Stepan Orevkov: An attempt to extend Levine’s theory from knots to links

Date: 2023-06-30

Time: 11:00 - 12:00

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Stepan Orevkov, University Toulouse-3


The maximal dimension of a totally isotropic subspace of a quadratic form is usually called the Witt index. We define the Witt coindex as n-2w where n is the rank and w is the Witt index. By a result of Laurence Taylor, the slice genus of a knot is bounded by the Witt coindex of its Seifert form.

Levine’s theory of algebraic concordance (combined with some results of Livingston) gives an algorithm to decide whether the coindex of a given Seifert form is zero or not (i.e. whether a given knot is algebraically slice).

For the slice Euler characteristic of a link we give an upper bound via the coindex. Under condition that $Delta_L(-1)ne 0$, we give an algorithm to decide whether the coindex is equal to 1 (which corresponds to links bounding an annulus in the 4-ball). More precisely, we show that any coindex-1 Seifert form is concordant to a certain Seifert form which uniquely determined by the Alexander polynomial.

On a joint work with Vincent Florens.