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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 …

Sophisticated mathematical concepts typically shed light on sophisticated mathematics. But in a few cases they also apply to elementary mathematics in an interesting way. I find such examples particul …

Morse theory investigates the topology of compact manifolds using critical points of real-valued functions $f\colon\, M\to \mathbb{R}$. Motivated by problems in dynamical systems, Novikov (Multivalued …

In Chapter 3 of Chris Schommer-Pries's PhD thesis, surfaces with corners (more generally, manifolds with faces) are equipped with extra structure called a "halation", with precisely the goal of making …

I think my paper with A. Carmi Daniel Moskovich, Avishy Y. Carmi, Tales told by coloured tangles, Int. J. Unconv. Comput. 12(1) 71-105 (2016), journal version, arXiv:1511.04919 is largely just this. …

This answer and its comments, is the reference you are searching for: https://mathoverflow.net/a/22384/2051 The paper in which the term appears appears is http://arxiv.org/abs/math.DG/0104196

Question: Given a `hard' diagram of a knot, with over a hundred crossings, what is the best algorithm and software tool to simplify it? Will it also simplify virtual knot diagrams, tangle diagrams, a …

There are several invariants whose "natural" domain is a category of disoriented tangles, that is tangles which are piecewise-oriented, but which contain points called `disorientations' at which the o …

As pointed out by Ian Agol in the comments, the Seifert-Weber space is the 5-fold cyclic branched cover of the Whitehead link complement. You can therefore: Untie the Whitehead link using $\pm 1$ fr …

It's a bit dated, but I found Neuwirth's book very useful, containing useful material not easily found in other sources: Neuwirth, L. P. (1965). Knot groups (No. 56). Princeton University Press. A …

The classical PL Reidemeister Theorem reads: Reidemeister Theorem: Two knots in $S^3$ are PL ambient isotopic if and only if any diagram of one can be transformed into a diagram of the other by Reide …

The Generalized Riemann Hypothesis (GRH) influences Complexity Theory. In particular, Pascal Koiran proved that the truth of the GRH implies that the problem of "whether a set of polynomial equations …

I've seen the following quote many times on the internet, and have used it myself. It is usually attributed to Grothendieck. It is better to have a good category with bad objects than a bad category …

Perhaps you might be interested in Section 4.3 of Linear Algebra and Geometry by Shafarevich and Reznikov (which is my favourite Linear Algebra textbook, by the way), in which a complex structure on a …

Peter Roegen works on this problem, with the practical goal of effectively identifying certain knotted proteins. His descriptors (not "invariants", because open curves are topologically unknotted) are …