Slavik Jablan and Radmila Sazdanovich
The Mathematical Institute, Yugoslavia
3:30 Thursday, 27 Nov 2003, in CoLab (P8493)
Knot Theory Program: LinKnot
Abstract
The Mathematica-based Windows knot theory program LinKnot is the extension of the program Knot2000 (K2K) written by M.Ochiai and N.Imafuji. LinKnot is the knot theory program that works with knots and links (KLs) given in the Conway notation. Conway symbols are an input used for creating Dowker codes or P-data (the main input for K2K functions). Instead of a graphical input or Dowker codes, for the first time in a computer program you can use human-comprehensive Conway notation of KLs represented as a Mathematica string and work with links, and not only with knots. For all KLs there is no restriction on the number of crossings. The program provides also the complete data base of alternating KLs with at most 12 crossings, and non-alternating KLs with at most 11 crossings, and the data base of basic polyhedra with at most 20 crossings.
By using it, it is possible to draw KLs, calculate all polynomial invariants of KLs, work with braids, reduce KLs, etc. For the first time, it is possible to compute unknotting and unlinking numbers, calculated according to Bernhard-Jablan Conjecture. For all alternating KLs you can compute minimum Dowker codes, find all non-isomorphic projections, work with the graphs of KLs, compute linking numbers, breaking and spliting numbers, signatures, and many other KL invariants.
The main property of the program is a possibility to use it as a tool in experimenting with KLs, for computing properties connected with infinite classes of KLs (KL families) and make new conjectures in knot theory. For example, except the famous Nakanishi-Bleiler exapmle of a knot 514 with the unknotting number projection gap, we discovered an infinite collection of such KLs.
The Mathematical Institute
Knez Mihailova 35, P.O. Box 367
11001 Belgrade, Yugoslavia
jablans@yahoo.com