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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 …

This is a nice question. There is actually quite a bit of work which has been done along these lines, although we are a very long way from having a good understanding of how a theory of finite-type in …

I'd like to advertise Which presentations of (non)planar algebras give rise to knots? (see also Noah Snyder's comment there). It is rare that one can provide a response to an MO question simply by lin …

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 …

If your polymer chains are open (embedded closed line segment in 3-space), then I wouldn't recommend using global knot invariants (Alexander polynomial, Jones polynomial) because they will not make an …

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 …

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 …

I would argue that most of knot theory has no need for a knot diagram. Algebraic topology- Start from a presentation of the knot group (can be anything, needn't be from a Wirtinger of Dehn presentat …

L.P. Neuwirth, Knot Groups, Annals of Mathematics Studies, Princeton University Press, 1965.

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 …

In the late 1970's and in the 1980's, Michael Freedman showed a relationship between the topological surgery problem in 4-dimensions, the slice problem for links, and the classification of non-simply- …

I'm interested in the calculation of $H_n (G,\mathbb{Z})$ for G a finitely generated abelian group, particularly for $n=3$. It's carried out in the 1954/55 Séminaire Henri Cartan, titled "Algèbres d'E …

This is more general than what you ask for, but the following paper by Ryan Budney is deeply relevant: JSJ-decompositions of knot and link complements in the 3-sphere. L'enseignement Mathe'matique (2 …

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 …

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 …