Mirror-Curves and Knot Mosaics

Slavik V. Jablan; Ljiljana Radovic; Radmila Sazdanovic; Ana Zekovic

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arXiv · 1106.3784

Mirror-Curves and Knot Mosaics

Slavik V. Jablan·Ljiljana Radovic·Radmila Sazdanovic·Ana Zekovic

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Abstract

Inspired by the paper on quantum knots and knot mosaics [23] and grid diagrams (or arc presentations), used extensively in the computations of Heegaard-Floer knot homology [2,3,7,24], we construct the more concise representation of knot mosaics and grid diagrams via mirror-curves. Tame knot theory is equivalent to knot mosaics [23], mirror-curves, and grid diagrams [3,7,22,24]. 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<5), and describe an efficient algorithm for computing the Kauffman bracket and L-polynomials [18,19,20] directly from mirror-curve representations.

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Slavik V. Jablan, Ljiljana Radovic, Radmila Sazdanovic, Ana Zekovic. 2011-12-28. Mirror-Curves and Knot Mosaics. https://arxiv.org/abs/1106.3784

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