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It was proven, I think by Mijatović EDIT: by Waldhausen, that there is an effective algorithm for distinguishing knot complements (the effective constants were found by Coward and Lackenby). The bound, if I understand correctly, uses something called Culler-Shalen seminorms and separating manifolds. I'm curious about whether there is a survey on this that is accessible to the general mathematical public?

Also it is surprising to me that this result did not make it into the hall of fame of great mathematical achievements along with the four-color theorem, the classification of finite simple groups, etc. Was it not surprising at the time? What is the history here?

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    $\begingroup$ Waldhausen gave the first algorithm to distinguish knots in the 1970’s using normal surface theory. $\endgroup$
    – Andy Putman
    Commented Oct 28, 2018 at 23:58
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    $\begingroup$ You misunderstand. The algorithm you descibe is more recent. Waldhausen’s work long predates all the stuff you describe. As for why it is not more famous, who knows? Everyone working on 3-manifolds is very aware of it. Some knot theorists don’t know about it, but that’s because they are focused on other kinds of questions. $\endgroup$
    – Andy Putman
    Commented Oct 29, 2018 at 3:38

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Technically the homeomorphism problem was resolved by Hemion, once he found an algorithm for the conjugacy problem in the mapping class groups. But this was the final step in a solution to the homoemorphism problem for Haken 3-manifolds conceived of by Haken and Waldhausen.

The geometrization theorem of Thurston gave another way of classifying knots via their complements. Knots have a canonical JSJ decomposition, and then each piece of this decomposition is either Seifert-fibered or hyperbolic. In principle, this gives another method to solve the homeomorphism problem, where one need only implement normal surface theory for tori, and solve the homemorphism problem for tori (which is much easier than the general case). The homeomorphism problem for the hyperbolic pieces is reduced to the isometry problem via Mostow rigidity. This approach has been implemented in the program SnapPea and its descendent SnapPy. However, these programs are not guaranteed to succeed, although in practice they work well on small examples. You can find a description of this program in Jeff Weeks' papers and in the documentation.

Note also that the conjugacy problem for mapping class groups of surfaces now has other solutions (also based on ideas of Thurston), some of which have been implemented (such as flipper). But I don't know if the Haken/Waldhausen/Hemion approach has been fully implemented (although see the program Regina for partial implementation). Many of these programs may be found at computop.org.

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