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11
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 58 (8 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.
Computing the Tutte polynomial on graphs of bounded cliquewidth
 SIAM Journal on Discrete Mathematics
, 2006
"... Abstract. The Tutte polynomial is a notoriously hard graph invariant, and efficient algorithms for it are known only for a few special graph classes, like for those of bounded treewidth. The notion of cliquewidth extends the definition of cograhs (graphs without induced P4), and it is a more gener ..."
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Cited by 12 (0 self)
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Abstract. The Tutte polynomial is a notoriously hard graph invariant, and efficient algorithms for it are known only for a few special graph classes, like for those of bounded treewidth. The notion of cliquewidth extends the definition of cograhs (graphs without induced P4), and it is a more general notion than that of treewidth. We show a subexponential algorithm (running in time exp O(n 1−ε) ) for computing the Tutte polynomial on graphs of bounded cliquewidth. In fact, our algorithm computes the more general Upolynomial.
Bounds on the Chromatic Polynomial and on the Number of Acyclic Orientations of a Graph
 Combinatorica
, 1996
"... An upper bound is given on the number of acyclic orientations of a graph, in terms of the spectrum of its Laplacian. It is shown that this improves upon the previously known bound, which depended on the degree sequence of the graph. Estimates on the new bound are provided. A lower bound on the numbe ..."
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Cited by 8 (0 self)
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An upper bound is given on the number of acyclic orientations of a graph, in terms of the spectrum of its Laplacian. It is shown that this improves upon the previously known bound, which depended on the degree sequence of the graph. Estimates on the new bound are provided. A lower bound on the number of acyclic orientations of a graph is given, with the help of the probabilistic method. This argument can take advantage of structural properties of the graph: it is shown how to obtain stronger bounds for smalldegree graphs of girth at least five, than are possible for arbitrary graphs. A simpler proof of the known lower bound for arbitrary graphs is also obtained. Both the upper and lower bounds are shown to extend to the general problem of bounding the chromatic polynomial from above and below along the negative real axis. 1 XEROX Palo Alto Research Center, 3333 Coyote Hill Road, CA 94304. Partially supported by the NSF under grant CCR9404113. Most of this research was done while th...
Does the Jones Polynomial Detect Unknottedness?
 Math
, 1997
"... There were many attempts to settle the question whether there exist nontrivial knots with trivial Jones polynomial. In this paper we show that such a knot must have crossing number at least 18. ..."
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Cited by 7 (2 self)
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There were many attempts to settle the question whether there exist nontrivial knots with trivial Jones polynomial. In this paper we show that such a knot must have crossing number at least 18.
Tutte polynomials of tensor products of signed graphs and their applications in knot theory
 J. Knot Theory Ramifications
"... Abstract. It is wellknown that the Jones polynomial of an alternating knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. Relying on the results of Bollobás and Riordan, we introduce a generalization of Kauffman’s Tutte polynomial of si ..."
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Abstract. It is wellknown that the Jones polynomial of an alternating knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. Relying on the results of Bollobás and Riordan, we introduce a generalization of Kauffman’s Tutte polynomial of signed graphs for which describing the effect of taking a signed tensor product of signed graphs is very simple. We show that this Tutte polynomial of a signed tensor product of signed graphs may be expressed in terms of the Tutte polynomials of the original signed graphs by using a simple substitution rule. Our result enables us to compute the Jones polynomials of some large nonalternating knots. The combinatorics used to prove our main result is similar to Tutte’s original way of counting “activities” and specializes to a new, perhaps simpler proof of the known formulas for the ordinary Tutte polynomial of the tensor product of unsigned graphs or matroids. 1.
RELATIVE TUTTE POLYNOMIALS FOR COLORED GRAPHS AND VIRTUAL KNOT THEORY
, 909
"... Abstract. We introduce the concept of a relative Tutte polynomial of colored graphs. We show that this relative Tutte polynomial can be computed in a way similar to the classical spanning tree expansion used by Tutte in his original paper on this subject. We then apply the relative Tutte polynomial ..."
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Abstract. We introduce the concept of a relative Tutte polynomial of colored graphs. We show that this relative Tutte polynomial can be computed in a way similar to the classical spanning tree expansion used by Tutte in his original paper on this subject. We then apply the relative Tutte polynomial to virtual knot theory. More specifically, we show that the Kauffman bracket polynomial (hence the Jones polynomial) of a virtual knot can be computed from the relative Tutte polynomial of its face (Tait) graph with some suitable variable substitutions. Our method offers an alternative to the ribbon graph approach, using the face graph obtained from the virtual link diagram directly. 1.
Subexponentially Computable Truncations of Jonestype Polynomials
 GRAPH STRUCTURE THEORY, CONTEMPORARY MATHEMATICS VOL. 147, AMERICAN MATHEMATICAL SOCIETY, PROVIDENCE, RHODE ISLAND
, 1993
"... We show that an essential part of the new (Jonestype) polynomial link invariants can be computed in subexponential time. This is in a sharp contrast to the result of Jaeger, Vertigan and Welsh that computing the whole polynomial and most of its evaluations is #Phard. ..."
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Cited by 1 (0 self)
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We show that an essential part of the new (Jonestype) polynomial link invariants can be computed in subexponential time. This is in a sharp contrast to the result of Jaeger, Vertigan and Welsh that computing the whole polynomial and most of its evaluations is #Phard.
On Some Hard Problems on Matroid Spikes
, 2005
"... Spikes form an interesting class of 3connected matroids of branchwidth 3. We show that some computational problems are hard on spikes with given matrix representations over infinite fields. Namely, the question whether a given spike is the free spike is coNPhard (though the property itself is de ..."
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Spikes form an interesting class of 3connected matroids of branchwidth 3. We show that some computational problems are hard on spikes with given matrix representations over infinite fields. Namely, the question whether a given spike is the free spike is coNPhard (though the property itself is definable in monadic secondorder logic); and the task to compute the Tutte polynomial of a spike is #Phard (even though that can be solved efficiently on all matroids of bounded branchwidth which are represented over a finite field).
Tutte Polynomials of Signed Graphs and Jones Polynomials of Some Large Knots
"... It is wellknown that the Jones polynomial of a knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. In this paper, we study the Tutte polynomials for signed graphs. We show that if a signed graph is constructed from a simpler graph via k ..."
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It is wellknown that the Jones polynomial of a knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. In this paper, we study the Tutte polynomials for signed graphs. We show that if a signed graph is constructed from a simpler graph via kthickening or kstretching, then its Tutte polynomial can be expressed in terms of the Tutte polynomial of the original graph, thus enabling us to compute the Jones polynomials for some (special) large nonalternating knots.