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I wonder if knot theory is related to any other topic in mathematics. I've not read much about it, but it seems to be living isolated.

I also wonder if there any particular mathematical background we should achieve in order to dive through this topic?

My last question is: Is Knot theory useful for the pure mathematician? Why? Why not?

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    $\begingroup$ It is a subfield of topology. $\endgroup$ – James Dec 10 '14 at 16:19
  • $\begingroup$ @James, algebraic topology? $\endgroup$ – Fawzy Hegab Dec 10 '14 at 16:24
  • $\begingroup$ People like calculating, say, fundamental groups of knot complements, sure, but there are lots of other things too. $\endgroup$ – James Dec 10 '14 at 18:49
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    $\begingroup$ There's a lot of algebraic topology involved, but there's also quite a bit of geometric topology--- look at hyperbolic 3-manifolds, for example. $\endgroup$ – anomaly Dec 11 '14 at 5:38
  • $\begingroup$ @Maths: geometric topology, and low-dimensional topology (en.wikipedia.org/wiki/Geometric_topology, en.wikipedia.org/wiki/Low-dimensional_topology). But also "quantum topology" - I don't want to link to the Wikipedia article here because it's not very good. $\endgroup$ – Qiaochu Yuan Dec 11 '14 at 5:46
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Knot theory is in fact one of the least isolated areas of mathematics! Let me attempt to tell some of the history here, although I certainly don't guarantee I'll even name most of the relevant authors.

In 1984 Jones discovered a knot invariant, the Jones polynomial, in the course of investigating some structures related to the theory of von Neumann algebras. The Jones polynomial admits a simple definition in terms of a so-called "skein relation" on knot diagrams, but it can also be defined in terms of an interesting representation of the braid groups involving a family of algebras, the Temperley-Lieb algebras, which originally arose in some problems in statistical mechanics.

There is a superficially similar invariant called the Alexander polynomial, which can also be defined in terms of a skein relation, but which also admits a purely "3-dimensional" definition which does not depend on a choice of knot diagram. Atiyah asked whether it was similarly possible to give a purely "3-dimensional" definition of the Jones polynomial; without one it's quite unclear what kind of information about a knot the Jones polynomial is really telling you.

In 1989 Witten proposed such a "3-dimensional" definition using a quantum field theory called Chern-Simons theory. The Jones polynomial appears as a certain path integral in the theory, which at least at the time was unfortunately only defined at a physical level of rigor. Nevertheless, Witten's paper sparked an enormous amount of mathematical activity which is still ongoing today. This is part of the work for which Witten won the Fields medal, becoming the first physicist to do so.

Among other things, Witten's proposal suggests that the Jones polynomial admits a natural generalization to an invariant of 3-manifolds and sparked a lot of work in topological quantum field theory (e.g. Reshetikhin-Turaev) relating this story to (among other things) the theory of quantum groups, loop groups, and affine Lie algebras, which are of great interest in representation theory.

In 2000 Khovanov discovered a categorification of the Jones polynomial to a more powerful knot and link invariant called Khovanov homology (roughly analogous to the more classical categorification of the Betti numbers to homology). Attempts to understand Khovanov homology have led to lots of interesting work in representation theory and also in symplectic geometry due to its resemblance to Floer homology. I am not at all familiar with the literature here, and it's too big for me to attempt to do justice to.

In the same way that the Jones polynomial is related to 3-dimensional quantum field theory, Khovanov homology turns out to be related to 4-dimensional quantum field theory. This is a huge subject on its own, which these days is perhaps most famously known for appearing in the story of the geometric Langlands program.

I could just keep going, but in any case I hope I've made my point.

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    $\begingroup$ It is indeed a strange idea that knot theory is an isolated subject! :-) $\endgroup$ – Mariano Suárez-Álvarez Dec 11 '14 at 5:41
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    $\begingroup$ @MarianoSuárez-Alvarez: Really? When I was an undergraduate who knew nothing other than calculus, linear algebra, and (the basics of) analysis and abstract algebra, the idea of an entire field devoted to studying knots seemed completely out of left field. Only after someone told me that studying knots is the same as studying embeddings of the circle $S^1$ did I begin to see any sort of connection. $\endgroup$ – Jesse Madnick Dec 11 '14 at 6:30
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Knot invariants, in particular categorifications of quantum knot invariants, arise from a breathtaking range of different fields of pure mathematics: There is Khovanov homology, building on 2D topological field theories, there is Khovanov-Rozansky homology, relating to to commutative algebra and singularity theory, there is Cautis-Kamnitzer's algebro-geometric categorication of quantum ${\mathfrak s}{\mathfrak l}_n$-invariant, there is Mazorchuk-Stroppel-Sussan's invariant using representation theory of Lie algebras, there is Lipshitz-Sarkar's invariant with values in homotopy types of spectra, and there is Seidel-Smith's construction involving Floer homology. Probably I still missed some... but the above list already shows that the source of knot invariants is amazingly rich and diverse, and connecting these different approaches is an active topic of research. In particular, I would definitely (k)not say that knot theory is a subfield of topology.

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