Markov trace on Funar algebra
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2012
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| _version_ | 1866913595471691776 |
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| author | Orevkov, Stepan |
| author_facet | Orevkov, Stepan |
| contents | Funar algebra $K_\infty=K_\infty(α,β;k)$ is the quotient of the group algebra over a ring $k$ of the braid group $B_\infty$ by two cubic relations: $σ_1^3-ασ_1^2+βσ_1-1=0$ and another one which involves $σ_1$ and $σ_2$. The universal Markov trace on $K_\infty$ is the quotient map $t$ of $K_\infty(α,β,k[u,v])$ to its quotient (as a $k[u,v]$-module) by trace relations $xy=yx$ and by Markov relations $σ_nx=ux$, $σ_n^{-1}x=vx$ for $x\in K_n$. It is easy to check that the quotient is of the form $k[u,v]/I$ for some ideal $I$ (i. e. that the trace $t$ is determined by $t(1)$). We give an algorithm to compute the ideal $I$ and we present the result of computations in some special cases. In the last section we discuss some properties of the resulting link invariant. This invariant for $β=0$, $k=GF(37)[α]$ detects the chirality of the knots $10_{48}$ and $10_{91}$ and it distinguish many other pairs of knots with equal HOMFLY polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1206_0765 |
| institution | arXiv |
| publishDate | 2012 |
| record_format | arxiv |
| spellingShingle | Markov trace on Funar algebra Orevkov, Stepan Geometric Topology Representation Theory 57M27 Funar algebra $K_\infty=K_\infty(α,β;k)$ is the quotient of the group algebra over a ring $k$ of the braid group $B_\infty$ by two cubic relations: $σ_1^3-ασ_1^2+βσ_1-1=0$ and another one which involves $σ_1$ and $σ_2$. The universal Markov trace on $K_\infty$ is the quotient map $t$ of $K_\infty(α,β,k[u,v])$ to its quotient (as a $k[u,v]$-module) by trace relations $xy=yx$ and by Markov relations $σ_nx=ux$, $σ_n^{-1}x=vx$ for $x\in K_n$. It is easy to check that the quotient is of the form $k[u,v]/I$ for some ideal $I$ (i. e. that the trace $t$ is determined by $t(1)$). We give an algorithm to compute the ideal $I$ and we present the result of computations in some special cases. In the last section we discuss some properties of the resulting link invariant. This invariant for $β=0$, $k=GF(37)[α]$ detects the chirality of the knots $10_{48}$ and $10_{91}$ and it distinguish many other pairs of knots with equal HOMFLY polynomials. |
| title | Markov trace on Funar algebra |
| topic | Geometric Topology Representation Theory 57M27 |
| url | https://arxiv.org/abs/1206.0765 |