There are non homotopic framed homotopies of long knots
Thomas Fiedler

TL;DR
This paper constructs a nontrivial cohomology class in the space of long knots, showing the existence of non-contractible loops of knots that involve crossing changes, and conjectures a new polynomial invariant for certain knots.
Contribution
It introduces a new cohomology class in the essential diagram space of long knots, revealing nontrivial loops involving crossing changes and proposing a novel knot polynomial for unknotting number one knots.
Findings
Existence of non-contractible loops in the essential diagram space
Construction of a new cohomology class extending the Kauffman bracket
Conjecture of a new knot polynomial for knots with unknotting number one
Abstract
Let 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 be the closure of the union of all singular knots in with exactly one ordinary double point and such that the two resolutions represent the same (non singular) knot type. We call the {\em inessential walls} and we call the {\em essential diagram space}. We construct a non trivial class in by an extension of the Kauffman bracket. This implies in particular that there are loops in which consist of regular isotopies of knots together with crossing changings and which are not contractible in (leading to the title of the…
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Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Advanced Combinatorial Mathematics
