[Paper Review] Skein algebras of surfaces

Józef H. Przytycki, Adam S. Sikora|arXiv (Cornell University)|Feb 24, 2016

Geometric and Algebraic Topology26 references3 citations

TL;DR

This paper establishes that the Kauffman bracket skein algebra of any oriented surface with marked boundary points is free of zero divisors and has a center generated by knots parallel to unmarked boundary components. Using filtrations induced by pants decompositions, the authors prove the skein algebra is Noetherian and Ore, with key results derived from analyzing the associated graded algebras and Dehn-Thurston coordinates on curves.

ABSTRACT

We show that the Kauffman bracket skein algebra of any oriented surface F (possibly with marked points in its boundary) has no zero divisors and that its center is generated by knots parallel to the unmarked components of the boundary of F. Furthermore, we show that skein algebras are Noetherian and Ore. Our proofs rely on certain filtrations of skein algebras induced by pants decompositions of surfaces. We prove some basic algebraic properties of the associated graded algebras along the way.

Motivation & Objective

• To establish fundamental algebraic properties of skein algebras of surfaces, including absence of zero divisors and structure of the center.
• To show that skein algebras are Noetherian and satisfy the Ore condition, ensuring good ring-theoretic behavior.
• To characterize the center of the reduced skein algebra as generated by knots parallel to unmarked boundary components.
• To develop a filtration on skein algebras via pants decompositions to analyze the structure of the associated graded algebras.
• To prove that skein algebras embed into quantum Teichmüller spaces and relate to TQFT and AJ-conjecture.

Proposed method

• The authors use a filtration on the skein algebra induced by a pants decomposition of the surface, which allows them to analyze the associated graded algebra.
• They define Dehn-Thurston coordinates for curves and use them to parametrize skein elements via intersection and twist numbers.
• The proof relies on analyzing the multiplication of links via concatenation in annuli and pairs of pants, using skein relations and recursive decomposition.
• They define the reduced skein algebra as a quotient by the ideal generated by arcs parallel to the boundary, simplifying the structure for center analysis.
• The key technical tool is the map rΩ, which tracks how multiplication by a fixed link Ω affects the coordinates of a curve, and proving it is injective via monomial leading term analysis.
• The injectivity of rΩ and lΩ (left multiplication) is established using lexicographic monomial ordering and properties of Laurent polynomials in A.

Experimental results

Research questions

• RQ1 What is the structure of the center of the skein algebra of a surface with marked boundary points?
• RQ2 Does the skein algebra of a surface have zero divisors, and under what conditions is it a domain?
• RQ3 How do pants decompositions and Dehn-Thurston coordinates help in analyzing the skein algebra’s algebraic structure?
• RQ4 Is the skein algebra Noetherian and does it satisfy the Ore condition, ensuring it admits a ring of fractions?
• RQ5 Can the center of the reduced skein algebra be explicitly described in terms of boundary-parallel curves?

Key findings

• The skein algebra of any marked surface (F,B) ≠ (S¹×I, ∅) has no zero divisors if A⁴ⁿ − 1 is not a zero divisor in R for all n > 0.
• The center of the reduced skein algebra RS(F,B) is the R-algebra of polynomials in the knots Kγ for unmarked boundary components γ of F.
• The skein algebra S(F,B) is both Noetherian and an Ore domain, meaning it admits a classical ring of fractions.
• The map rΩ: R[x₁,…,xᵤ,xᵤ₊₁±¹,…,x_d₊ₛ±¹] → R[x₁,…,xᵤ,xᵤ₊₁±¹,…,x_d₊ₛ±¹] induced by multiplication by a fixed link Ω is injective, proven via monomial leading term analysis.
• The injectivity of rΩ and lΩ (left multiplication) relies on the fact that the leading term of rΩ(w) matches that of w up to a power of A, ensuring injectivity.
• The structure of the associated graded algebra under the pants filtration allows the authors to reduce the problem to analyzing coordinate changes in Dehn-Thurston parameters.