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I remember reading somewhere that there are just a few contributions to knot theory which do not involve knot diagrams. Hence my question:

Does anybody know about papers concerning knot theory which work directly with knots in 3D, without using planar (projections, for example) diagrams?

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I think almost none of classical knot theory uses knot diagrams. So almost all of Burde-Zieschang, for example. This is a bit of a sad question for me... is this what things look like now; knots are just 3-dimensional representations of knot diagrams?? –  Daniel Moskovich Aug 6 '12 at 18:21
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Marius, yes of course you can compute the Alexander polynomial without using a projection -- frequently this is the most efficient way to do it. The Alexander polynomial of torus knots are most easily computed using the Seifert fibering of the knot complement. This gives you a beautiful cell decomposition of it with one cell in dimensions 0 and 2, and two cells in dimension 1. –  Ryan Budney Aug 6 '12 at 23:57
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I think this is a good question, in that because there's such a relatively high volume of knot theory through the eyes of planar diagrams, it can easily give the impression that this is all there is. Frequently people will give surveys of knot theory and never mention Schubert, Seifert, Waldhausen or Thurston. I've been present in the audience at least three times for such surveys. There's of course two reasons for this: (1) new invariants originally couched in the language of planar diagrams and (2) this is a low-overhead approach to knot theory, so it's easier to draw in undergrads. –  Ryan Budney Aug 7 '12 at 0:18
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I think most of three manifold topology is about doing knot theory without diagrams. Only asking for one paper seems out of proportion to the amount of literature dedicated to doing knot theory without diagrams. For instance, Dave Gabai's proof of property R, Gordan and Luecke's solution to the knot complement problem. Thurston's work on geometrization, the whole field of normal surface theory.... –  Charlie Frohman Aug 7 '12 at 1:32
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Haken also should be mentioned here since the only known algorithm for telling one knot from the other goes back to his theory of normal surfaces and has nothing to do with knot diagrams. Incidentally, distinguishing knots is the original raison d'être for the entire field of knot invariants and none of them so far can do it (for arbitrary knots). –  Misha Aug 7 '12 at 4:18

6 Answers 6

up vote 8 down vote accepted

Here is an example close to my heart. We show that the second coefficient of the Conway polynomial can be defined by counting quadrisecants. (Collinearities of $4$ points on the knot.)

Quadrisecant on knot

Indeed, much of classical knot theory is done without resorting to planar diagram calculus. For example, the Alexander polynomial can be calculated from a Seifert surface for a knot, and most properties of the Alexander polynomial are best proved without trying to resort to diagrams. (There are exceptions.) Like Daniel Moskovich alludes to, the Alexander polynomial of a slice knot factorizes as $f(t)f(t^-1)$, and the proof does not have anything to do with diagrams. More modern developments are also often diagram-free. Ozsvath and Szabo's Heegard-Floer homology was originally developed without diagrams. Only later was it figured out how to interpret the theory using planar diagrams, which was regarded as a good thing, since it made the problem of calculating it algorithmic, although incredibly computationally intensive.

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@Ryan: Thanks you sir! –  Jim Conant Aug 7 '12 at 12:30
    
"Thanks"-->"Thank" –  Jim Conant Aug 7 '12 at 21:04
    
You're welcomes. :) –  Ryan Budney Aug 7 '12 at 22:00

Knot theory was fundamentally connected to 3-manifold theory (making planar diagrams relatively peripheral) by Schubert, in his pair of papers, where he proved the prime factorization of knots and then realized the proofs could all be re-done in the language of incompressible tori in the knot complement. This was later extended to a classification of knots by Waldhausen, although his classification wasn't very easy to use. But these tools evolved into what is called "Geometrization" and now there's a relatively practical classification of knots via these techniques: To every knot there is an associated rooted tree, where the vertices are decorated by various links in $S^3$ whose complements have the geometry of hyperbolic 3-space $H^3$ or the fibred geometry $H^2 \times \mathbb R$. In practice this is a very usable classification for small knots, since the software SnapPea can quickly find the hyperbolic structures on small knots (when the structure exists).

More recently, the computer software Regina computes the Alexander polynomials of knots (and any 3-manifold whose 1st Betti number is one) directly from a triangulation. In the case of knots, it's the triangulation of the complement that Regina uses. Regina has no knowledge of a planar diagram of the knot. One advantage of this is Regina computes the entire Alexander module, so it can also compute things like Milnor signatures and Tristram-Levine invariants. Whenever Regina 5.0 comes out it will also be able to compute things like signatures of 4-manifolds, also directly from triangulations. Regina is slowly building-up all the tools to be able to perform the "geometrization classification of knots", and hopefully it will be fairly quick for all knots with say less than 100 crossings in the not too distant future.

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Thank you Ryan! And thanks everybody for the instructive comments. –  Marius Buliga Aug 7 '12 at 9:00

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 presentation), and calculate and define Alexander polynomials, twisted Alexander polynomials, higher Alexander polynomials, twisted and untwisted torsions, Blanchfield linking pairings, signatures of all sorts, Cochran-Teichner-Orr concordance obstructions... No knot diagram ever makes any essential appearance.
  • Geometry- Hyperbolic volume, JSJ decompositions, and so on and so forth. This is 3-manifold topology applied to knots. No diagram ever appears essentially.
  • $3$-manifold other- Normal surface theory, character varieties, thin position, Morse-Novikov number$\ldots$

Anything that generalizes to $n$--knots can and will have no dependence on Reidemeister's Theorem; as will anything that generalizes to $3$--manifolds... and these two classes together surely cover most of knot theory however one measures it. The only place that knot diagrams seem important, I think, is in Quantum Topology, where one works with Skein relations, but even there, as Qiaochu's answer indicates (indeed also in Vassiliev's original approach), there are significant aspects of the story which are independent of the existence of knot diagrams.

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@Daniel: I guess the same argument would imply that geometry has no need for pictures since everything that could be proven with pictures, could be proven without them (actually, I heard this argument from some algebraists). Similarly, one could argue that no need for poetry since everything that could be said poetically could be said in prose as well... –  Misha Aug 7 '12 at 14:04
    
@Misha: Knot theory papers are improved by good exposition, which may or may not include the use of figures. But if none of the arguments use Reidemeister's Theorem or any mathematical statement about knot diagrams, then diagrams of knots are an expositional flourish and nothing more. –  Daniel Moskovich Aug 7 '12 at 14:43
    
@Daniel: "Anything ... will have no dependence...", "surely cover most of knot theory ...", "the only place ..." and so on look to me as authority arguments. I am glad that my question receives such heated, say, comments, that simply means that the matter is not so obvious. I did an authority-based experiment on Scholar: "knot invariant -diagram" gives 15900 items, "knot invariant diagram" gives 11800 items and "knot invariant" gives 27200. –  Marius Buliga Aug 8 '12 at 10:16

@article{freedman1994mobius, title={M{\"o}bius energy of knots and unknots}, author={Freedman, M.H. and He, Z.X. and Wang, Z.}, journal={The Annals of Mathematics}, volume={139}, number={1}, pages={1--50}, year={1994}, publisher={JSTOR} }

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As far as quantum invariants, there is Witten's Quantum Field Theory and the Jones Polynomial, which gives a 3-dimensional definition of the Jones polynomial (usually defined, as you say, using 2-dimensional diagrams) at a physical level of rigor. I understand that this paper was extremely influential, but I'm not familiar with the details. More recently, Witten's Fivebranes and Knots gives a 5-dimensional definition of Khovanov homology at a physical level of rigor.

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Here's a recent example.

http://arxiv.org/pdf/0911.2518v1.pdf

Adam McDougall constructs a "diagramless" homology theory that ends up being essentially equivalent to Khovanov homology.

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