On values of $\mathfrak{sl}_3$ weight system on chord diagrams whose intersection graph is complete bipartite
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866929730212593664 |
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| author | Yang, Zhuoke |
| author_facet | Yang, Zhuoke |
| contents | Each knot invariant can be extended to singular knots according to the skein rule. A Vassiliev invariant of order at most $n$ is defined as a knot invariant that vanishes identically on knots with more than $n$ double points. A chord diagram encodes the order of double points along a singular knot. A Vassiliev invariant of order $n$ gives rise to a function on chord diagrams with $n$ chords. Such a function should satisfy some conditions in order to come from a Vassiliev invariant. A weight system is a function on chord diagrams that satisfies so-called 4-term relations. Given a Lie algebra $\mathfrak{g}$ equipped with a non-degenerate invariant bilinear form, one can construct a weight system with values in the center of the universal enveloping algebra $U(\mathfrak{g})$. In this paper, we calculate $\mathfrak{sl}_3$ weight system for chord diagram whose intersection graph is complete bipartite graph $K_{2,n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_00888 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On values of $\mathfrak{sl}_3$ weight system on chord diagrams whose intersection graph is complete bipartite Yang, Zhuoke Combinatorics Geometric Topology Each knot invariant can be extended to singular knots according to the skein rule. A Vassiliev invariant of order at most $n$ is defined as a knot invariant that vanishes identically on knots with more than $n$ double points. A chord diagram encodes the order of double points along a singular knot. A Vassiliev invariant of order $n$ gives rise to a function on chord diagrams with $n$ chords. Such a function should satisfy some conditions in order to come from a Vassiliev invariant. A weight system is a function on chord diagrams that satisfies so-called 4-term relations. Given a Lie algebra $\mathfrak{g}$ equipped with a non-degenerate invariant bilinear form, one can construct a weight system with values in the center of the universal enveloping algebra $U(\mathfrak{g})$. In this paper, we calculate $\mathfrak{sl}_3$ weight system for chord diagram whose intersection graph is complete bipartite graph $K_{2,n}$. |
| title | On values of $\mathfrak{sl}_3$ weight system on chord diagrams whose intersection graph is complete bipartite |
| topic | Combinatorics Geometric Topology |
| url | https://arxiv.org/abs/2102.00888 |