The unbearable hardness of unknotting

Arnaud de Mesmay; Yo'av Rieck; Eric Sedgwick; Martin Tancer

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arXiv · 1810.03502

The unbearable hardness of unknotting

Arnaud de Mesmay·Yo'av Rieck·Eric Sedgwick·Martin Tancer

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Abstract

We prove that deciding if a diagram of the unknot can be untangled using at most $k$ Riedemeister moves (where $k$ is part of the input) is NP-hard. We also prove that several natural questions regarding links in the $3$-sphere are NP-hard, including detecting whether a link contains a trivial sublink with $n$ components, computing the unlinking number of a link, and computing a variety of link invariants related to four-dimensional topology (such as the $4$-ball Euler characteristic, the slicing number, and the $4$-dimensional clasp number).

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Arnaud de Mesmay, Yo'av Rieck, Eric Sedgwick, Martin Tancer. 2018-10-08. The unbearable hardness of unknotting. https://arxiv.org/abs/1810.03502

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