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 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)

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

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

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

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)

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 iff there exists a finite sequence of Reidemeister moves that converts it to the trivial knot diagram. But how many steps?

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.

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.

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.

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.

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

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)

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?

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).

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

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

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