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The most convincing example I have found of "two basepoints being better than one" is the incorrect statement of the main result of the following paper: Garoufalidis, Stavros, and Andrew Kricker. "A …

Le and Murakami (HERE and HERE) discovered several previously unknown relations between multiple zeta values through the study of quantum invariants of knots. Further relations were later discovered t …

The Witten-Reshetikhin-Turaev approach to constructing quantum topological invariants of $3$-manifolds is to define them on framed links and to prove invariance under Kirby moves. There is a paper of …

Appendix 3 of Eisenbud's "Commutative Algebra" is the best short treatment I know. I find it fantastic. It clearly and concisely covers a surprising number of topics in homological algebra.

I don't think that there has been a tremendous amount of progress in understanding the Kontsevich Invariant of a knot in the last decade or so. It appears that essential new ideas may be needed in ord …

This question was answered by Misha in a comment. Nothing useful can be said about the handle decomposition of $N$, because, among other things, if an $n$-manifold $N$ has boundary $M$, then the conn …

I'm not an expert and this might be wrong, but I think that Cerf theory should be impossible for orbifolds, and therefore all that comes from it, e.g. Kirby Calculus. Could somebody who knows please c …

I think you're describing Spanier. Everyone I know who has seriously studied from Spanier swears by it- it's an absolute classic. The approach is exactly as you describe- algebraic topology for grown …

My recommendation would be the book of Freedman and Quinn, Topology of 4-manifolds. It's hands-on, very very good, and suitable I think for a reader of your background. Indeed, I would strongly recom …

Roughly speaking, algebraic topology works by reducing questions about topological objects such as manifolds and cell to questions about chain complexes. On the topological side, although in the PL …

T. Ohtsuki's Problems on invariants of knots and $3$--manifolds sounds to me like what you are looking for. Updates for problems in it, since it was published in 2002, are here. In my opinion, the bi …

Very much so. There are a number of small industries centred around studying equivalence classes of knot diagrams generated by a set of moves. The study of claspers. For example, $C_k$-moves are a sp …

The field of L-theory, the algebraic K-theory of quadratic forms, lies at the intersection of algebraic topology and of number theory. My impression is that it is an underpopulated discipline partiall …

Any closed connected n-manifold admits a Morse function f with one critical point of index zero and one critical point of index n (see e.g. Matsumoto's "Introduction to Morse Theory", Theorem 3.35). I …

This question has already been answered, but there's a tiny piece of intuition which I'd like to make explicit: If you're thinking about a manifold in the PL world, surgery might look a bit arbitrary …