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Register a new userThe C-complex clasp number of links
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In the 1980s Daryl Cooper introduced the notion of a C-complex (or clasp-complex) bounded by a link and explained how to compute signatures and polynomial invariants using a C-complex. Since then this was extended by works of Cimasoni, Florens, Mellor, Melvin, Conway, Toffoli, Friedl, and others to compute other link invariants. Informally a C-complex is a union of surfaces which are allowed to intersect each other in clasps. The purpose of the current paper is to study the minimal number of clasps amongst all C-complexes bounded by a fixed link $L$. This measure of complexity is related to the number of crossing changes needed to reduce $L$ to a boundary link. We prove that if $L$ is a 2-component link with nonzero linking number, then the linking number determines the minimal number of clasps amongst all C-complexes. In the case of 3-component links, the triple linking number provides an additional lower bound on the number of clasps in a C-complex.
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Kronheimer and Mrowka asked whether the difference between the four-dimensional clasp number and the slice genus can be arbitrarily large. This question is answered affirmatively by studying a knot invariant derived from equivariant singular instanton theory, and which is closely related to the Chern--Simons functional. This also answers a conjecture of Livingston about slicing numbers. Also studied is the singular instanton Fr{o}yshov invariant of a knot. If defined with integer coefficients, this gives a lower bound for the unoriented slice genus, and is computed for quasi-alternating and torus knots. In contrast, for certain other coefficient rings, the invariant is identified with a multiple of the knot signature. This result is used to address a conjecture by Poudel and Saveliev about traceless $SU(2)$ representations of torus knots. Further, for a concordance between knots with non-zero signature, it is shown that there is a traceless representation of the concordance complement which restricts to non-trivial representations of the knot groups. Finally, some evidence towards an extension of the slice-ribbon conjecture to torus knots is provided.
We provide an algorithm to determine whether a link L admits a crossing change that turns it into a split link, under some fairly mild hypotheses on L. The algorithm also provides a complete list of all such crossing changes. It can therefore also determine whether the unlinking number of L is 1.
It is shown that every knot or link is the set of complex tangents of a 3-sphere smoothly embedded in the three-dimensional complex space. We show in fact that a one-dimensional submanifold of a closed orientable 3-manifold can be realised as the set of complex tangents of a smooth embedding of the 3-manifold into the three-dimensional complex space if and only if it represents the trivial integral homology class in the 3-manifold. The proof involves a new application of singularity theory of differentiable maps.
We give asymptotically sharp upper bounds for the Khovanov width and the dealternation number of positive braid links, in terms of their crossing number. The same braid-theoretic technique, combined with Ozsvath, Stipsicz, and Szabos Upsilon invariant, allows us to determine the exact cobordism distance between torus knots with braid index two and six.
It is shown that if the exterior of a link L in the three sphere admits a genus 2 Heegaard splitting, then L has Generalized Property R.
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