A Dehornoy-Type Ordering on Bridge Positions of Links

Abstract

By a theorem of Hovland, bridge isotopy classes of $n$-bridge positions of links in $S^3$ correspond bijectively to Hilden double cosets $P_{2n}=H_{2n}\backslash B_{2n}/H_{2n}$. We use this correspondence to introduce Dehornoy-type orderings on bridge positions. After fixing a proper complexity function on $B_{2n}$, we assign to each double coset its Dehornoy-minimal braid among the elements of minimal complexity. Comparing these representatives by the Dehornoy order gives a strict total order ${\prec }_{D,n}$, and comparing them first by complexity and then by the Dehornoy order gives a strict well-order ${◃}_{D,n}$. For a link type $L$ and $n\ge b(L)$, we single out a canonical $n$-bridge position ${\mathrm{CanPlat}}_D^{(n)}(L)$: it is the ${◃}_{D,n}$-least element of the set of $n$-bridge positions of $L$, and the ${\prec }_{D,n}$-least element among those of minimal complexity, and its canonical braid coincides with the globally Dehornoy-minimal braid of minimal complexity representing $L$. We show by an example that, in general, it need not be least for ${\prec }_{D,n}$ on the whole set. Finally, we show that the fixed-level bridge finiteness conjecture is equivalent to boundedness of the complexities of the canonical representatives, and to the existence of a ${◃}_{D,n}$-greatest $n$-bridge position.

arXiv comment11 pages. Substantially revised version. Corrected the false minimality assertion in the original Theorem 4.5 by introducing a complexity-refined Dehornoy well-order. Added an explicit counterexample to the original claim and updated the proofs and references