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Skeins, $q$-series, and modularity

Sunghyuk Park

TL;DR

The paper extends BPS $q$-series to skein modules by inserting line defects, proving a Melvin–Morton–Rozansky type expansion whose leading term factors through the Alexander polynomial and link data, and showing the result depends only on the skein class. It constructs a two-variable $q$-series module $V_{x,q}(S^3\setminus K)$ and proves a surgery-invariance theorem via twisted spin$^c$ structures, enabling a map from SL$_2$ skein modules of complements to $q$-series that respects boundary actions. By relating these series to state-integrals through the non-compact quantum dilogarithm, the work establishes holomorphic quantum modularity with explicit cocycles and automorphy factors, and it discusses a Langlands duality perspective for skein modules via these dual $q$-series. Concrete knot and surgery examples illustrate the module structure, Habiro-type expansions, and descendants, highlighting the analytic structure and potential physical interpretations in Kapustin–Witten theory. The results lay groundwork for a unified framework connecting skein theory, quantum modularity, and Langlands duality in 3-manifold topology, with practical impact on the study of 3-manifold invariants and their categorifications.

Abstract

We study BPS $q$-series associated to 3-manifolds decorated by a line defect along an embedded link. We prove that these $q$-series depend only on the class of the link in the skein module, thereby defining a homomorphism from the skein module to the space of $q$-series. The image of this homomorphism is conjectured to be holomorphically quantum modular, which suggests a new approach to Langlands duality for skein modules through $q$-series.

Skeins, $q$-series, and modularity

TL;DR

The paper extends BPS -series to skein modules by inserting line defects, proving a Melvin–Morton–Rozansky type expansion whose leading term factors through the Alexander polynomial and link data, and showing the result depends only on the skein class. It constructs a two-variable -series module and proves a surgery-invariance theorem via twisted spin structures, enabling a map from SL skein modules of complements to -series that respects boundary actions. By relating these series to state-integrals through the non-compact quantum dilogarithm, the work establishes holomorphic quantum modularity with explicit cocycles and automorphy factors, and it discusses a Langlands duality perspective for skein modules via these dual -series. Concrete knot and surgery examples illustrate the module structure, Habiro-type expansions, and descendants, highlighting the analytic structure and potential physical interpretations in Kapustin–Witten theory. The results lay groundwork for a unified framework connecting skein theory, quantum modularity, and Langlands duality in 3-manifold topology, with practical impact on the study of 3-manifold invariants and their categorifications.

Abstract

We study BPS -series associated to 3-manifolds decorated by a line defect along an embedded link. We prove that these -series depend only on the class of the link in the skein module, thereby defining a homomorphism from the skein module to the space of -series. The image of this homomorphism is conjectured to be holomorphically quantum modular, which suggests a new approach to Langlands duality for skein modules through -series.
Paper Structure (25 sections, 21 theorems, 156 equations, 8 figures, 3 tables)

This paper contains 25 sections, 21 theorems, 156 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Melvin-Morton-Rozansky with skeins
  3. Brief review of the finite state sum model
  4. Proof of Theorem \ref{['mainthm:MMR']}
  5. Proof of Lemma \ref{['lem:perturbed-random-walk']}
  6. From skeins to $q$-series
  7. Brief review of inverted state sum
  8. Proof of Theorem \ref{['mainthm:skein-q-series']}
  9. Boundary action
  10. Examples
  11. Surgery
  12. Twisted spin$^c$ structures
  13. Proof of Theorem \ref{['mainthm:surgery']}
  14. $R_{\chi}$-module structure vs surgery
  15. Examples
  16. ...and 10 more sections

Key Result

Theorem A

In the large-color asymptotic where $\hbar \rightarrow 0$, $n\rightarrow \infty$ while $u := n\hbar$ is fixed, where $x := e^u$, $\chi_m(z) := \frac{z^{\frac{m}{2}}-z^{-\frac{m}{2}}}{z^{\frac{1}{2}}-z^{-\frac{1}{2}}}$, $\Delta_K(x)$ denotes the Alexander polynomial of $K$, and $P_d(x) \in \mathbb{Q}[x^{\pm \frac{1}{2}}]$ are some polynomials determined by $K$ and $L$.The right-hand side should be

Figures (8)

  • Figure 1: $K \sqcup L$ as a colored $(1, 1)$-tangle
  • Figure 2: Highway-and-cars diagram
  • Figure 3: Random walk (Markov chain) example; $a_1, \cdots, a_{11}$ label the internal arcs of $K$ minus the local maxima and minima, $c_1, \cdots, c_7$ label the crossings, and $s_1, \cdots, s_4$ denote the spins on arcs of $L$.
  • Figure 4: The trefoil knot with Wilson line defect $W_n$
  • Figure 5: The figure-eight knot with Wilson line defect $W_n$
  • ...and 3 more figures

Theorems & Definitions (49)

  • Theorem A
  • Remark 1
  • Theorem B
  • Remark 2: Grading
  • Theorem C
  • Theorem D
  • Lemma 1.1
  • proof : Proof of Lemma \ref{['lem:truncation-bound']}
  • Lemma 1.2
  • Lemma 1.3
  • ...and 39 more