Package to generate Braid diagrams

KnotData does this kind of thing:

KnotData[{4, 1}, "BraidImage"]

Update:

I've been working on this and made a package with a single public function: to make these braid diagrams. Here's a semi-permanent link to the paclet file which can be installed with PacletInstall in Mathematica.

To load up the package after installing, use Needs[Braids`] followed by ?BraidDiagram to see a full description of how to use the BraidDiagram function. This is a somewhat outdated example of what it looks like: