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Minimal grid diagrams of the prime knots with crossing number 14 and arc index 13, 14

Hwa Jeong Lee, Alexander Stoimenow, Hun Kim, Minchae Kim, Songwon Ryu, Dongju Shin, Joon Hyeok Choi, Woo Jin Choi, Jin Seong Park, Gyo Taek Jin

TL;DR

This work analyzes the distribution of arc index $\ alpha(K)$ among prime knots with crossing number $14$ and provides explicit minimal grid diagrams for arc indices $13$ and $14$. It combines the knot-spoke method with the filtered spanning tree approach to convert diagrams into arc presentations, using $c(D)+2$ pages and reducing by $3$ to reach $\ alpha(K)=13$, demonstrated concretely on $14n10$. The authors report $8{,}027$ knots with arc index $13$ and $15{,}735$ with arc index $14$, accompanied by complete grid-diagram lists; a four-stage Dowker–Thistlethwaite code workflow enables systematic construction and verification. These results advance practical enumeration of minimal grid diagrams for high-crossing knots and supply actionable tools for computational knot theory and planar representations.

Abstract

There are 46,972 prime knots with crossing number 14. Among them 19,536 are alternating and have arc index 16. Among the non-alternating knots, 17, 477, and 3,180 have arc index 10, 11, and 12, respectively. The remaining 23,762 have arc index 13 or 14. There are none with arc index smaller than 10 or larger than 14. We obtained 8,027 knots having arc index 13 and 15,735 knots having arc index 14. We show them by their minimal grid diagrams.

Minimal grid diagrams of the prime knots with crossing number 14 and arc index 13, 14

TL;DR

This work analyzes the distribution of arc index among prime knots with crossing number and provides explicit minimal grid diagrams for arc indices and . It combines the knot-spoke method with the filtered spanning tree approach to convert diagrams into arc presentations, using pages and reducing by to reach , demonstrated concretely on . The authors report knots with arc index and with arc index , accompanied by complete grid-diagram lists; a four-stage Dowker–Thistlethwaite code workflow enables systematic construction and verification. These results advance practical enumeration of minimal grid diagrams for high-crossing knots and supply actionable tools for computational knot theory and planar representations.

Abstract

There are 46,972 prime knots with crossing number 14. Among them 19,536 are alternating and have arc index 16. Among the non-alternating knots, 17, 477, and 3,180 have arc index 10, 11, and 12, respectively. The remaining 23,762 have arc index 13 or 14. There are none with arc index smaller than 10 or larger than 14. We obtained 8,027 knots having arc index 13 and 15,735 knots having arc index 14. We show them by their minimal grid diagrams.
Paper Structure (6 sections, 3 theorems, 4 equations, 8 figures, 1 table)

This paper contains 6 sections, 3 theorems, 4 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. The knot-spoke method and the filtered spanning tree method
  3. A construction using $14n10$
  4. Four stages of the work
  5. Minimal Grid Diagrams of Arc Index 13
  6. The case of arc index 14

Key Result

Theorem 1

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

Figures (8)

  • Figure 1: An arc presentation of the figure eight knot
  • Figure 2: A grid diagram and its corresponding arc presentation
  • Figure 3: Minimal grid diagrams of the knot $14n10$
  • Figure 4: Diagram of $14n10$ and its non-alternating edges SnapPy
  • Figure 5: A filtered spanning tree and the resulting spokes
  • ...and 3 more figures

Theorems & Definitions (3)

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