Markov conjugation and stabilization moves

Let $(B_n{)}_{n\ge 0}$ be the braid groups of the Artin presentation (The braid group by Artin presentation), with generators ${\sigma }_1,\dots ,{\sigma }_{n-1}$ and the two Artin relations, and let

${\iota }_n\text{\hspace{0.17em}⁣}:B_n⟶B_{n+1},{\iota }_n({\sigma }_i):={\sigma }_i(1\le i\le n-1),$

be the standard inclusion homomorphism. It is well defined: the defining relators of $B_n$ are among the defining relators of $B_{n+1}$ under the assignment ${\sigma }_i\mapsto {\sigma }_i$, so Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group extends the assignment to a homomorphism. Its image is generated by ${\sigma }_1,\dots ,{\sigma }_{n-1}$, and we write ${\iota }_n(\beta )$ for it; whether this homomorphism is injective is not needed anywhere on this page, and no subgroup identification is made. For the geometric description, use the base configuration $Q^{(n+1)}$ of The elementary geometric half twist, its support disc, and its opposite. Express $\beta$ as an Artin word and replace its letters by the half twists with the same indices at $Q^{(n+1)}$. The letters with $i<n$ fix the rightmost point, since their support discs exclude it. This gives a representative of ${\iota }_n(\beta )$ under The Artin presentation surjects onto the geometric braid group; it is independent of the word as a geometric braid class. Thus adding the strand on the right includes expressing the old braid at the first $n$ target basepoints. Those points differ from $Q^{(n)}$, so the original parametrized strands are not literally left untouched. The final half twist is the target ${\sigma }_n$.

On the disjoint union ${⨆}_{n\ge 0}{B}_n$ define two kinds of elementary Markov moves.

Conjugation. For $\beta ,\gamma \in B_n$, replace $\beta$ by $\gamma \beta {\gamma }^{-1}$, an element of the same group $B_n$. The number of strands does not change. When $\beta$ is a geometric braid and $\gamma$ the class of a braid in $G_n$, this is realized on closures by reading $\gamma$ around the axis, as the closure-preservation lemma below shows.

Stabilization and destabilization. For $n\ge 1$ and $\beta \in B_n$ the positive stabilization is $\beta {\sigma }_n:={\iota }_n(\beta )\cdot {\sigma }_n\in B_{n+1}$ and the negative stabilization is $\beta {\sigma }_n^{-1}:={\iota }_n(\beta )\cdot {\sigma }_n^{-1}\in B_{n+1}$; the inverse operations, from $B_{n+1}$ to $B_n$, are called destabilizations. Both signs occur, and the sign is the sign of the exponent of the new generator. By The Artin presentation surjects onto the geometric braid group and The elementary geometric half twist, its support disc, and its opposite the product $\beta {\sigma }_n^{\pm 1}$ is the class of the target-basepoint geometric word just described, composed with the target half twist ${\sigma }_n^{\pm 1}$ in its fixed product position; that half twist fixes the first $n-1$ target points. Inserting the generator inside a factorization need not produce a conjugate of that fixed stabilization. It nevertheless gives a Markov sequence: if $\beta =uv$, old conjugation by $v$ gives $vu$, its right stabilization is $(vu){\sigma }_n^{\pm 1}$, and conjugation by $v^{-1}$ gives $u{\sigma }_n^{\pm 1}v$. Only moving a factor past the whole word is cyclic conjugacy. At $n=0$ no ${\sigma }_0$ exists, so the empty braid has only conjugation moves and is an isolated Markov class.

Markov equivalence. Two braids $\beta ,{\beta }^{\mathrm{\prime }}$ (possibly with different numbers of strands) are Markov equivalent when they can be connected by a finite sequence of moves each of which is a conjugation, a positive or negative stabilization, or a destabilization. Markov equivalence is an equivalence relation on ${⨆}_n{B}_n$: it is reflexive, symmetric by allowing inverses of the listed moves, and transitive by concatenating sequences.

Convention on orientation and side. All braids are read with the product and side conventions fixed on the pages geometric-braids-and-artin-generators and artin-presentation-completeness-and-braid-combing: the rightmost factor is traversed first in geometric time, and stabilization always adds the new strand on the right. The definition is choice-free: no choice principle is used in the Artin presentation, in the inclusion, or in the identification of ${\sigma }_n$ with the elementary geometric half twist.