The complexity of determining knot genus in a fixed three-manifold
Mehdi Yazdi (Oxford)
Abstract:
The genus of a knot in a three-manifold is defined to be the minimum genus of a compact, orientable surface bounding that knot, if such a surface exists. In particular a knot can be untangled if and only if it has genus zero. We consider the computational complexity of determining knot genus. Such problems have been studied by several mathematicians; among them are the works of Hass-Lagarias-Pippenger, Agol-Hass-Thurston, Agol and Lackenby. For a fixed three-manifold the knot genus problem asks, given a knot \(K\) and an integer \(g\), whether the genus of \(K\) is equal to \(g\). Marc Lackenby proved that the knot genus problem for the three-sphere lies in NP. In joint work with Lackenby, we prove that this can be generalised to any fixed, compact, orientable three-manifold, answering a question of Agol-Hass-Thurston from 2002.
algebraic topologydifferential geometrydynamical systemsgroup theorygeometric topologysymplectic geometry
Audience: researchers in the topic
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