<p>We construct the linear representation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\hat{\beta _ n}^{d,g,e}: SM_n \rightarrow M_n(\mathbb {Z}[t^{\pm 1},g,d,e])$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mover accent="true"> <msub> <mi>β</mi> <mi>n</mi> </msub> <mo stretchy="false">^</mo> </mover> <mrow> <mi>d</mi> <mo>,</mo> <mi>g</mi> <mo>,</mo> <mi>e</mi> </mrow> </msup> <mo>:</mo> <mi>S</mi> <msub> <mi>M</mi> <mi>n</mi> </msub> <mo stretchy="false">→</mo> <msub> <mi>M</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mrow> <mo stretchy="false">[</mo> <msup> <mi>t</mi> <mrow> <mo>±</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mi>g</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>e</mi> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which extends the Burau representation of the braid group <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$B_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> to the singular braid monoid <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$SM_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>M</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. We show that any extension of the Burau representation to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$SM_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>M</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> is of the form <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\hat{\beta _ n}^{d,g,e}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mover accent="true"> <msub> <mi>β</mi> <mi>n</mi> </msub> <mo stretchy="false">^</mo> </mover> <mrow> <mi>d</mi> <mo>,</mo> <mi>g</mi> <mo>,</mo> <mi>e</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>. We show that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\hat{\beta _ n}^{d,g,e}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mover accent="true"> <msub> <mi>β</mi> <mi>n</mi> </msub> <mo stretchy="false">^</mo> </mover> <mrow> <mi>d</mi> <mo>,</mo> <mi>g</mi> <mo>,</mo> <mi>e</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> is reducible to a reduced representation <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\hat{\beta _n^r}: SM_n \rightarrow M_{n-1}(\mathbb {Z}[t^{\pm 1},g,d,e]))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <msubsup> <mi>β</mi> <mi>n</mi> <mi>r</mi> </msubsup> <mo stretchy="false">^</mo> </mover> <mo>:</mo> <mi>S</mi> <msub> <mi>M</mi> <mi>n</mi> </msub> <mo stretchy="false">→</mo> <msub> <mi>M</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mrow> <mo stretchy="false">[</mo> <msup> <mi>t</mi> <mrow> <mo>±</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mi>g</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>e</mi> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Additionally, we study whether or not extensions of the Burau representation to <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$SB_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>B</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, the singular braid group, exist.</p>

Extension of Burau representation to the monoid of singular braids

期刊论文
发表日期：

2026年1月12日

查看全文

Rana S. Kahil

摘要

We construct the linear representation $\hat{\beta _ n}^{d,g,e}: SM_n \rightarrow M_n(\mathbb {Z}[t^{\pm 1},g,d,e])$ ${\hat{β_{n}}}^{d, g, e} : S M_{n} \to M_{n} (Z [t^{\pm 1}, g, d, e])$ , which extends the Burau representation of the braid group $B_n$ $B_{n}$ to the singular braid monoid $SM_n$ $S M_{n}$ . We show that any extension of the Burau representation to $SM_n$ $S M_{n}$ is of the form $\hat{\beta _ n}^{d,g,e}$ ${\hat{β_{n}}}^{d, g, e}$ . We show that $\hat{\beta _ n}^{d,g,e}$ ${\hat{β_{n}}}^{d, g, e}$ is reducible to a reduced representation $\hat{\beta _n^r}: SM_n \rightarrow M_{n-1}(\mathbb {Z}[t^{\pm 1},g,d,e]))$ $\hat{β_{n}^{r}} : S M_{n} \to M_{n - 1} (Z [t^{\pm 1}, g, d, e]))$ . Additionally, we study whether or not extensions of the Burau representation to $SB_n$ $S B_{n}$ , the singular braid group, exist.