<p>The knot <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1952_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$8_{18}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mn>8</mn> <mn>18</mn> </msub> </math></EquationSource> </InlineEquation> is the first non-arborescent hyperbolic knot. In 2020, Paoluzzi and Porti found its <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1952_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{SL}(2,\mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>SL</mtext> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-character variety with the aid of a computer, but many details were omitted. In this paper, we determine the character variety by a software-free procedure, which is easy to follow and enlightening. Along the way, we develop an efficient method for working with simultaneous conjugacy classes of four elements of <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1952_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{SL}(2,\mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>SL</mtext> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

The $\textrm{SL}(2,\mathbb {C})$-character variety of $8_{18}$

期刊论文
发表日期：

2025年8月21日

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Haimiao Chen

摘要

The knot $8_{18}$ $8_{18}$ is the first non-arborescent hyperbolic knot. In 2020, Paoluzzi and Porti found its $\textrm{SL}(2,\mathbb {C})$ $SL (2, C)$ -character variety with the aid of a computer, but many details were omitted. In this paper, we determine the character variety by a software-free procedure, which is easy to follow and enlightening. Along the way, we develop an efficient method for working with simultaneous conjugacy classes of four elements of $\textrm{SL}(2,\mathbb {C})$ $SL (2, C)$ .

The \(\textrm{SL}(2,\mathbb {C})\)-character variety of \(8_{18}\)

摘要