A preprint of Lins and Lins appeared on the arXiv today, posing a challenge [LL]. In this post, I’m going to discuss that challenge, and describe a recent algorithm of Scott–Short [SS] which may point towards an answer.

**The Lins–Lins challenge**

The theory of 3-manifolds is now very advanced, and we can even say in a certain sense that we understand ‘all’ 3-manifolds (as I discussed in an earlier post). But that understanding is very theoretical; the Lins–Lins challenge is to put this theory into practice.

They ask: ‘Are the two closed, hyperbolic 3-manifolds given by Dehn surgery on the following two framed links homeomorphic?’

(I’ve taken the liberty of copying the diagrams from their paper.)