Exact integral formulas for volumes of two-bridge knot cone-manifolds
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914586205093888 |
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| author | Tran, Anh T. Yadav, Nisha |
| author_facet | Tran, Anh T. Yadav, Nisha |
| contents | 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 $C(2n,-2n)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_02095 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exact integral formulas for volumes of two-bridge knot cone-manifolds Tran, Anh T. Yadav, Nisha Geometric Topology Primary 57K10, 57K32 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 $C(2n,-2n)$ 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. |
| title | Exact integral formulas for volumes of two-bridge knot cone-manifolds |
| topic | Geometric Topology Primary 57K10, 57K32 |
| url | https://arxiv.org/abs/2510.02095 |