We calculate Jones polynomials <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3531_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$V(H_r,t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mo stretchy="false">(</mo> <msub> <mi>H</mi> <mi>r</mi> </msub> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for a family of alternating knots and links <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3531_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$H_r$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mi>r</mi> </msub> </math></EquationSource> </InlineEquation> with arbitrarily many crossings r, by computing the Tutte polynomials <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3531_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="115" /> </InlineMediaObject> <EquationSource Format="TEX">$T(G_+(H_r),x,y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>G</mi> <mo>+</mo> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>H</mi> <mi>r</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for the associated graphs <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3531_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$G_+(H_r)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>G</mi> <mo>+</mo> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>H</mi> <mi>r</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and evaluating these with <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3531_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$x=-t$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>=</mo> <mo>-</mo> <mi>t</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3531_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$y=-1/t$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>y</mi> <mo>=</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mi>t</mi> </mrow> </math></EquationSource> </InlineEquation>. Our method enables us to circumvent the generic feature that the computational complexity of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3531_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$V(L_r,t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mo stretchy="false">(</mo> <msub> <mi>L</mi> <mi>r</mi> </msub> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for a knot or link <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3531_Article_IEq8.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$L_r$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>r</mi> </msub> </math></EquationSource> </InlineEquation> for generic t grows exponentially rapidly with r. We also study the accumulation set of the zeros of these polynomials in the limit of infinitely many crossings, <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3531_Article_IEq9.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$r \rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>.

Jones Polynomials and their Zeros for a Family of Knots and Links

期刊论文
发表日期：

2025年10月25日

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Yue Chen,
Robert Shrock

摘要

We calculate Jones polynomials $V(H_r,t)$ $V (H_{r}, t)$ for a family of alternating knots and links $H_r$ $H_{r}$ with arbitrarily many crossings r, by computing the Tutte polynomials $T(G_+(H_r),x,y)$ $T (G_{+} (H_{r}), x, y)$ for the associated graphs $G_+(H_r)$ $G_{+} (H_{r})$ and evaluating these with $x=-t$ $x = - t$ and $y=-1/t$ $y = - 1 / t$ . Our method enables us to circumvent the generic feature that the computational complexity of $V(L_r,t)$ $V (L_{r}, t)$ for a knot or link $L_r$ $L_{r}$ for generic t grows exponentially rapidly with r. We also study the accumulation set of the zeros of these polynomials in the limit of infinitely many crossings, $r \rightarrow \infty $ $r \to \infty$ .