket. This implies in particular that there are loops in ${\mathcal{M}}_{ess}$ which consist of regular isotopies of knots together with crossing changings and which are not contractible in ${\mathcal{M}}_{ess}$ (leading to the title of the paper). We conjecture that our construction gives rise to a new knot polynomial for knots of unknotting number one.

15 pages, 14 figures v3: exposition improved, proofs completed

References in corpus (1)

• Knot polynomials via one parameter knot theory