I am relatively new to the world of braids/knots so really sorry if this question is simple. However, I am not able to find if there is any theorem/procedure that determines if a closed braid, given its representation in the Artin braid group, is a link or an unlink. Or, any theorem that says this cannot be determined? I have read the textbook A Study of Braids by Kunio Murasugi to get familiar with the concepts of Alexander's Theorem.
Any suggestion or recommendation of literature is really appreciated.


Edit: To be more specific, I am looking for some specific papers/algorithms (that can be efficiently implemented in computers), using which, I can determine if the closure of a braid (in the form of generators $\sigma\_1$, $\sigma\_2$, ...) gives a trivial knot/link or non-tirvial knot/link.



Determine if a closed braid is a link/unlink
A braid gives a braid closure. This can be drawn as a knot (or link) diagram. There are then various approaches to solve the unknot (or unlink) recognition problem given a diagram. This begins with work of Haken, and then work of Hass, Lagarias, and Pippenger, and then work of Lackenby.


There is also another line of research, which is perhaps more in line with your stated interests (in braids). Namely, Birman with various co-authors and then work of Dynnikov work more directly with the presentation of the knot or link as a braid.


You can find a discussion and references at the Wikipedia page on the [unknotting problem](https://en.wikipedia.org/wiki/Unknotting_problem).


