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Knots and Links - Introduction and Complexity Results

Knots and Links - Introduction and Complexity Results. Krishnaram Kenthapadi 11/27/2002. Outline. Definition Classification Representation Knot Triviality Splitting Problem Genus Problem Open Questions. Definitions.

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Knots and Links - Introduction and Complexity Results

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  1. Knots and Links - Introduction and Complexity Results Krishnaram Kenthapadi 11/27/2002

  2. Outline • Definition • Classification • Representation • Knot Triviality • Splitting Problem • Genus Problem • Open Questions

  3. Definitions • Knot– A closed curve embedded in space as a simple (non-self-intersecting) polygon with finitely many edges. (Informally, a thin elastic string with extremities glued together)

  4. Definitions • Link– A finite collection of simple polygons disjointly embedded in 3-dimensional space. • Individual polygons -components of link • Knot – A link with one component

  5. Classification of Knots • Isotopy is a deformation of knots s.t. • Piecewise linear & continuous • Polygon remains simple throughout • Defines an equivalence relation • Knots in a single plane are equivalent • Trivial knots

  6. Computational Representation • Polygonal Representation in 3-D space • List the vertices of each polygon in order • Link diagram representing a 2-D projection • Extra labeling for crosses • Both are polynomial time equivalent

  7. Unknotting Problem • Instance : A link diagram D • Question : Is D a knot diagram that represents the trivial knot? • This problem is in NP. (Hass, Lagarias & Pippenger, 1999)

  8. Unknotting Problem • Haken’s Algorithm (1961): Runs in exponential time. • Reidemeister moves : Combinatorial transformations on the knot diagram that don’t change the equivalence class of the knot. • A knot diagram is unknotted iffthere exists a finite sequence of Reidemeister moves that converts it to the trivial knot diagram. • But how many steps?

  9. Splitting Problem • Instance : A link diagram D • Question : Is the link represented by D splittable? • Splittable : the polygons can be separated by piecewise linear isotopy. • This problem is in NP.

  10. Genus of a surface • Any oriented surface without boundary can be obtained from a sphere by adding “handles”. • Genus = Number of handles • Eg: Genus of Sphere is 0, Torus is 1, etc.

  11. Genus of a surface • Genus is also the number of surfaces along which a surface can be cut while leaving it connected. • Surface with boundary : Glue a disk to each component of the boundary (“capping off”) and then obtain the genus.

  12. Genus of a knot • Informally, the degree of “knottedness” of a curve. • S(K) – class of all orientable spanning surfaces for a knot K, ie, surfaces embedded in the manifold, with a single boundary component that coincides with K. • S(K) is non-empty for any knot in 3-sphere (Seifert, 1935). Seifert also showed a construction.

  13. Genus of a knot • Genus(K) = min{Genus(s) | s \in S(K)} if S(K) is non-empty; otherwise Genus(K) is infinity ().

  14. 3-Manifold Knot Genus • Instance: A triangulated 3-manifold M, a knot K and a natural number,g. • Question: Is genus(K) <= g ? • Size of instance : Number of tetrahedra in M and log(g). • This problem is NP-complete. (Agol, Hass & Thurston, 2002)

  15. 3-Manifold Knot Genus • NP- hard: By reduction from an NP-complete problem, ONE-IN-THREE-SAT. • ONE-IN-THREE-SAT: • Instance: A set U of variables and a collection C of clauses (of three literals each) over U. • Question: Is there a truth assignment for U s.t. each clause in C has exactly one true literal?

  16. A Special Case • A knot is trivial iff its genus is zero. • Hence, Unknotting problem is a special case of 3-Manifold Knot Genus (with the input, g = 0).

  17. Recap • Definition of knots & links. • Classification – knot isotopy • Computational Representation • polygonal (3D) • link diagram (2D) • Knot Triviality is in NP • Splitting Problem is in NP • Genus Problem is NP-complete

  18. Open Problems • Is 3-SPHERE KNOT GENUS NP-hard? • Is determining genus of a knot in 3-Manifold in NP? • Amounts to showing a lower bound • If “yes”, UNKNOTTING problem is in both NP and co-NP

  19. References • C. C. Adams, The Knot Book. An elementary introduction to the mathematical theory of knots, W. H. Freeman, New York 1994. • V.V.Prasolov, Intuitive Topology, American Mathematical Society, 1995. • J. Hass, J. C. Lagarias and N. Pippenger, The computational complexity of Knot and Link problems", Journal of the ACM, 46 (1999) 185-211. • I. Agol, J. Hass and W.P. Thurston, The Computational Complexity of Knot Genus and Spanning Area, Proceedings of STOC 2002.

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