A resistance invariant of special alternating links
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914326261006336 |
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| author | Jablonowski, Michal |
| author_facet | Jablonowski, Michal |
| contents | We introduce a new numerical invariant for special, reduced, alternating diagrams of oriented knots and links, defined in terms of the Laplacian matrix of the associated Tait graph. For a special alternating diagram, the Laplacian encodes both the combinatorics of the checkerboard graph and the crossing signs. While its spectrum depends on the chosen diagram, we show that a specific quadratic trace expression involving the Laplacian and its Moore-Penrose pseudoinverse is invariant under flype moves. The invariant admits an interpretation in terms of total effective resistance of the associated weighted graph viewed as an electrical network. Explicit computations for pairs of flype-related diagrams demonstrate that, although the Laplacian characteristic polynomials differ, the invariant FP coincides. Values for several prime alternating knots are provided. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_12226 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A resistance invariant of special alternating links Jablonowski, Michal Geometric Topology 57K10 (primary), 05C50 (secondary) We introduce a new numerical invariant for special, reduced, alternating diagrams of oriented knots and links, defined in terms of the Laplacian matrix of the associated Tait graph. For a special alternating diagram, the Laplacian encodes both the combinatorics of the checkerboard graph and the crossing signs. While its spectrum depends on the chosen diagram, we show that a specific quadratic trace expression involving the Laplacian and its Moore-Penrose pseudoinverse is invariant under flype moves. The invariant admits an interpretation in terms of total effective resistance of the associated weighted graph viewed as an electrical network. Explicit computations for pairs of flype-related diagrams demonstrate that, although the Laplacian characteristic polynomials differ, the invariant FP coincides. Values for several prime alternating knots are provided. |
| title | A resistance invariant of special alternating links |
| topic | Geometric Topology 57K10 (primary), 05C50 (secondary) |
| url | https://arxiv.org/abs/2602.12226 |