Exact Variance and Refined Limit Laws for Two-Bridge Signatures
Overview
Two-bridge knots can be drawn with two overpasses; their signature is an integer knot invariant. In the word model used by Cohen, Baker, Dam, Felber, Madras, Saha and Thackrah, some knots are counted twice. The paper determines the exact signature variance at every crossing number c and proves that the variance minus c tends to -14/3, answering their numerical Conjecture 5.1 in arXiv 2604.21107. It also gives increasingly accurate approximations for individual signature probabilities and for rare large signatures. For uniform knot classes identified with their mirror images, choosing the sign independently and fairly gives a different symmetric model; the paper derives its parity-dependent variance and probability corrections while retaining the exact sampling weights.
Original public questions
Conjecture 5.1: signature variance of two-bridge knots
Moshe Cohen, Cody Baker, Henry Dam, Rebecca Felber, Neal Madras, Ritvik Saha and Daisy Thackrah: A central limit theorem for the signatures of 2-bridge knots — Conjecture 5.1, arXiv v1; exact refinement of the numerical variance question. Theorem 1.2 and Section 4 supply the prior signature counts and mixture method.
Moshe Cohen, Adam M. Lowrance, Neal Madras and Steven Raanes: Average signature and 4-genus of 2-bridge knots — arXiv v2; word and mirror-class sampling conventions and palindrome enumeration.
Original abstract (English)
The signature counts of two-bridge knots admit exact binomial formulas, and their normalized distributions satisfy a central limit theorem. We use these formulas to determine the finite-crossing-number mean and variance in the partially double-counted word model. In particular, the variance minus the crossing number converges to -14/3, giving an exact answer to Conjecture 5.1, the numerical variance question in A central limit theorem for the signatures of 2-bridge knots by Cohen, Baker, Dam, Felber, Madras, Saha and Thackrah (arXiv:2604.21107). A single analytic factor determines every fixed-order cumulant correction and a uniform local expansion to any fixed order. We give its first correction explicitly, together with a relative binomial approximation on the full support, sharp interior large deviations and endpoint asymptotics. For uniform mirror classes, an independent fair choice of sign gives a canonical symmetric distribution. We prove its parity-dependent variance and local corrections, with the sampling normalization retained exactly, and obtain the same large-deviation rate for both models.
Mathematical review
Paper Close marks a working paper prepared and internally reviewed before release. Please treat it as an ordinary scholarly preprint: further checking or discussion may lead to corrections. It does not indicate complete formal verification or external peer review.
Review standard