The computational complexity of knot genus and spanning area

Agol, Ian; Hass, Joel; Thurston, William P.

doi:10.1090/s0002-9947-05-03919-x

Cited by 79 publications

(172 citation statements)

References 24 publications

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“…In this section, we explain how the work of Agol, Hass, and Thurston [1] shows that the decision problem variant of OBCP is NP-complete, and then use this to prove that the OHCP-D is also NP-complete. Precisely, consider the following decision problem:…”

Section: Np-completeness Of the Obcp And The Ohcpmentioning

confidence: 99%

“…Agol, Hass, and Thurston [1] considered the following discrete version. Take K to be a subcomplex of the 1-skeleton of a triangulation of Y , where each simplex has a fixed geometric shape corresponding to a simplex in R 3 with rational edge lengths.…”

Section: Least Area Surfaces Bounded By a Knotmentioning

confidence: 99%

“…The areas of the 2-simplices of M can now be used to define the area of this surface, which we denote Area( f ). We now consider: The proof that Least Spanning Area is NP-hard is essentially the same as in [1], and that it is in NP will follow from the desingularization procedure discussed below.…”

Section: Least Area Surfaces Bounded By a Knotmentioning

confidence: 99%

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The least spanning area of a knot and the optimal bounding chain problem

Dunfield

Hirani

2011

Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry

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Two fundamental objects in knot theory are the minimal genus surface and the least area surface bounded by a knot in a 3-dimensional manifold. When the knot is embedded in a general 3-manifold, the problems of finding these surfaces were shown to be NP-complete and NPhard respectively. However, there is evidence that the special case when the ambient manifold is R 3 , or more generally when the second homology is trivial, should be considerably more tractable. Indeed, we show here that a natural discrete version of the least area surface can be found in polynomial time.The precise setting is that the knot is a 1-dimensional subcomplex of a triangulation of the ambient 3-manifold. The main tool we use is a linear programming formulation of the Optimal Bounding Chain Problem (OBCP), where one is required to find the smallest norm chain with a given boundary. While the decision variant of OBCP is NPcomplete in general, we give conditions under which it can be solved in polynomial time. We then show that the least area surface can be constructed from the optimal bounding chain using a standard desingularization argument from 3-dimensional topology.We also prove that the related Optimal Homologous Chain Problem is NP-complete for homology with integer coefficients, complementing the corresponding result of Chen and Freedman for mod 2 homology.

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Section: Np-completeness Of the Obcp And The Ohcpmentioning

confidence: 99%

Section: Least Area Surfaces Bounded By a Knotmentioning

confidence: 99%

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The least spanning area of a knot and the optimal bounding chain problem

Dunfield

Hirani

2011

Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry

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“…Both constants are optimal; the 10 on the left is approximated asymptotically by links in a certain iterative pattern, described in [Agol and Thurston 2004]. The constant V 8 on the right makes the inequality sharp for the Borromean rings.…”

Section: The Determinant and Twist Number Of Alternating Diagramsmentioning

confidence: 99%

“…It is only the optimal choice of constants, due to I. Agol and D. Thurston [2004], that goes into Theorem 1.1 for a better estimate. The same is true of Proposition 6.3, which uses the proof of Theorem 1.1 to improve the inequality in the special case of arborescent (Conwayalgebraic) links.…”

Section: Introductionmentioning

confidence: 99%