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Minimal grid diagrams of the prime knots with crossing number 13 and arc index 13

Hwa Jeong Lee, Yoonsang Lee, Chanmin Lee, Yeseo Park, Hun Kim, Gyo Taek Jin

Abstract

We give a list of minimal grid diagrams of the 13 crossing prime nonalternating knots which have arc index 13. There are 9,988 prime knots with crossing number 13. Among them 4,878 are alternating and have arc index 15. Among the other nonalternating knots, 49, 399, 1,412 and 3,250 have arc index 10, 11, 12, and 13, respectively. We used the Dowker-Thistlethwaite code of the 3,250 knots provided by the program Knotscape to generate spanning trees of the corresponding knot diagrams to obtain minimal arc presentations in the form of grid diagrams.

Minimal grid diagrams of the prime knots with crossing number 13 and arc index 13

Abstract

We give a list of minimal grid diagrams of the 13 crossing prime nonalternating knots which have arc index 13. There are 9,988 prime knots with crossing number 13. Among them 4,878 are alternating and have arc index 15. Among the other nonalternating knots, 49, 399, 1,412 and 3,250 have arc index 10, 11, 12, and 13, respectively. We used the Dowker-Thistlethwaite code of the 3,250 knots provided by the program Knotscape to generate spanning trees of the corresponding knot diagrams to obtain minimal arc presentations in the form of grid diagrams.
Paper Structure (4 sections, 3 theorems, 3 equations, 5 figures, 1 table)

This paper contains 4 sections, 3 theorems, 3 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. The knot-spoke method and the filtered spanning tree method
  3. Filtered spanning tree method adapted to non-alternating knots
  4. Minimal Grid Diagrams of Arc Index 13

Key Result

Theorem 1

A prime link $L$ is non-alternating if and only if

Figures (5)

  • Figure 1: A minimal grid diagram of the knot $13n3003$
  • Figure 2: Diagram of $13n3003$ and its non-alternating edges SnapPy
  • Figure 3: A filtered spanning tree and the resulting spokes
  • Figure 4: Destabilization at two places
  • Figure 5: Bottom 6 edges moved to the top and then left 5 edges moved to the right

Theorems & Definitions (3)

  • Theorem 1: Jin-Park2010
  • Theorem 2: BP2000
  • Theorem 3: MB1998