Studying Links via Plats

Between 1990 and 2006, Birman and Menasco wrote a sequence of papers with the running major title “Stabilisation in the braid groups”, that investigated the question of what stabilisation accomplishes in Markov’s theorem. One key result coming out of that sequence of papers was the discovery of a new isotopy --  the exchange move – which also preserves braid index and writhe, but could jump between conjugacy classes. Thus, it was established that using just a sequence of braid isotopies, exchange moves and destabilisations, one could go from any closed braid representative of the unlink to the representative of minimal index – the trivial closed braid. It was also shown that using just braid isotopies and exchange moves, one could take a closed braid representing a split or composite link to one where it was “obviously” split or composite, respectively.

In the setting of plat presentations of links, we now ask the same question: what does stabilisation accomplish? We introduce two new isotopy moves that preserve the bridge index: the pocket move and the flip move, which can be thought of as being analogous to an exchange move of the closed braid setting. Equipped with these isotopies, we prove results analogous to Birman-Menasco’s results, in the plat setting.