On the n-loop Kontsevich invariant of knots having the same Alexander polynomial
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912089367379968 |
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| author | Yamaguchi, Kouki |
| author_facet | Yamaguchi, Kouki |
| contents | The $n$-loop Kontsevich invariant of knots takes its value in the completion of the space of $n$-loop open Jacobi diagrams, which is an infinite dimensional vector space. Since the 1-loop part is presented by the Alexander polynomial, we are interested in the image of the $\geq 2$-loop Kontsevich invariant of knots having the same Alexander polynomial. In this paper, we show that for $n\geq 2$ the subspace generated by the image of the $n$-loop Kontsevich invariant of genus $\leq g$ knots having the same Alexander polynomial is finite dimensional. Further, we give some concrete calculations about those subspaces and dimensions in several simple cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_20458 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the n-loop Kontsevich invariant of knots having the same Alexander polynomial Yamaguchi, Kouki Geometric Topology 57K10, 57K16 The $n$-loop Kontsevich invariant of knots takes its value in the completion of the space of $n$-loop open Jacobi diagrams, which is an infinite dimensional vector space. Since the 1-loop part is presented by the Alexander polynomial, we are interested in the image of the $\geq 2$-loop Kontsevich invariant of knots having the same Alexander polynomial. In this paper, we show that for $n\geq 2$ the subspace generated by the image of the $n$-loop Kontsevich invariant of genus $\leq g$ knots having the same Alexander polynomial is finite dimensional. Further, we give some concrete calculations about those subspaces and dimensions in several simple cases. |
| title | On the n-loop Kontsevich invariant of knots having the same Alexander polynomial |
| topic | Geometric Topology 57K10, 57K16 |
| url | https://arxiv.org/abs/2410.20458 |