A Non-Abelian Generalization of the Alexander Polynomial from Quantum $\mathfrak{sl}_3$
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2020
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866917350454853632 |
|---|---|
| author | Harper, Matthew |
| author_facet | Harper, Matthew |
| contents | One construction of the Alexander polynomial is as a quantum invariant associated with representations of restricted quantum $\mathfrak{sl}_2$ at a fourth root of unity. We generalize this construction to define a link invariant $Δ_{\mathfrak{g}}$ for any semisimple Lie algebra $\mathfrak{g}$ of rank $n$, taking values in $n$-variable Laurent polynomials. Focusing on the case $\mathfrak{g}=\mathfrak{sl}_3$, we establish a direct relation between $Δ_{\mathfrak{sl}_3}$ and the Alexander polynomial. We show that certain parameter evaluations of $Δ_{\mathfrak{sl}_3}$ recover the Alexander polynomial on knots, despite the $R$-matrix not satisfying the Alexander-Conway skein relation at these points. We tabulate $Δ_{\mathfrak{sl}_3}$ for all knots up to seven crossings and various other examples, including the Kinoshita-Terasaka knot and Conway knot mutant pair which are distinguished by this invariant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2008_06983 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A Non-Abelian Generalization of the Alexander Polynomial from Quantum $\mathfrak{sl}_3$ Harper, Matthew Quantum Algebra Geometric Topology Representation Theory 57K16, 17B37 One construction of the Alexander polynomial is as a quantum invariant associated with representations of restricted quantum $\mathfrak{sl}_2$ at a fourth root of unity. We generalize this construction to define a link invariant $Δ_{\mathfrak{g}}$ for any semisimple Lie algebra $\mathfrak{g}$ of rank $n$, taking values in $n$-variable Laurent polynomials. Focusing on the case $\mathfrak{g}=\mathfrak{sl}_3$, we establish a direct relation between $Δ_{\mathfrak{sl}_3}$ and the Alexander polynomial. We show that certain parameter evaluations of $Δ_{\mathfrak{sl}_3}$ recover the Alexander polynomial on knots, despite the $R$-matrix not satisfying the Alexander-Conway skein relation at these points. We tabulate $Δ_{\mathfrak{sl}_3}$ for all knots up to seven crossings and various other examples, including the Kinoshita-Terasaka knot and Conway knot mutant pair which are distinguished by this invariant. |
| title | A Non-Abelian Generalization of the Alexander Polynomial from Quantum $\mathfrak{sl}_3$ |
| topic | Quantum Algebra Geometric Topology Representation Theory 57K16, 17B37 |
| url | https://arxiv.org/abs/2008.06983 |