# Knot Diagrams of Treewidth Two

H.L. Bodlaender, Benjamin Burton, Fedor Fomin, Alexander Grigoriev

### Abstract

In this paper, we study knot diagrams for which the underlying graph has treewidth two. We give a linear time algorithm for the following problem: given a knot diagram of treewidth two, does it represent the unknot? We also show that for a link diagram of treewidth two we can test in linear time if it represents the unlink. From the algorithm, it follows that a diagram of the unknot of treewidth 2 can always be reduced to the trivial diagram with at most n (un)twist and (un)poke Reidemeister moves.
Original language English Cornell University Library, US arXiv.org at Cornell University Library 19 1904.03117v2 Published - 8 Apr 2019

### Publication series

Series cs.DS

### Cite this

Bodlaender, H. L., Burton, B., Fomin, F., & Grigoriev, A. (2019). Knot Diagrams of Treewidth Two. Cornell University Library, US: arXiv.org at Cornell University Library. cs.DS
Bodlaender, H.L. ; Burton, Benjamin ; Fomin, Fedor ; Grigoriev, Alexander. / Knot Diagrams of Treewidth Two. Cornell University Library, US : arXiv.org at Cornell University Library, 2019. 19 p. (cs.DS).
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Bodlaender, HL, Burton, B, Fomin, F & Grigoriev, A 2019, Knot Diagrams of Treewidth Two. cs.DS, vol. 1904.03117v2, arXiv.org at Cornell University Library, Cornell University Library, US.

Knot Diagrams of Treewidth Two. / Bodlaender, H.L.; Burton, Benjamin; Fomin, Fedor; Grigoriev, Alexander.

Cornell University Library, US : arXiv.org at Cornell University Library, 2019. 19 p. (cs.DS).

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T1 - Knot Diagrams of Treewidth Two

AU - Bodlaender, H.L.

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AU - Fomin, Fedor

AU - Grigoriev, Alexander

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PY - 2019/4/8

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N2 - In this paper, we study knot diagrams for which the underlying graph has treewidth two. We give a linear time algorithm for the following problem: given a knot diagram of treewidth two, does it represent the unknot? We also show that for a link diagram of treewidth two we can test in linear time if it represents the unlink. From the algorithm, it follows that a diagram of the unknot of treewidth 2 can always be reduced to the trivial diagram with at most n (un)twist and (un)poke Reidemeister moves.

AB - In this paper, we study knot diagrams for which the underlying graph has treewidth two. We give a linear time algorithm for the following problem: given a knot diagram of treewidth two, does it represent the unknot? We also show that for a link diagram of treewidth two we can test in linear time if it represents the unlink. From the algorithm, it follows that a diagram of the unknot of treewidth 2 can always be reduced to the trivial diagram with at most n (un)twist and (un)poke Reidemeister moves.

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Bodlaender HL, Burton B, Fomin F, Grigoriev A. Knot Diagrams of Treewidth Two. Cornell University Library, US: arXiv.org at Cornell University Library, 2019. 19 p. (cs.DS).