A Dehornoy-Type Ordering on Plat Presentation Classes

For each integer $n\ge 1$, after fixing a proper complexity function on the braid group ${\text{B}}_{2n}$, we use the Dehornoy order to define a strict total order on the set $${\mathcal{P}}_{2n}=H_{2n}\mathrm{\backslash }{\text{B}}_{2n}\mathrm{/}{H}_{2n}$$ 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 ${\text{CanPlat}}_D^{(n)}(\mathcal{L})$ 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.