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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.

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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.

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    $\begingroup$ Thank you for giving me an overview of the problem and references. As a robotics engineer focusing on applying the theory into practice, I probably do not have the luxury of time to delve into the abundant literature. So, may I ask, if there is some specific papers/algorithms you would recommend, 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? $\endgroup$
    – Muqing Cao
    Commented Oct 10, 2022 at 4:45
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    $\begingroup$ There is an algorithm implemented in the 3-manifolds software "Regina". You triangulate the link exterior using the Thurston/Weeks algorithm, then using (almost) normal surface theory you do the connect-sum decomposition and the Haken unknot recognition algorithm. It's not a theoretically fast algorithm, but in practice it's very effective. $\endgroup$
    – Ryan Budney
    Commented Oct 14, 2022 at 5:26
  • $\begingroup$ There is a way to use Snappy to recognise the unknot (or unlink) from a braid. This is explained in the fourth green block here: snappy.math.uic.edu/spherogram.html $\endgroup$
    – Sam Nead
    Commented Oct 14, 2022 at 19:21
  • $\begingroup$ If you have a particular example braid you want checked let me know,. I will write the first bits of code and you can then use that to check further examples. $\endgroup$
    – Sam Nead
    Commented Oct 14, 2022 at 19:23
  • $\begingroup$ Thank you Sam and Ryan for your answers, very helpful! It would be totally great if you can write a bit of example code, here is an example 4-braid: $\sigma_2\sigma_3^{-1}\sigma_1^{-1}\sigma_2\sigma_3^{-1}\sigma_1^{-1}\sigma_2^{-1}\sigma_2^{-1}\sigma_1\sigma_3^{-1}$. $\endgroup$
    – Muqing Cao
    Commented Oct 15, 2022 at 4:36

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