Twisted Blanchfield pairings and twisted signatures II: Relation to Casson-Gordon invariants
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866918286319419392 |
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| author | Borodzik, Maciej Conway, Anthony Politarczyk, Wojciech |
| author_facet | Borodzik, Maciej Conway, Anthony Politarczyk, Wojciech |
| contents | This paper studies twisted signature invariants and twisted linking forms, with a view towards obstructions to knot concordance. Given a knot $K$ and a representation $ρ$ of the knot group, we define a twisted signature function $σ_{K,ρ} \colon S^1 \to \mathbb{Z}$. This invariant satisfies many of the same algebraic properties as the classical Levine-Tristram signature $σ_K$. When the representation is abelian, $σ_{K,ρ}$ recovers $σ_K$, while for appropriate metabelian representations, $σ_{K,ρ}$ is closely related to the Casson-Gordon invariants. Additionally, we prove satellite formulas for $σ_{K,ρ}$ and for twisted Blanchfield forms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1809_08791 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Twisted Blanchfield pairings and twisted signatures II: Relation to Casson-Gordon invariants Borodzik, Maciej Conway, Anthony Politarczyk, Wojciech Geometric Topology 57K10 This paper studies twisted signature invariants and twisted linking forms, with a view towards obstructions to knot concordance. Given a knot $K$ and a representation $ρ$ of the knot group, we define a twisted signature function $σ_{K,ρ} \colon S^1 \to \mathbb{Z}$. This invariant satisfies many of the same algebraic properties as the classical Levine-Tristram signature $σ_K$. When the representation is abelian, $σ_{K,ρ}$ recovers $σ_K$, while for appropriate metabelian representations, $σ_{K,ρ}$ is closely related to the Casson-Gordon invariants. Additionally, we prove satellite formulas for $σ_{K,ρ}$ and for twisted Blanchfield forms. |
| title | Twisted Blanchfield pairings and twisted signatures II: Relation to Casson-Gordon invariants |
| topic | Geometric Topology 57K10 |
| url | https://arxiv.org/abs/1809.08791 |