The Braid Indices of the Reverse Parallel Links of Alternating Knots
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916371369033728 |
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| author | Diao, Yuanan Morton, Hugh |
| author_facet | Diao, Yuanan Morton, Hugh |
| contents | The braid indices of most links remain unknown as there is no known universal method that can be used to determine the braid index of an arbitrary knot. This is also the case for alternating knots. In this paper, we show that if $K$ is an alternating knot, then the braid index of any reverse parallel link of $K$ can be precisely determined. More precisely, if $D$ is a reduced diagram of $K$, $v_+(D)$ ($v_-(D)$) is the number of regions in the checkerboard shading of $D$ for which all crossings are positive (negative), $w(D)$ is the writhe of $D$, then the braid index of a reverse parallel link of $K$ with framing $f$, denoted by $\mathbb{K}_f$, is given by the following precise formula $$\textbf{b}(\mathbb{K}_f)=\left\{ \begin{array}{ll} c(D)+2+a(D)-f, &\ {\rm if}\ f < a(D),\\ c(D)+2, &\ {\rm if}\ a(D)\le f \le b(D),\\ c(D)+2-b(D)+f, &\ {\rm if}\ f > b(D),\\ \end{array} \right.
$$ where $a(D)=-v_-(D)+w(D)$ and $b(D)=v_+(D)+w(D)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_05579 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Braid Indices of the Reverse Parallel Links of Alternating Knots Diao, Yuanan Morton, Hugh Geometric Topology Algebraic Topology 57K10, 57K31 The braid indices of most links remain unknown as there is no known universal method that can be used to determine the braid index of an arbitrary knot. This is also the case for alternating knots. In this paper, we show that if $K$ is an alternating knot, then the braid index of any reverse parallel link of $K$ can be precisely determined. More precisely, if $D$ is a reduced diagram of $K$, $v_+(D)$ ($v_-(D)$) is the number of regions in the checkerboard shading of $D$ for which all crossings are positive (negative), $w(D)$ is the writhe of $D$, then the braid index of a reverse parallel link of $K$ with framing $f$, denoted by $\mathbb{K}_f$, is given by the following precise formula $$\textbf{b}(\mathbb{K}_f)=\left\{ \begin{array}{ll} c(D)+2+a(D)-f, &\ {\rm if}\ f < a(D),\\ c(D)+2, &\ {\rm if}\ a(D)\le f \le b(D),\\ c(D)+2-b(D)+f, &\ {\rm if}\ f > b(D),\\ \end{array} \right. $$ where $a(D)=-v_-(D)+w(D)$ and $b(D)=v_+(D)+w(D)$. |
| title | The Braid Indices of the Reverse Parallel Links of Alternating Knots |
| topic | Geometric Topology Algebraic Topology 57K10, 57K31 |
| url | https://arxiv.org/abs/2302.05579 |