Cobordism distance on the projective space of the knot concordance group

Charles Livingston

doi:10.4153/s0008414x23000408

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arXiv · 2107.08992

Cobordism distance on the projective space of the knot concordance group

Charles Livingston

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Abstract

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.

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Charles Livingston. 2021-07-19. Cobordism distance on the projective space of the knot concordance group. https://doi.org/10.4153/s0008414x23000408

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