Theory of linear and integer programming
Theory of linear and integer programming
Discrete Applied Mathematics
Complexity: knots, colourings and counting
Complexity: knots, colourings and counting
Journal of the ACM (JACM)
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Hilbert Bases, Caratheodory's Theorem and Combinatorial Optimization
Proceedings of the 1st Integer Programming and Combinatorial Optimization Conference
The Computational Complexity of Knot and Link Problems
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
3-manifold knot genus is NP-complete
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Computing Linking Numbers of a Filtration
WABI '01 Proceedings of the First International Workshop on Algorithms in Bioinformatics
Algorithms for Normal Curves and Surfaces
COCOON '02 Proceedings of the 8th Annual International Conference on Computing and Combinatorics
The parametrized complexity of knot polynomials
Journal of Computer and System Sciences - Special issue on Parameterized computation and complexity
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Fast algorithms for computing Jones polynomials of certain links
Theoretical Computer Science
Hardness of embedding simplicial complexes in Rd
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Morphing polyhedra with parallel faces: Counterexamples
Computational Geometry: Theory and Applications
LATA '09 Proceedings of the 3rd International Conference on Language and Automata Theory and Applications
The complexity of the normal surface solution space
Proceedings of the twenty-sixth annual symposium on Computational geometry
Maximal admissible faces and asymptotic bounds for the normal surface solution space
Journal of Combinatorial Theory Series A
A tree traversal algorithm for decision problems in knot theory and 3-manifold topology
Proceedings of the twenty-seventh annual symposium on Computational geometry
Estimating Jones and Homfly polynomials with one clean qubit
Quantum Information & Computation
Estimating Jones polynomials is a complete problem for one clean qubit
Quantum Information & Computation
Planarity of knots, register automata and logspace computability
LATA'11 Proceedings of the 5th international conference on Language and automata theory and applications
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
Tracing compressed curves in triangulated surfaces
Proceedings of the twenty-eighth annual symposium on Computational geometry
Computing the Crosscap Number of a Knot Using Integer Programming and Normal Surfaces
ACM Transactions on Mathematical Software (TOMS)
Computing closed essential surfaces in knot complements
Proceedings of the twenty-ninth annual symposium on Computational geometry
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We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, ie., capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, UNKNOTTING PROBLEM is in NP. We also consider the problem, SPLITTING PROBLEM of determining whether two or more such polygons can be split, or continuously deformed without self-intersection 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 worst-case 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.