Asymptotics for the Number of Simple (4a + 1)-knots of Genus 1

DOI10.1093/IMRN/RNAA315zbMATH Open1503.57003arXiv1905.04369OpenAlexW3112624723MaRDI QIDQ5075187

Publication date: 10 May 2022

Published in: IMRN. International Mathematics Research Notices (Search for Journal in Brave)

Abstract: We investigate the asymptotics of the total number of simple

4 a + 1

-knots with Alexander polynomial of the form

m t^{2} + (1 - 2 m) t + m

for some

m i n [- X, X]

. Using Kearton and Levine's classification of simple knots, we give equivalent algebraic and arithmetic formulations of this counting question. In particular, this count is the same as the total number of

m a t h b b Z [1 / m]

-equivalence classes of binary quadratic forms of discriminant

1 - 4 m

, for

m

running through the same range. Our heuristics, based on the Cohen-Lenstra heuristics, suggest that this total is asymptotic to

X^{3 / 2} / l o g X

, and the largest contribution comes from the values of

m

that are positive primes. Using sieve methods, we prove that the contribution to the total coming from

m

prime is bounded above by

O (X^{3 / 2} / l o g X)

, and that the total itself is

o (X^{3 / 2})

.

Full work available at URL: https://arxiv.org/abs/1905.04369

zbMATH Keywords

quadratic forms sieve Alexander polynomials simple knots

Mathematics Subject Classification ID

Asymptotic results on counting functions for algebraic and topological structures (11N45) Applications of sieve methods (11N36) Quadratic forms (reduction theory, extreme forms, etc.) (11H55) Knot theory (57K10)