nLab plat closure

Idea

The plat closure of a braid of an even number of strands is the link obtained by connecting each consecutive pair of incoming strands and each consecutive pair of outgoing strands.

There is a maximal subgroup $H_{2n} \subset Br_{2n}$ of the braid group, called the Hilden subgroup, such that the plat closure is invariant under pre- & postcomposition with elements in $H_{2n}$, hence such that it descends to an operation on the double coset

$$PlatClosure \;\colon\; H_{2n}\backslash Br_{2n}/H_{2n} \longrightarrow Links_{/isotopy} \,.$$

This situation generalizes to framed braids whose plat closures are framed links.

Related concepts

• braid, link

• framed braid, framed link

References

Review:

• Stephen James Tawn: Plat closure of braids, PhD thesis, Warwick (2008) [pdf]

and for framed braids:

• Anastasios Kokkinakis: Framed Braid Equivalences [arXiv:2503.05342]