Exact integral formulas for volumes of two-bridge knot cone-manifolds

TL;DR

This work derives exact integral formulas for the hyperbolic and spherical volumes of cone-manifolds on S^3 whose singular loci are two-bridge knots from three infinite families: C(2n, 2), C(2n, 3), and C(2n, -2n). By leveraging nonabelian SL_2(C) representations, Riley-type polynomials, and Chebyshev polynomials of the second kind, the authors reduce holonomy conditions to algebraic relations f_n^2(y) + A^2 = (1 + A^2) g_n(y) (with A = cot(α/2)) and then express volumes as explicit integrals. The resulting formulas give precise hyperbolic and spherical volumes in terms of integrals of rational functions with integration limits determined by roots of the Riley-type equations, extending prior implicit or numerical approaches and generalizing Mednykh's exact results to new infinite knot families. The paper thus clarifies the link between knot group representations, special polynomials, and geometric transitions in cone-manifold volumes, providing practical computable expressions for a broad class of two-bridge knot cone-manifolds.

Abstract

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.

Figures (3)

• Figure 1: The rational knot $C(a_1,a_2, \ldots,a_k)$. The upper/lower one corresponds to odd/even $k$.
• Figure 2: $C(2n,2p+1)$
• Figure 3: $C(2n,2p)$

Theorems & Definitions (29)

• Theorem 1
• Theorem 2
• Proposition 3.1
• proof
• Lemma 3.2
• Lemma 3.3
• Proposition 3.4
• proof
• Proposition 3.5
• proof