Mock Seifert matrices and unoriented algebraic concordance
| Autor: | Boden, Hans U., Karimi, Homayun |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | A mock Seifert matrix is an integral square matrix representing the Gordon-Litherland form of a pair $(K,F)$, where $K$ is a knot in a thickened surface and $F$ is an unoriented spanning surface for $K$. Using these matrices, we introduce a new notion of unoriented algebraic concordance, as well as a new group denoted $\mathcal{m} \mathcal{G}^{\mathbb Z}$ and called the unoriented algebraic concordance group. This group is abelian and infinitely generated. There is a surjection $\lambda \colon \mathcal{v} \mathcal{C} \to \mathcal{m} \mathcal{G}^{\mathbb Z}$, where $\mathcal{v} \mathcal{C} $ denotes the virtual knot concordance group. Mock Seifert matrices can also be used to define new invariants, such as the mock Alexander polynomial and mock Levine-Tristram signatures. These invariants are applied to questions about virtual knot concordance, crosscap numbers, and Seifert genus for knots in thickened surfaces. For example, we show that $\mathcal{m} \mathcal{G}^{\mathbb Z}$ contains a copy of ${\mathbb Z}^\infty \oplus ({\mathbb Z}/2)^\infty \oplus({\mathbb Z}/4)^\infty.$ Comment: 34 pages, 8 figures. Revisions made to section 2 |
| Databáze: | arXiv |
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