Finite-type (Vassiliev) invariants, quantum invariants, and perturbative invariants of knotted objects and of manifolds.

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Algebraic Proof of 4-Colour Theorem?

4-Colour Theorem. Every planar graph is 4-colourable. This theorem of course has a well-known history. It was first proven by Appel and Haken in 1976, but their proof was met with skepticism ...
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Is there a volume conjecture for closed 3-manifolds?

A typical statement of the volume conjecture, for instance in Murakami's survey 1002.0126, is Conjecture: For $K$ a knot in $S^3$, the N-th colored Jones polynomials are related to the volume of ...
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Is there a canonical Hopf structure on the center of a universal enveloping algebra?

Let $\mathfrak g$ be a finite-dimensional Lie algebra over $\mathbb C$. Define $\mathcal Z(\mathfrak g)$ to be the center of the universal enveloping algebra $\mathcal U\mathfrak g$, and define $(\...
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Casson's invariant and the trivial connection contribution to witten's 3-manifold invariant

This question might turn out not to make any sense, but here it is: Witten's (and Reshetikhin and Turaev's) 3-manifold invariant can be "defined" as an integral over the space of connections on the ...
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Why is the Alexander polynomial a quantum invariant?

When we think of quantum invariants, we usually think of the Jones polynomial or of the coloured HOMFLYPT. But (arguably) the simplest example of a quantum invariant of a knot or link is its Alexander ...