Cobordism distance on the projective space of the knot concordance group
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866912124555493376 |
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| author | Livingston, Charles |
| author_facet | Livingston, Charles |
| contents | The cobordism distance on the knot concordance group is used to define a measure of how close two knots are to being linearly dependent. Roughly stated, d(K,J) is defined by minimizing the cobordism distance between pairs of knots in cyclic subgroups containing K and J. When made precise, this leads to the definition of the projective space of the knot concordance group. We explore basic properties of this projective space and its integer-valued metric by considering torus knots. Twist knots are used to show that the associated simplicial complex contains an infinite set of simplices of arbitrarily large dimension. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2107_08992 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Cobordism distance on the projective space of the knot concordance group Livingston, Charles Geometric Topology 57K10 The cobordism distance on the knot concordance group is used to define a measure of how close two knots are to being linearly dependent. Roughly stated, d(K,J) is defined by minimizing the cobordism distance between pairs of knots in cyclic subgroups containing K and J. When made precise, this leads to the definition of the projective space of the knot concordance group. We explore basic properties of this projective space and its integer-valued metric by considering torus knots. Twist knots are used to show that the associated simplicial complex contains an infinite set of simplices of arbitrarily large dimension. |
| title | Cobordism distance on the projective space of the knot concordance group |
| topic | Geometric Topology 57K10 |
| url | https://arxiv.org/abs/2107.08992 |