The Computational Complexity of Knot Genus and Spanning Area

We investigate the computational complexity of some problems in three-dimensional topology and geometry. We show that the problem of determining a bound on the genus of a knot in a 3-manifold, is NP-complete. Using similar ideas, we show that deciding whether a curve in a metrized PL 3-manifold bounds a surface of area less than a given constant C is NP-hard.

[1]  J. Lagarias,et al.  The number of Reidemeister moves needed for unknotting , 1998, math/9807012.

[2]  J. Hass.  Algorithms for recognizing knots and 3-manifolds , 1997, math/9712269.

[3]  Jeffrey C. Lagarias,et al.  The computational complexity of knot and link problems , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[4]  William Jaco,et al.  Algorithms for the complete decomposition of a closed $3$-manifold , 1995 .

[5]  Geoffrey Hemion.  The Classification of Knots and 3-Dimensional Spaces , 1992 .

[6]  D. Welsh,et al.  On the computational complexity of the Jones and Tutte polynomials , 1990, Mathematical Proceedings of the Cambridge Philosophical Society.

[7]  András Sebö,et al.  Hilbert Bases, Caratheodory's Theorem and Combinatorial Optimization , 1990, IPCO.

[8]  J. Hyam Rubinstein,et al.  PL equivariant surgery and invariant decompositions of 3-manifolds , 1989 .

[9]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[10]  Thomas J. Schaefer,et al.  The complexity of satisfiability problems , 1978, STOC.

[11]  Horst Schubert,et al.  Bestimmung der Primfaktorzerlegung von Verkettungen , 1961 .

[12]  Edwin E. Moise,et al.  Affine structures in 3-manifolds, V, The triangulation theorem and Hauptvermutung , 1952 .

[13]  H. Seifert,et al.  Über das Geschlecht von Knoten , 1935 .

[14]  M. Dehn,et al.  Über die Topologie des dreidimensionalen Raumes , 1910 .

[15]  J. Rubinstein,et al.  An Algorithm to Recognize the 3-Sphere , 1995 .

[16]  Abigail Thompson,et al.  THIN POSITION AND THE RECOGNITION PROBLEM FOR S3 , 1994 .

[17]  Abigail Thompson,et al.  Thin Position and the Recognition Problem for $\bold{S^3}$ , 1994 .

[18]  Dominic Welsh,et al.  The Complexity of Knots , 1993 .

[19]  D. Welsh.  Complexity: Knots, Colourings and Counting: Link polynomials and the Tait conjectures , 1993 .

[20]  Dominic Welsh,et al.  Knots and braids: Some algorithmic questions , 1991, Graph Structure Theory.

[21]  Ulrich Oertel,et al.  An algorithm to decide if a 3-manifold is a Haken manifold , 1984 .

[22]  William Jaco,et al.  Lectures on three-manifold topology , 1980 .

[23]  John Stillwell,et al.  Knots and Braids , 1980 .

[24]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[25]  Michael O. Rabin,et al.  Recursive Unsolvability of Group Theoretic Problems , 1958 .

[26]  Hellmuth Kneser,et al.  Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten. , 1929 .