Mirror-curves and knot mosaics

Abstract

AbstractInspired 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 3×3 and p×2 (p≤4), and describe an efficient algorithm for computing the Kauffman bracket and L-polynomials (Jablan and Sazdanović, 2007 [8], Kauffman, 2006 [11], Kauffman, 1987 [12]) directly from mirror-curve representations