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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 …

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 …

This question is inspired by an answer of Tim Porter. Ronnie Brown pioneered a framework for homotopy theory in which one may consider multiple basepoints. These ideas are accessibly presented in his …

Let $X$ be a connected CW complex. One can ask to what extent $H_\ast(X)$ determines $\pi_1(X)$. For example, it determines its abelianization, because the Hurewicz Theorem implies that $H_1(X)$ is is …

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 …

Given two morphisms between chain complexes $f_\bullet,g_\bullet\colon\,C_\bullet\longrightarrow D_\bullet$, a chain homotopy between them is a sequence of maps $\psi_n\colon\,C_n\longrightarrow D_{n …

In my mind, algebraic topology is comprised of two components: Chain complex information, which is intrinsic information concerning how your object may be built up out of simple "lego blocks". Charac …

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.

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 …

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 …

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 …