There are non homotopic framed homotopies of long knots
| dc.creator | Fiedler, Thomas | |
| dc.date | 2007-10-23 | |
| dc.date | 2009-03-10 | |
| dc.date.accessioned | 2026-07-07T12:50:06Z | |
| dc.date.available | 2026-07-07T12:50:06Z | |
| dc.description | 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 $cl(Σ^{(1)}_{iness})$ 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 $Σ^{(1)}_{iness}$ the {\em inessential walls} and we call $\mathcal {M}_{ess} = \mathcal {M} \setminus cl(Σ^{(1)}_{iness})$ 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. | |
| dc.description | 15 pages, 14 figures v3: exposition improved, proofs completed | |
| dc.identifier | https://arxiv.org/abs/0710.4253 | |
| dc.identifier | http://arxiv.org/abs/0710.4253 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/222587 | |
| dc.subject | Geometric Topology | |
| dc.subject | 57M25 | |
| dc.title | There are non homotopic framed homotopies of long knots | |
| dc.type | text |