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The Braid Indices of Pretzel Links: A Comprehensive Study, Part I

Yuanan Diao, Claus Ernst, Gabor Hetyei

Abstract

The determination of the braid index of an oriented link is generally a hard problem. In the case of alternating links, some significant progresses have been made in recent years which made explicit and precise braid index computations possible for links from various families of alternating links, including the family of all alternating Montesinos links. However, much less is known for non-alternating links. For example, even for the non-alternating pretzel links, which are special (and simpler) Montesinos links, the braid index is only known for a very limited few special cases. In this paper and its sequel, we study the braid indices for all non-alternating pretzel links by a systematic approach. We classify the pretzel links into three different types according to the Seifert circle decompositions of their standard link diagrams. More specifically, if $D$ is a standard diagram of an oriented pretzel link $\mathcal{L}$, $S(D)$ is the Seifert circle decomposition of $D$, and $C_1$, $C_2$ are the Seifert circles in $S(D)$ containing the top and bottom long strands of $D$ respectively, then $\mathcal{L}$ is classified as a Type 1 (Type 2) pretzel link if $C_1\not=C_2$ and $C_1$, $C_2$ have different (identical) orientations. In the case that $C_1=C_2$, then $\mathcal{L}$ is classified as a Type 3 pretzel link. In this paper, we present the results of our study on Type 1 and Type 2 pretzel links. Our results allow us to determine the precise braid index for any non-alternating Type 1 or Type 2 pretzel link. Since the braid indices are already known for all alternating pretzel links from our previous work, it means that we have now completely determined the braid indices for all Type 1 and Type 2 pretzel links.

The Braid Indices of Pretzel Links: A Comprehensive Study, Part I

Abstract

The determination of the braid index of an oriented link is generally a hard problem. In the case of alternating links, some significant progresses have been made in recent years which made explicit and precise braid index computations possible for links from various families of alternating links, including the family of all alternating Montesinos links. However, much less is known for non-alternating links. For example, even for the non-alternating pretzel links, which are special (and simpler) Montesinos links, the braid index is only known for a very limited few special cases. In this paper and its sequel, we study the braid indices for all non-alternating pretzel links by a systematic approach. We classify the pretzel links into three different types according to the Seifert circle decompositions of their standard link diagrams. More specifically, if is a standard diagram of an oriented pretzel link , is the Seifert circle decomposition of , and , are the Seifert circles in containing the top and bottom long strands of respectively, then is classified as a Type 1 (Type 2) pretzel link if and , have different (identical) orientations. In the case that , then is classified as a Type 3 pretzel link. In this paper, we present the results of our study on Type 1 and Type 2 pretzel links. Our results allow us to determine the precise braid index for any non-alternating Type 1 or Type 2 pretzel link. Since the braid indices are already known for all alternating pretzel links from our previous work, it means that we have now completely determined the braid indices for all Type 1 and Type 2 pretzel links.
Paper Structure (8 sections, 11 theorems, 62 equations, 18 figures)

This paper contains 8 sections, 11 theorems, 62 equations, 18 figures.

Table of Contents

  1. Introduction
  2. preparations for establishing the lower bounds
  3. Braid index lower bounds for Type 1 pretzel links
  4. Braid index lower bounds for Type 2 pretzel links
  5. preparations for establishing the upper bounds
  6. Braid index upper bounds for Type 1 pretzel links
  7. Braid index upper bounds for Type 2 pretzel links
  8. Ending Remarks

Key Result

Theorem 1.3

If $\mathcal{L}\in P_{1}(2\alpha_1+1,\ldots, 2\alpha_{\kappa^+}+1; -(2\beta_1+1),\ldots, -(2\beta_{\kappa^-}+1))$, then the braid index of $\mathcal{L}$, denoted by $\textbf{b}(\mathcal{L})$, is given by the following formulas: and where $\tau$ ($\tau^\prime$) is the number of $\beta_i$'s ($\alpha_j$'s) that equal to zero and it is understood that $\sum \alpha_j=0$ if $\kappa^+=0$ and $\sum \bet

Figures (18)

  • Figure 1: Left: The general structure of a Montesinos link: each circle in the picture contains a (2 string) rational tangle; Middle: a vertical strip of 2 right-handed half twists; Right: a vertical strip of 2 left-handed half twists.
  • Figure 2: From left to right: Examples of Type 1, 2 and 3 pretzel link diagrams and their corresponding Seifert circle decompositions.
  • Figure 3: The sign convention at a crossing of an oriented link and the splitting of the crossing: the crossing in $D_+$ ($D_-$) is positive (negative) and is assigned $+1$ ($-1$) in the calculation of the writhe of the link diagram.
  • Figure 4: A mutation move applied to two adjacent vertical strips in a pretzel link diagram.
  • Figure 5: The non alternating pretzel link $P_1(5,5;-1)$ is equivalent to the alternating pretzel link diagram $P_1(3,3,1;0)$.
  • ...and 13 more figures

Theorems & Definitions (40)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Example 1.5
  • Remark 1.6
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • ...and 30 more