High crossing knot complements with few tetrahedra

Abstract:

It is well known that given a diagram of a knot \(K\) with \(n\) crossings, one can construct a triangulation of \(S^3 - K\) with at most \(4n\) tetrahedra. A natural question is then: given a triangulation of a knot complement with \(t\) tetrahedra, is the minimum crossing number (for a diagram) of K bounded by a linear or polynomial function in \(t\)? We will answer the question in the negative by constructing a family of hyperbolic knot complements where for each knot \(K_n\) in \(S^3\) whose the minimum crossing number goes as a function of \(O(b^n)\) for \(b > 1\), but the minimum number of tetrahedra in a triangulation of \(S^3 - K_n\) is bounded above by \(O(n)\). Similar constructions exist for torus and satellite knot complements.

This is joint work with Robert Haraway.

algebraic topologydifferential geometrydynamical systemsgroup theorygeometric topologysymplectic geometry

Audience: researchers in the topic

( slides )

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Geometry and topology online

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