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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2409.12858 |
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| _version_ | 1866913508367532032 |
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| author | Howards, Hugh Kindred, Thomas Moore, W. Frank Tolbert, John |
| author_facet | Howards, Hugh Kindred, Thomas Moore, W. Frank Tolbert, John |
| contents | All checkerboard surfaces for a given knot in $S^3$ are related by isotopy and "kinking" and "unkinking" moves, which change the surfaces' Goeritz matrices like this: $G\leftrightarrow G\oplus [\pm1]=\left[\begin{smallmatrix} G&\mathbf{0}\\ \mathbf{0}^T&\pm1 \end{smallmatrix}\right]$. We call two symmetric integer matrices "kink-equivalent" if they are related by "kinking'' and "unkinking'' moves $G\leftrightarrow G\oplus [\pm1]$ and unimodular congruence. We prove constructively that every nonsingular symmetric integer matrix is kink-equivalent to a positive-definite matrix and to a negative-definite matrix, and we give bounds on the number of moves required. This has several implications, e.g. every knot in $S^3$ is "alternating up to fake unkinking moves" and every simply connected, closed, topological 4-manifold with nonsingular intersection pairing has a positive blow-up that is homeomorphic to a negative blow-up of a positive-definite, simply connected, closed, topological 4-manifold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_12858 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Kink-equivalence of matrices, spanning surfaces, 4-manifolds, and quadratic forms Howards, Hugh Kindred, Thomas Moore, W. Frank Tolbert, John Geometric Topology Number Theory 57K10, 57K40, 15A63, 11E20 All checkerboard surfaces for a given knot in $S^3$ are related by isotopy and "kinking" and "unkinking" moves, which change the surfaces' Goeritz matrices like this: $G\leftrightarrow G\oplus [\pm1]=\left[\begin{smallmatrix} G&\mathbf{0}\\ \mathbf{0}^T&\pm1 \end{smallmatrix}\right]$. We call two symmetric integer matrices "kink-equivalent" if they are related by "kinking'' and "unkinking'' moves $G\leftrightarrow G\oplus [\pm1]$ and unimodular congruence. We prove constructively that every nonsingular symmetric integer matrix is kink-equivalent to a positive-definite matrix and to a negative-definite matrix, and we give bounds on the number of moves required. This has several implications, e.g. every knot in $S^3$ is "alternating up to fake unkinking moves" and every simply connected, closed, topological 4-manifold with nonsingular intersection pairing has a positive blow-up that is homeomorphic to a negative blow-up of a positive-definite, simply connected, closed, topological 4-manifold. |
| title | Kink-equivalence of matrices, spanning surfaces, 4-manifolds, and quadratic forms |
| topic | Geometric Topology Number Theory 57K10, 57K40, 15A63, 11E20 |
| url | https://arxiv.org/abs/2409.12858 |