On the Quantum Modularity Conjecture for Knots

Quantum topology is considered to be initiated by the discovery of the Jones polynomial in 1984,  followed with observations of numerous links to physics. In the late '80s,  Atiyah,  Segal,  and Witten established an intrinsic definition of the Jones polynomial using SU(2) Chern-Simons theory,  revealing the rich connections of the Jones polynomial with the physical world. Successive findings around the Jones polynomial emerged,  including one famous conjecture that is the main topic of this thesis,  the Quantum Modularity Conjecture. 

In 1995,  R. Kashaev introduced a knot invariant using the quantum dilogarithm function,  which for a hyperbolic knot K is conjectured to have an exponential growth rate,  a conjecture known as the Volume Conjecture. In 2001,  H. Murakami and J. Murakami discovered that Kashaev's invariant is equal to the value of the N-colored Jones polynomial at N-th roots of the unity. With this,  D. Zagier observed a modular relation between the values of the N-colored Jones polynomial at different roots of the unity and extended the statement of Volume Conjecture to a modular relation of the functions. The extended statement is known as the Quantum Modularity Conjecture  (QMC).

More recently,  J. E. Andersen and R. Kashaev introduced the Teichmuller TQFT based on Chern-Simons theory with infinite dimensional gauge groups,  promoting the quantum Teichmuller theory to a TQFT of categroids. On further investigation into values of the Teichmuller TQFT on knot complements of the 4-1 knot and the 5-2 knot,  S. Garoufalidis and D. Zagier discovered phenomena suggesting deep relationships between the state integral from the Teichmuller TQFT and QMC. Furthermore,  their observation also suggested rich connections with several other topics,  such as the Dimofte-Gaiotto-Gukov index and the quantum spin network.

This thesis will mainly focus on introducing the construction of the Teichmuller TQFT and the contents of QMC,  and demonstrate their connections by listing the observations made by S. Garoufalidis and D. Zagier and more recent results along with elementary proofs for some of them from joint work of the author,  N. An and S. Garoufalidis.

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