An analogue of Turaev comultiplication for knots in non-orientable thickening of a non-orientable surface
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929442567225344 |
|---|---|
| author | Tarkaev, Vladimir |
| author_facet | Tarkaev, Vladimir |
| contents | This paper concerns pseudo-classical knots in the non-orientable manifold $\hatΣ =Σ\times [0,1]$, where $Σ$ is a non-orientable surface and a knot $K \subset \hatΣ$ is called pseudo-classical if $K$ is orientation-preserving path in $\hatΣ$. For this kind of knot we introduce an invariant $Δ$ that is an analogue of Turaev comultiplication for knots in a thickened orientable surface. As its classical prototype, $Δ$ takes value in a polynomial algebra generated by homotopy classes of non-contractible loops on $Σ$, however, as a ground ring we use some subring of $\mathbb{C}$ instead of $\mathbb{Z}$. Then we define a few homotopy, homology and polynomial invariants, which are consequences of $Δ$, including an analogue of the affine index polynomial. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20715 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An analogue of Turaev comultiplication for knots in non-orientable thickening of a non-orientable surface Tarkaev, Vladimir Geometric Topology 57M25, 57M27 This paper concerns pseudo-classical knots in the non-orientable manifold $\hatΣ =Σ\times [0,1]$, where $Σ$ is a non-orientable surface and a knot $K \subset \hatΣ$ is called pseudo-classical if $K$ is orientation-preserving path in $\hatΣ$. For this kind of knot we introduce an invariant $Δ$ that is an analogue of Turaev comultiplication for knots in a thickened orientable surface. As its classical prototype, $Δ$ takes value in a polynomial algebra generated by homotopy classes of non-contractible loops on $Σ$, however, as a ground ring we use some subring of $\mathbb{C}$ instead of $\mathbb{Z}$. Then we define a few homotopy, homology and polynomial invariants, which are consequences of $Δ$, including an analogue of the affine index polynomial. |
| title | An analogue of Turaev comultiplication for knots in non-orientable thickening of a non-orientable surface |
| topic | Geometric Topology 57M25, 57M27 |
| url | https://arxiv.org/abs/2407.20715 |