$U_q(\mathfrak{gl}(m|n))$ bounds on the minimal genus of virtual links
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914221853245440 |
|---|---|
| author | Chrisman, Micah Davis, Killian Poudel, Anup |
| author_facet | Chrisman, Micah Davis, Killian Poudel, Anup |
| contents | For links $L \subset Σ\times [0,1]$, where $Σ$ is a closed orientable surface, we define a $U_q(\mathfrak{gl}(1|1))$ Reshetikhin-Turaev invariant with coefficients in $\mathbb{Z}[H_1(Σ)]$. This invariant turns out to be equivalent to an infinite cyclic version of the Carter-Silver-Williams (CSW) polynomial. The importance of the CSW polynomial is that half its symplectic rank gives strong lower bounds on the virtual genus. Recall that the virtual genus of a virtual link $J$ is the smallest genus of all closed orientable surfaces $Σ$ on which $J$ can be represented by a link diagram on $Σ$. Here we generalize the CSW lower bound to all quantum supergroups $U_q(\mathfrak{gl}(m|n))$ with $m,n>0$. For $(m,n)=(1,1)$, the $U_q(\mathfrak{gl}(m|n))$ bound is the same as the CSW bound. However, changing the value of the pair $(m,n)$ can give lower bounds better than those available from other known methods. We compare the $U_q(\mathfrak{gl}(m|n))$ lower bounds to those coming from the CSW polynomial, the surface bracket, the arrow polynomial, hyperbolicity, and the Gordon-Litherland determinant test. As a first application, we show that the Seifert genus of homologically trivial knots in thickened surfaces is not additive under the connected sum operation of virtual knots. As a second application, we prove that the Jaeger-Kauffman-Saleur invariant of a virtual knot is always realizable as the Alexander polynomial of an infinite cyclic cover of a knot complement in some $Σ\times [0,1]$, but is not always so on a surface of minimal genus. This is accomplished with a generalization of the $Zh$-construction, called the homotopy $Zh$-construction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22401 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $U_q(\mathfrak{gl}(m|n))$ bounds on the minimal genus of virtual links Chrisman, Micah Davis, Killian Poudel, Anup Geometric Topology Primary: 57K12, 57K14 Secondary: 17B10 For links $L \subset Σ\times [0,1]$, where $Σ$ is a closed orientable surface, we define a $U_q(\mathfrak{gl}(1|1))$ Reshetikhin-Turaev invariant with coefficients in $\mathbb{Z}[H_1(Σ)]$. This invariant turns out to be equivalent to an infinite cyclic version of the Carter-Silver-Williams (CSW) polynomial. The importance of the CSW polynomial is that half its symplectic rank gives strong lower bounds on the virtual genus. Recall that the virtual genus of a virtual link $J$ is the smallest genus of all closed orientable surfaces $Σ$ on which $J$ can be represented by a link diagram on $Σ$. Here we generalize the CSW lower bound to all quantum supergroups $U_q(\mathfrak{gl}(m|n))$ with $m,n>0$. For $(m,n)=(1,1)$, the $U_q(\mathfrak{gl}(m|n))$ bound is the same as the CSW bound. However, changing the value of the pair $(m,n)$ can give lower bounds better than those available from other known methods. We compare the $U_q(\mathfrak{gl}(m|n))$ lower bounds to those coming from the CSW polynomial, the surface bracket, the arrow polynomial, hyperbolicity, and the Gordon-Litherland determinant test. As a first application, we show that the Seifert genus of homologically trivial knots in thickened surfaces is not additive under the connected sum operation of virtual knots. As a second application, we prove that the Jaeger-Kauffman-Saleur invariant of a virtual knot is always realizable as the Alexander polynomial of an infinite cyclic cover of a knot complement in some $Σ\times [0,1]$, but is not always so on a surface of minimal genus. This is accomplished with a generalization of the $Zh$-construction, called the homotopy $Zh$-construction. |
| title | $U_q(\mathfrak{gl}(m|n))$ bounds on the minimal genus of virtual links |
| topic | Geometric Topology Primary: 57K12, 57K14 Secondary: 17B10 |
| url | https://arxiv.org/abs/2512.22401 |