Blanchfield pairings and twisted Blanchfield pairings of torus knots
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912893307453440 |
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| author | Yanagida, Koki |
| author_facet | Yanagida, Koki |
| contents | We give explicit matrix presentations of the Blanchfield pairing and certain twisted Blanchfield pairings of the $(m,n)$-torus knot $T(m,n)$. Our method uses a taut identity realizing a genus-two Heegaard splitting of the manifold $X_{T(m,n)}$ obtained from $S^3$ by $0$-surgery along $T(m,n)$. The taut identity allows us to construct a chain complex of $X_{T(m,n)}$ with few generators. As a result, we obtain explicit matrix presentations of the Blanchfield pairing of $T(m,n)$. Moreover, for each Casson-Gordon type metabelian representation and for suitable roots of unity $ξ$ depending on the representation, we describe the $(t-ξ)$-primary part of the associated twisted Alexander module and give an explicit description of the restriction of the twisted Blanchfield pairing to this primary summand. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_07575 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Blanchfield pairings and twisted Blanchfield pairings of torus knots Yanagida, Koki Geometric Topology 57K10 We give explicit matrix presentations of the Blanchfield pairing and certain twisted Blanchfield pairings of the $(m,n)$-torus knot $T(m,n)$. Our method uses a taut identity realizing a genus-two Heegaard splitting of the manifold $X_{T(m,n)}$ obtained from $S^3$ by $0$-surgery along $T(m,n)$. The taut identity allows us to construct a chain complex of $X_{T(m,n)}$ with few generators. As a result, we obtain explicit matrix presentations of the Blanchfield pairing of $T(m,n)$. Moreover, for each Casson-Gordon type metabelian representation and for suitable roots of unity $ξ$ depending on the representation, we describe the $(t-ξ)$-primary part of the associated twisted Alexander module and give an explicit description of the restriction of the twisted Blanchfield pairing to this primary summand. |
| title | Blanchfield pairings and twisted Blanchfield pairings of torus knots |
| topic | Geometric Topology 57K10 |
| url | https://arxiv.org/abs/2602.07575 |