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<p>We provide exact integral formulas for hyperbolic and spherical volumes of cone-manifolds whose underlying space is the 3-sphere and whose singular set belongs to three infinite families of two-bridge knots: <i>C</i>(2<i>n</i>,&#xa0;2) (twist knots), <i>C</i>(2<i>n</i>,&#xa0;3), and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(C(2n,-2n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>,</mo> <mo>-</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for any non-zero integer <i>n</i>. Our formulas express volumes as integrals of explicit rational functions involving Chebyshev polynomials of the second kind, with integration limits determined by roots of algebraic equations. This extends previous work where only implicit formulas requiring numerical approximation were known.</p>

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Exact Integral Formulas for Volumes of Two-Bridge Knot Cone-Manifolds

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月25日

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  • Anh T. Tran,
  • Nisha Yadav

摘要

We provide exact integral formulas for hyperbolic and spherical volumes of cone-manifolds whose underlying space is the 3-sphere and whose singular set belongs to three infinite families of two-bridge knots: C(2n, 2) (twist knots), C(2n, 3), and \(C(2n,-2n)\) C ( 2 n , - 2 n ) for any non-zero integer n. Our formulas express volumes as integrals of explicit rational functions involving Chebyshev polynomials of the second kind, with integration limits determined by roots of algebraic equations. This extends previous work where only implicit formulas requiring numerical approximation were known.