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10
The computational Complexity of Knot and Link Problems
 J. ACM
, 1999
"... We consider the problem of deciding whether a polygonal knot in 3dimensional Euclidean space is unknotted, capable of being continuously deformed without selfintersection so that it lies in a plane. We show that this problem, unknotting problem is in NP. We also consider the problem, unknotting pr ..."
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Cited by 55 (6 self)
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We consider the problem of deciding whether a polygonal knot in 3dimensional Euclidean space is unknotted, capable of being continuously deformed without selfintersection so that it lies in a plane. We show that this problem, unknotting problem is in NP. We also consider the problem, unknotting problem of determining whether two or more such polygons can be split, or continuously deformed without selfintersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in PSPACE. We also give exponential worstcase running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.
Bounds On The Complex Zeros Of (Di)Chromatic Polynomials And PottsModel Partition Functions
 Chromatic Roots Are Dense In The Whole Complex Plane, Combinatorics, Probability and Computing
"... I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros ..."
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Cited by 45 (11 self)
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I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros of the Pottsmodel partition function ZG(q, {ve}) in the complex antiferromagnetic regime 1 + ve  ≤ 1. The proof is based on a transformation of the Whitney–Tutte–Fortuin–Kasteleyn representation of ZG(q, {ve}) to a polymer gas, followed by verification of the Dobrushin–Koteck´y–Preiss condition for nonvanishing of a polymermodel partition function. I also show that, for all loopless graphs G of secondlargest degree ≤ r, the zeros of PG(q) lie in the disc q  < C(r) + 1. KEY WORDS: Graph, maximum degree, secondlargest degree, chromatic polynomial,
Spanning trees on graphs and lattices in d dimensions
 J. PHYS. A
, 2000
"... The problem of enumerating spanning trees on graphs and lattices is considered. We obtain bounds on the number of spanning trees NST and establish inequalities relating the numbers of spanning trees of different graphs or lattices. A general formulation is presented for the enumeration of spanning t ..."
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Cited by 14 (3 self)
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The problem of enumerating spanning trees on graphs and lattices is considered. We obtain bounds on the number of spanning trees NST and establish inequalities relating the numbers of spanning trees of different graphs or lattices. A general formulation is presented for the enumeration of spanning trees on lattices in d ≥ 2 dimensions, and is applied to the hypercubic, bodycentered cubic, facecentered cubic, and specific planar lattices including the kagomé, diced, 488 (bathroomtile), Union Jack, and 31212 lattices. This leads to closedform expressions for NST for these lattices of finite sizes. We prove a theorem concerning the classes of graphs and lattices L with the property that NST ∼ exp(nzL) as the number of vertices n → ∞, where zL is a finite nonzero constant. This includes the bulk limit of lattices in any spatial dimension, and also sections of lattices whose lengths in some dimensions go to infinity while others are finite. We evaluate zL exactly for the lattices we considered, and discuss the dependence of zL on d and the lattice coordination number. We also establish a relation connecting zL to the free energy of the critical Ising model for planar lattices L.
Protein Folding, Spin Glass and Computational Complexity
 In Proceedings of the 3rd DIMACS Workshop on DNA Based Computers, held at the University of Pennsylvania, June 23 – 25
, 1997
"... . A reduction from "Ground State of Spin Glass" in statistical mechanics to a minimumenergy model of protein folding is made, which shows that the latter is NPcomplete (high complexity) . The reduction approximates true folding of a protein. The method also enables to show that even if the backbone ..."
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Cited by 9 (0 self)
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. A reduction from "Ground State of Spin Glass" in statistical mechanics to a minimumenergy model of protein folding is made, which shows that the latter is NPcomplete (high complexity) . The reduction approximates true folding of a protein. The method also enables to show that even if the backbone of the protein is fixed, the folding of the sidechains is NPcomplete. In a separate second part, the possibility of synthesizing proteins to solve arbitrary instances of the spin glass problem is speculated upon. 1. Introduction The motivation for this work is the speculation of exploiting nature's capability of protein folding to solve computationally intractable problems. One way of investigating this idea is to encode known NPcomplete problems in terms of protein folding. The main content of this paper is to do this for the spin glass problem. We construct a protein that achieves the encoding, i.e., the folded protein provides a solution to spin glass. More precisely, albeit incident...
The computational complexity of knot genus and spanning area
 electronic), arXiv: math.GT/0205057. MR MR2219001
"... Abstract. We show that the problem of deciding whether a polygonal knot in a closed threedimensional manifold bounds a surface of genus at most g is NPcomplete. We also show that the problem of deciding whether a curve in a PL manifold bounds a surface of area less than a given constant C is NPha ..."
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Cited by 6 (0 self)
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Abstract. We show that the problem of deciding whether a polygonal knot in a closed threedimensional manifold bounds a surface of genus at most g is NPcomplete. We also show that the problem of deciding whether a curve in a PL manifold bounds a surface of area less than a given constant C is NPhard. 1.
THE NEIGENVALUE PROBLEM AND TWO APPLICATIONS
"... Abstract. We consider the classification problem for compact Lie groups G ⊂ U(n) which are generated by a single conjugacy class with a fixed number N of distinct eigenvalues. We give an explicit classification when N = 3, and apply this to extract information about Galois representations and braid ..."
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Cited by 1 (1 self)
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Abstract. We consider the classification problem for compact Lie groups G ⊂ U(n) which are generated by a single conjugacy class with a fixed number N of distinct eigenvalues. We give an explicit classification when N = 3, and apply this to extract information about Galois representations and braid group representations. 1.
The complexity of the Weight Problem for permutation and matrix groups
"... Given a metric d on a permutation group G, the corresponding weight problem is to decide whether there exists an element π ∈ G such that d(π,e) = k, for some given value k. Here we show that this problem is NPcomplete for many wellknown metrics. An analogous problem in matrix groups, eigenvaluef ..."
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Cited by 1 (1 self)
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Given a metric d on a permutation group G, the corresponding weight problem is to decide whether there exists an element π ∈ G such that d(π,e) = k, for some given value k. Here we show that this problem is NPcomplete for many wellknown metrics. An analogous problem in matrix groups, eigenvaluefree problem, and two related problems in permutation groups, the maximum and minimum weight problems, are also investigated in this paper.
Linkless and flat embeddings in 3space and the Unknot problem (Extended Abstract)
 SCG'10
, 2010
"... ..."
Classical Ising
, 2009
"... View the table of contents for this issue, or go to the journal homepage for more ..."
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