Markov's theorem for braid closures

Markov equivalence implies isotopy of closures. Let $\beta$ and ${\beta }^{\mathrm{\prime }}$ be Markov equivalent, so there is a finite sequence of braids $\beta ={\beta }_0,{\beta }_1,\dots ,{\beta }_k={\beta }^{\mathrm{\prime }}$ in which each consecutive pair is related by a conjugation, a stabilization or a destabilization. By [F1] each single step preserves the oriented closure up to ambient isotopy; composing the finitely many ambient isotopies gives an equivalence of $\widehat{\beta }$ and $\widehat{{\beta }^{\mathrm{\prime }}}$.

Isotopy of closures implies Markov equivalence. Conversely, assume $\widehat{\beta }$ and $\widehat{{\beta }^{\mathrm{\prime }}}$ are equivalent oriented links. If either is empty, both are empty and [F5] forces both braid strand counts to be zero; they are the unique element of $B_0$ and hence Markov equivalent. Otherwise both strand counts are positive. Use the closed braid diagrams obtained from the closure construction applied to $\beta$ and ${\beta }^{\mathrm{\prime }}$, and read each diagram at its original cutting ray. The braids read this way are $\beta$ and ${\beta }^{\mathrm{\prime }}$ themselves, up to braid isotopy. Apply [F3] to these two diagrams to obtain their Markov equivalence. If a different cutting ray is used, [F2] gives a conjugate braid, which lies in the same Markov class. Thus $\beta$ and ${\beta }^{\mathrm{\prime }}$ are Markov equivalent.

Conclusion. Steps 1.1 and 1.2 give the equivalence. AC discharges the countable-choice hypothesis of [F1] and supplies the AC hypotheses of the factorization result [F3] and the conjugacy bridge [F2]. The Reidemeister theorem [F4] is used inside [F3], rather than to identify a braid read at its original cutting ray.