Mirror-curves and knot mosaics

  • Authors:

  • Slavik Jablan;Ljiljana Radović;Radmila Sazdanović;Ana Zeković

  • Affiliations:

  • The Mathematical Institute, Knez Mihailova 36, P.O.Box 367, 11001 Belgrade, Serbia;University of Niš, Faculty of Mechanical Engineering, A. Medvedeva 14, 18 000 Niš, Serbia;Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104-6395, USA;Zeta System, Golsvortijeva 1, 11000 Belgrade, Serbia

  • Venue:
  • Computers & Mathematics with Applications
  • Year:
  • 2012

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Abstract

Inspired by the paper on quantum knots and knot mosaics (Lomonaco and Kauffman, 2008 [18]) and grid diagrams (or arc presentations), used extensively in the computations of Heegaard-Floer knot homology (Bar-Natan, 0000 [16], Cromwell, 1995 [21], Manolescu et al., 2007 [22]), we construct the more concise representation of knot mosaics and grid diagrams via mirror-curves. Tame knot theory is equivalent to knot mosaics (Lomonaco and Kauffman, 2008 [18]), mirror-curves, and grid diagrams (Bar-Natan, 0000 [16], Cromwell, 1995 [21], Kuriya, 2008 [20], Manolescu et al., 2007 [22]). Hence, we introduce codes for mirror-curves treated as knot or link diagrams placed in rectangular square grids, suitable for software implementation. We provide tables of minimal mirror-curve codes for knots and links obtained from rectangular grids of size 3x3 and px2 (p@?4), and describe an efficient algorithm for computing the Kauffman bracket and L-polynomials (Jablan and Sazdanovic, 2007 [8], Kauffman, 2006 [11], Kauffman, 1987 [12]) directly from mirror-curve representations.