This book is the result of research initiatives formed during the workshop "Low Dimensional Topology and Number Theory XIII" at Kyushu University in 2022. It is also dedicated to the memory of Professor Toshie Takata, who has been a main figure of the session chairs for the series of annual workshops since 2009.

The activity was aimed at understanding and deepening recent developments of lively and fruitful interactions between low-dimensional topology and number theory over the past decades.

In this volume of proceedings, the reader will find research papers as well as survey articles, including open problems, at the interface between classical and quantum topology, and algebraic and analytic number theory, written by leading experts and active researchers in the respective fields.

Topics include, among others, the strong slope conjecture; Kashiwara-Vergne Lie algebra; braids and bered double branched covers of 3-manifolds; Temperley-Lieb-Jones category and conformal blocks; WRT invariants and false theta functions; the colored Jones polynomial of the gure-eight knot; potential functions and A-polynomials; l-adic Galois polylogarithms; Dijkgraaf-Witten invariants in Bloch groups; analogies between knots and primes in arithmetic topology; normalized Jones polynomials for rational links; Iwasawa main conjecture; Weber's class number problem.

The book provides a valuable resource for researchers and graduate students interested in topics related to both low-dimensional topology and number theory.

Autorentext

Masanori Morishita is professor of mathematics at Kyushu University, Fukuoka Japan.

He is one of the primary pioneers who established "Arithmetic Topology"-- a new branch of mathematics which is focused upon the analogy between knot theory and number theory. He authored the first systematic treatment of the subject in the book "Knots and Primes" (Universitext) published from Springer in 2012. Since 2009, he has organized a series of international annual meetings "Low dimensional topology and number theory" that enhances the community of mathematicians in the world who contribute to the active frontiers of the promising area interacting with topology and number theory.

Hiroaki Nakamura is professor of mathematics at Osaka University, Osaka Japan.

He is a world-leading figure in anabelian geometry and Galois-Teichmüller theory in arithmetic algebraic geometry. He is known as the first person who made a break-through on Grothendieck's conjecture in anabelian geometry by solving it in the case of genus 0, and he was awarded Autumn Prize of the Mathematical Society of Japan.

His outstanding contributions to mathematics are cross over number theory, algebraic geometry and topology. He is also an organizer of the international annual meetings "Low dimensional topology and number theory" and is enrolled in the scientific committee of "LPP-RIMS Arithmetic and Homotopic Galois Theory"-- CNRS France-Japan International Research Network.

Jun Ueki is a senior lecturer of mathematics at Ochanomizu University, Tokyo Japan. He is an active researcher, who is leading the young generation, in arithmetic topology. He made a pioneering contribution on a topological idelic theory for 3-manifolds, and his notable works range over arithmetic topology of branched covers of 3-manifolds in connection with Iwasawa theory, the profinite rigidity of twisted Alexander invariants, and modular knots.

He is also an organizer of the international annual meetings "Low dimensional topology and number theory".