In this paper we study the HOMFLYPT skein module of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$S^1 \times S^2\, \cong \, L(0, 1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>S</mi> <mn>1</mn> </msup> <mo>×</mo> <msup> <mi>S</mi> <mn>2</mn> </msup> <mspace width="0.166667em" /> <mo>≅</mo> <mspace width="0.166667em" /> <mi>L</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, denoted <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {S}(S^1 \times S^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">S</mi> <mo stretchy="false">(</mo> <msup> <mi>S</mi> <mn>1</mn> </msup> <mo>×</mo> <msup> <mi>S</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, using braid-theoretic techniques. We extend the Lambropoulou invariant, X, for links in the solid torus ST to links in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$S^1 \times S^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>S</mi> <mn>1</mn> </msup> <mo>×</mo> <msup> <mi>S</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, by solving an infinite system of equations of the form <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$X_{\widehat{a}} = X_{\widehat{bbm(a)}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>X</mi> <mover accent="true"> <mi>a</mi> <mo stretchy="true">^</mo> </mover> </msub> <mo>=</mo> <msub> <mi>X</mi> <mover accent="true"> <mrow> <mi>b</mi> <mi>b</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="true">^</mo> </mover> </msub> </mrow> </math></EquationSource> </InlineEquation>, where bbm(a) denotes all possible band moves applied to a, for all a in a basis of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {S}(ST)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">S</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We show that the free part of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathcal {S}(S^1 \times S^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">S</mi> <mo stretchy="false">(</mo> <msup> <mi>S</mi> <mn>1</mn> </msup> <mo>×</mo> <msup> <mi>S</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is generated by the empty link, while all other elements lies in the torsion submodule.

On the HOMFLYPT skein module of $S^1 \times S^2$ via braids

期刊论文
发表日期：

2025年12月23日

查看全文

Ioannis Diamantis

摘要

In this paper we study the HOMFLYPT skein module of $S^1 \times S^2\, \cong \, L(0, 1)$ $S^{1} \times S^{2} ≅ L (0, 1)$ , denoted $\mathcal {S}(S^1 \times S^2)$ $S (S^{1} \times S^{2})$ , using braid-theoretic techniques. We extend the Lambropoulou invariant, X, for links in the solid torus ST to links in $S^1 \times S^2$ $S^{1} \times S^{2}$ , by solving an infinite system of equations of the form $X_{\widehat{a}} = X_{\widehat{bbm(a)}}$ $X_{\hat{a}} = X_{\hat{b b m (a)}}$ , where bbm(a) denotes all possible band moves applied to a, for all a in a basis of $\mathcal {S}(ST)$ $S (S T)$ . We show that the free part of $\mathcal {S}(S^1 \times S^2)$ $S (S^{1} \times S^{2})$ is generated by the empty link, while all other elements lies in the torsion submodule.

On the HOMFLYPT skein module of \(S^1 \times S^2\) via braids

摘要