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<p>In this paper, we introduce a two-variable polynomial invariant <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1969_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">\(FP_D(t,s)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <msub> <mi>P</mi> <mi>D</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for virtual knots. It is defined via 1-smoothing at all classical crossings and using the polynomial <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1969_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(P_{\overline{D}}(s)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>P</mi> <mover> <mi>D</mi> <mo>¯</mo> </mover> </msub> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of flat virtual knots. We further introduce several properties of this polynomial and provide an example to illustrate its applications.</p>

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A two-variable polynomial of virtual knots via 1-smoothing

  • 期刊论文

  • 发表日期：

    2025年9月8日

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  • Liang Liang,
  • Jiaqi Lei

摘要

In this paper, we introduce a two-variable polynomial invariant \(FP_D(t,s)\) F P D ( t , s ) for virtual knots. It is defined via 1-smoothing at all classical crossings and using the polynomial \(P_{\overline{D}}(s)\) P D ¯ ( s ) of flat virtual knots. We further introduce several properties of this polynomial and provide an example to illustrate its applications.