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Fast Algorithm for Multiplication on the Skein Algebra of One-hole Torus

Sike Wang, Helen Wong

TL;DR

This work tackles the problem of multiplying elements in the Kauffman bracket skein algebra of the one-hole torus, where naive crossing-resolution is exponential. It introduces a five-term recursive discrepancy $D{p}{r}{q}{s}$ that measures deviation from the closed-torus Product-to-Sum formula and then leverages Dehn-twist reductions and symmetry operations to reduce computations to a polynomial-time algorithm. The main contribution is a comprehensive, polynomial-time framework with complexity $O(prqs^6)$ for the discrepancy (and hence for multiplication), along with closed-form formulas for selected cases. These results advance practical computation in quantum topology and enhance understanding of the multiplicative structure of $\\mathcal{S}^A(\\Sigma_{1,1})$, tying into the broader interplay between skein algebras, mapping class groups, and quantum invariants.

Abstract

The Kauffman bracket skein algebra of a surface is a generalization of the Jones polynomial invariant for links and plays a principal role in the Witten-Reshetikhin- Turaev topological quantum field theory. However, the multiplicative structure of the skein algebra is not well understood, with a priori exponential complexity. We consider the case of one-hole torus, and provide a polynomial algorithm for computing multiplication of any two skein elements. Some closed form formulas for multiplication of curves with low crossing number are also given.

Fast Algorithm for Multiplication on the Skein Algebra of One-hole Torus

TL;DR

This work tackles the problem of multiplying elements in the Kauffman bracket skein algebra of the one-hole torus, where naive crossing-resolution is exponential. It introduces a five-term recursive discrepancy that measures deviation from the closed-torus Product-to-Sum formula and then leverages Dehn-twist reductions and symmetry operations to reduce computations to a polynomial-time algorithm. The main contribution is a comprehensive, polynomial-time framework with complexity for the discrepancy (and hence for multiplication), along with closed-form formulas for selected cases. These results advance practical computation in quantum topology and enhance understanding of the multiplicative structure of , tying into the broader interplay between skein algebras, mapping class groups, and quantum invariants.

Abstract

The Kauffman bracket skein algebra of a surface is a generalization of the Jones polynomial invariant for links and plays a principal role in the Witten-Reshetikhin- Turaev topological quantum field theory. However, the multiplicative structure of the skein algebra is not well understood, with a priori exponential complexity. We consider the case of one-hole torus, and provide a polynomial algorithm for computing multiplication of any two skein elements. Some closed form formulas for multiplication of curves with low crossing number are also given.
Paper Structure (8 sections, 19 theorems, 38 equations, 3 algorithms)

This paper contains 8 sections, 19 theorems, 38 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background and Notation
  3. The Recursion Rule and transformations on the one-hole torus
  4. The Recursion Rule
  5. Reductions under Dehn twists
  6. Reducing discrepancy using symmetries
  7. Some example calculations
  8. Fast Algorithm for computing the Discrepancy

Key Result

Theorem 1.1

There is an algorithm to compute the product of two multicurves $m, m'$ in $\mathcal{S}^A(\Sigma_{1,1})$ in $\mathcal{O} (i(m,m')^6)$ steps, where $i(m, m')$ is the intersection number of $m$ and $m'$.

Theorems & Definitions (40)

  • Theorem 1.1
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Example 2.6
  • Lemma 2.7
  • Lemma 2.8
  • ...and 30 more