A Dehornoy-Type Ordering on Plat Presentation Classes

TL;DR

The paper introduces a Dehornoy-type order on plat presentation classes of links, providing a new structured approach to studying bridge positions and their canonical representatives in knot theory.

Contribution

It defines a total order on plat presentation classes using the Dehornoy order and relates canonical representatives to bridge positions, offering a new perspective on link positions.

Findings

Established a strict total order on plat presentation classes.

Connected canonical representatives with bridge positions.

Reformulated the fixed-level bridge finiteness conjecture.

Abstract

For each integer $n\ge 1$, after fixing a proper complexity function on the braid group $\B_{2n}$, we use the Dehornoy order to define a strict total order on the set \[ \mathcal P_{2n}=H_{2n}\backslash \B_{2n}/H_{2n} \] of $2n$--plat presentation classes. For a link type $\mathcal L$ with bridge number $b(\mathcal L)\le n$, this induces a strict total order on the subset $\mathcal P^{(n)}(\mathcal L)$ corresponding to bridge isotopy classes of $n$--bridge positions of $\mathcal L$. We also define a distinguished class $\CanPlat_D^{(n)}(\mathcal L)$ and show that the globally chosen Dehornoy canonical braid agrees with the cosetwise canonical representative of the associated Hilden double coset. As an application, we reformulate the fixed-level bridge finiteness conjecture in terms of boundedness of canonical representatives. This viewpoint supports the role of bridge positions as a structured finite-level model for studying the otherwise vast collection of geometric positions of a link.