[Paper Review] Algebraic Markov equivalence for links in 3-manifolds

Sofia Lambropoulou, Colin Rourke|ArXiv.org|May 26, 2004

Geometric and Algebraic Topology8 references4 citations

TL;DR

This paper establishes an algebraic classification of oriented links in 3-manifolds via extended Markov equivalence in mixed braid groups. It introduces twisted conjugation and combed band moves as algebraic moves on $B_{m,rown}$ that classify links up to isotopy in link complements and closed 3-manifolds, generalizing the classical Markov theorem using $L$-moves and providing a foundation for new invariants and skein module computations.

ABSTRACT

Let $B_n$ denote the classical braid group on $n$ strands and let the {\em mixed braid group} $B_{m,n}$ be the subgroup of $B_{m+n}$ comprising braids for which the first $m$ strands form the identity braid. Let $B_{m,\infty}=\cup_nB_{m,n}$. We will describe explicit algebraic moves on $B_{m,\infty}$ such that equivalence classes under these moves classify oriented links up to isotopy in a link complement or in a closed, connected, oriented 3--manifold. The moves depend on a fixed link representing the manifold in $S^3$. More precisely, for link complements the moves are: the two familiar moves of the classical Markov equivalence together with {\em `twisted' conjugation} by certain loops $a_i$. This means premultiplication by ${a_i}^{-1}$ and postmultiplication by a `combed' version of $a_i$. For closed 3--manifolds there is an additional set of {\it `combed' band moves} which correspond to sliding moves over the surgery link. The main tool in the proofs is the one-move Markov Theorem using {\it $L$--moves} \cite{LR} (adding in-box crossings). The resulting algebraic classification is a direct extension of the classical Markov Theorem that classifies links in $S^3$ up to isotopy, and potentially leads to powerful new link invariants, which have been explored in special cases by the first author. It also provides a controlled range of isotopy moves, useful for studying skein modules of 3--manifolds.

Motivation & Objective

• To generalize the classical Markov theorem to classify oriented links in 3-manifolds, including link complements and closed 3-manifolds, via algebraic braid equivalence.
• To identify the precise algebraic moves—twisted conjugation and combed band moves—that generate isotopy classes of links in these spaces.
• To provide a controlled algebraic framework using $L$-moves and braid group structures for studying skein modules and constructing new link invariants.

Proposed method

• Uses the mixed braid group $B_{m,n}$ and its direct limit $B_{m,rown}$ to represent links in 3-manifolds via mixed braids.
• Applies the one-move Markov Theorem using $L$-moves (adding in-box crossings) as the foundational algebraic move to generate isotopy equivalence.
• Introduces 'twisted conjugation' as a new move: premultiplication by $a_i^{-1}$ and postmultiplication by a 'combed' version of $a_i$, where $a_i$ are loops around surgery components.
• Defines 'combed band moves' for closed 3-manifolds, corresponding to sliding links over surgery components, with explicit algebraic formulations involving framing twists and braiding relations.
• Employs combing techniques via the $\mathcal{BR}$-relations to express band moves and twisted conjugations in terms of standard braid generators and fixed surgery braids.
• Reduces general band moves to a standard form via $L$-moves and loop conjugations, ensuring algebraic tractability and equivalence under $L$-equivalence.

Experimental results

Research questions

• RQ1 What algebraic moves on mixed braids classify oriented links up to isotopy in a link complement within $S^3$?
• RQ2 How can Markov equivalence be extended to classify links in closed, connected, oriented 3-manifolds obtained by surgery on a framed link?
• RQ3 What is the algebraic formulation of sliding a link over a surgery component in a 3-manifold, and how does it relate to braid group relations?
• RQ4 How do twisted conjugation and combed band moves arise naturally from the $L$-move framework in the context of 3-manifold link invariants?
• RQ5 Can the extended Markov equivalence be used to construct new invariants for skein modules of 3-manifolds?

Key findings

• The paper establishes that isotopy classes of links in a link complement $S^3 \setminus \widehat{B}$ are classified by equivalence under braid isotopy, Markov conjugation, and twisted conjugation by loops $a_i$, where the latter involves conjugation by a combed version of $a_i$.
• For closed 3-manifolds obtained by surgery along a framed link $\widehat{B}$, isotopy classes of links are classified by braid isotopy, Markov conjugation, twisted conjugation, and combed band moves corresponding to sliding over surgery components.
• The twisted conjugation moves are explicitly formulated as $a_i^{\pm 1} \leftrightarrow a_i^{\pm 1}$ for $i > k$, and $a_i^{\pm 1} \leftrightarrow \lambda_n (a_j \cdots) \lambda_n^{-1}$ for $i < k$, with $\lambda_n$ encoding the framing and braid structure.
• Combed band moves are shown to be equivalent to standard band moves up to $L$-equivalence and loop conjugation, and are expressed algebraically via substitutions involving $\sigma_{n+1}$, $t_{2,n}$, and combing elements $r_k$.
• For non-pure surgery braids, band moves can be normalized to occur at the top and rightmost position of the surgery component, with algebraic expressions involving $\lambda_n$, $a_i$, and $\sigma$-generators.
• The framework provides a direct extension of the classical Markov Theorem and enables the construction of new algebraic invariants for skein modules of 3-manifolds through controlled isotopy moves.