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Widths of links via diagram colorings

Ricky Lee, Puttipong Pongtanapaisan, Hanh Vo

Abstract

In this paper, we define invariants of links in terms of colorings of link diagrams and prove that these invariants coincide with various notions of widths of links with respect to the standard Morse function. Our formulations are advantageous because they are algorithmic and suitable for program implementations. As an application, we calculate the max-width of over 10000 links up to 14 crossings from the link table.

Widths of links via diagram colorings

Abstract

In this paper, we define invariants of links in terms of colorings of link diagrams and prove that these invariants coincide with various notions of widths of links with respect to the standard Morse function. Our formulations are advantageous because they are algorithmic and suitable for program implementations. As an application, we calculate the max-width of over 10000 links up to 14 crossings from the link table.
Paper Structure (14 sections, 10 theorems, 4 equations, 5 figures)

This paper contains 14 sections, 10 theorems, 4 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Original formulations of widths
  4. Wirtinger colorings
  5. Cut-split links
  6. Coloring from the height function
  7. Coloring by Height
  8. Embedding from link diagram coloring
  9. Computational results
  10. Applications and questions
  11. Tube embeddings
  12. Thin position of branched cover
  13. Thin position of graphs
  14. Questions

Key Result

Theorem 1.1

Let $L$ be a link in $S^3.$ Then, the lex-width of $L$ is equal to the Wirtinger lex-width.

Figures (5)

  • Figure 1: One can visualize the intersection of the level planes and the link by parallel lines in the link diagram. This figure shows that the link L11a496 has an embedding with trunk equals six.
  • Figure 2: The coloring move
  • Figure 3: A coloring sequence is demonstrated on L10n35 from left to right. The red, blue, and green strands generate the coloring until a special crossing of type I appears in the leftmost diagram on the second row. The purple coloring is added in the penultimate diagram of the second row. This implies that there is an embedding of L10n35 where a minimum is higher than a maximum, giving trunk(L10n35) = 6.
  • Figure 4: Another coloring sequence is demonstrated on L9a55 from left to right. The red, blue, and green strands generate the coloring until a special crossing of type II appears in the leftmost diagram on the second row. The purple coloring is added in the penultimate diagram of the second row. This implies that there is an embedding of L9a55 where a minimum is higher than a maximum, giving trunk(L9a55) = 6.
  • Figure 5: There are three local minima, but there is only one special crossing. However, Lemma \ref{['lem:nowiggle']} shows that if this happens then $L$ is not in thin position with respect to any notion of widths.

Theorems & Definitions (29)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Proposition 2.7
  • Definition 3.2
  • Definition 3.3
  • ...and 19 more