The Least Spanning Area of a Knot and the Optimal Bounding Chain Problem

📝 Abstract

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 NP-hard 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 NP-complete 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.

💡 Analysis

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 NP-hard 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 NP-complete 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.

📄 Content

The Least Spanning Area of a Knot and the Optimal Bounding Chain Problem Nathan M. Dunfield University of Illinois, Mathematics 1409 W. Green St. Urbana IL, 61801, USA Anil N. Hirani University of Illinois, Computer Science 201 N. Goodwin Ave. Urbana IL, 61801, USA Abstract 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 em- bedded in a general 3-manifold, the problems of finding these surfaces were shown to be NP-complete and NP- hard respectively. However, there is evidence that the spe- cial case when the ambient manifold is R3, or more gener- ally when the second homology is trivial, should be con- siderably more tractable. Indeed, we show here that a nat- ural 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 NP- complete 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 argu- ment 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. 1. Introduction A knot K is a simple closed loop in an ambient 3-dimen- sional manifold Y . Provided K is null-homologous, which is always the case if Y = R3, there is an embedded ori- entable surface S in Y whose boundary is K (equivalently S is a compact smooth orientable surface in Y without self-intersections and with boundary K ). A fundamental property of K is the minimal genus of such an S, which is denoted g(K ) (we take g(K ) = ∞if there are no such sur- faces). In the 1960s, Haken used normal surface theory to give an algorithm for computing g(K ), opening the door to a whole subfield of low-dimensional topology and lead- ing to the discovery of algorithms for determining a wide range of topological properties of 3-manifolds [18]. How- ever, algorithms based on normal surface theory are quite slow in practice [2–4], and there are very few results that have been verified via such normal surface computations [5, 10]. Moreover, in some cases the underlying problems have been shown to be fundamentally difficult. In their foundational work, Agol, Hass, and Thurston showed that the following decision problem is NP–complete [1]: 1.1 Knot Genus. Given an integer g0 and a knot K embed- ded in the 1-skeleton of a triangulation of a closed 3-mani- fold Y , is g(K ) ≤g0? While Knot Genus is NP–complete, when Y is orientable and the second Betti number b2(Y ) = rank ¡ H2(Y ;Z) ¢ is 0, for instance Y = S3, then this problem likely simplifies. While their project is not yet complete, Agol, Hass, and Thurston have developed a very promising approach to showing that when b2(Y ) = 0 there is a certificate for the complementary problem g(K ) ≥g0 which can be verified in polynomial time. This would mean that this special case of Knot Genus is also in co-NP, raising the possibility of a polynomial-time algorithm when b2(Y ) = 0. However, currently there are no known algorithms which exploit the fact that b2(Y ) = 0. Despite this, our long-term goal is 1.2 Conjecture. For orientable Y with b2(Y ) = 0 the Knot Genus problem is in P. Here, we study the related problem of finding the least area surface bounded by a knot. This problem has its ori- gin in classical differential geometry, as we now sketch starting with the case where the ambient manifold Y is R3. For a smooth knot K in R3 there is always a smooth embed- ded orientable surface S ⊂R3 with ∂S = K . By deep the- orems in Geometric Measure Theory, there always exists such a surface S0 of least area [19]. A least area surface S0 is necessarily minimal in that it has mean curvature 0 ev- erywhere, like the surface of a soap-bubble. It is typically impossible to find the least area surface analytically, and the first paper on numerical methods for approximating S0 appeared in 1927 [9]. An algorithm to deal with arbitrary K was first given by Sullivan [27] in 1990; see also [21–25] for alternate approaches and numerical experiments. Of course, one can consider this question for null- homologous knots in an arbitrary Riemannian 3-mani- 1 2018/10/29 arXiv:1012.3030v3 [cs.CG] 23 Mar 2011 fold Y , and one has the same existence theorems for least area surfaces when Y is closed. Agol, Hass, and Thurston considered a certain discrete version of this problem, and showed that the question of whether K bounds a surface of area ≤A0 is NP-hard [1]. Because they put no r

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