Some people seem to rejoice in knotiness. To non-topologists, it’s not clear why anyone would care about even a plain old knot in 
But what then, about links? High dimensional knots? Tangles? Braids? High dimensional links? Homotopy links? I’m sure we were a bit skeptical about the usefulness of these when we first saw them, but now we can just about accept them.
What about the next step? Knots and links in arbitrary manifolds? Singular knots? And then what about virtual knots? free knots? coloured knots? knotted trivalent graphs? What about these new objects of study like knotted handlebodies? Turaev’s topology of words (knotted words)?
How does one decide such a topic is interesting… why and when is extending a result about links in the 3-sphere to higher dimensional stuff or stuff in strange manifolds interesting? How does one become interested in it?
I know that I’m still a bit skeptical about virtual knots, for instance. But I’ve come to accept knotted trivalent graphs as natural… for rather strange reasons. How about all of you?
October 29, 2009
Which knotted objects are worthy of study?
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Low Dimensional Topology 
[i]To non-topologists, it’s not clear why anyone would care about even a plain old knot in S^3 (or a long knot), but to us it’s the most natural thing in the world. To them it would seem to specific, too specialized, not really interesting; but we know that they are wrong, right?[/i]
I don’t think that’s entirely true. Knots in S^3 are perhaps one of the easiest aspects of modern mathematics to motivate to a non-mathematician. Many people have the frustrating experience of an extension cord getting tangled and apparently knotted. Knotted hair. Braided hair. Knotting of molecules is increasingly studied in microbiology departments. But in a sense you’re right. These people by-and-large are interested in things like the isotopy problem, and low-dimensional topologists don’t spend much time talking about it. Sometimes knot theorists talk about related things, but by-and-large they’re not busy writing computer software to efficiently solve the isotopy problem.
High-dimensional knots have various motivations. Perhaps the best one nowadays is the connection to algebra via Jerry Levine’s work. ie: view knot theory as a geometric manifestation of certain aspects of the study of various types of bilinear forms, modules over Laurent polynomial rings, etc. In low-dimensional topology an analogue of this would be the use of things like Cayley graphs to study groups via presentations. A higher-dimensional analogue would be the classifying space construction and how it gives insight into groups via group cohomology, etc. In general, if you bring a different mechanism to view a subject, it tends to shed light on it.
Regarding the “next step”, knots and links in arbitrary manifolds have been studied for some time. There is a general framework for studing isotopy invariants that began with people like Haefliger and have been souped-up by Tom Goodwillie (Goodwillie’s framework is complete in that it gives a full set of isotopy invariants, one of the main difficulties is it’s difficult to “get your hands on” these invariants). This is in the codimension > 2 case. In codimension 2, you have analogues of many of the constructions from 3-dimensional knot theory available. In codimension 1, there’s still one of the biggest open problems in general knot theory open, the 4-dimensional Schoenflies problem. Does a smooth 3-sphere in the 4-sphere bound a smooth 4-ball on both sides?
Comment by Ryan Budney — October 29, 2009 @ 4:45 pm |
I must add to the previous commentary that knots and links in arbitrary manifolds, besides old age, have a very good reason to pretend to “naturalness” : knots in arbitrary 3-manifolds arise spontaneously when you study surgery, that is when you consider (generic) functions on 4-manifolds. If studying functions on a manifold is not natural to a topologist, I cannot see what will.
Comment by toto — October 30, 2009 @ 2:57 am |
Ryan: Perhaps, but what about the “next step” after that? What about virtual knots, or free knots?
toto: Thanks for your reply… I’m not sure I understand though. In that context, you have framed surgery links, and you are modifying them by Kirby moves or other moves, so is the knot isotopy problem really relevant?
Comment by Daniel Moskovich — October 30, 2009 @ 3:16 pm |
Hi Daniel,
I’m a little confused by your first question. Is there a particular next step you’re interested in?
Regarding the 2nd, IMO the motivation for virtual knots is pretty literal. They’re a very special case of knots in 3-manifolds, knots in SxI for S a surface. IMO the most natural way for this to occur is when you have a knot in a 3-manifold M that has a Heegaard splitting. In Rubinstein-Scharlemann sweep-out terminology, the sweep-out looks like SxI but where you have an equivalence relation on the boundary (the boundary gets crushed down to two graphs, I think RS call those graphs the “spine” of the sweep-out, or something like that). So a knot in M can be perturbed to be disjoint from the spine, so it’s in the interior of SxI, ie: a virtual knot. So the upshot of this I suppose is, through the eyes of Heegaard splittings you can view knot theory in 3-manifolds as virtual knots mod certain global moves associated to passing a strand over a spine. This is sort of analogous to the Markov theorem for representing links by closed braids.
Aside from one or two papers titles on the arXiv (papers which I haven’t looked at) I don’t think I’ve encountered the terminology “free knot” before. A Google search on “free knot” returns something associated to World of Warcraft.
-ryan
Comment by Ryan Budney — October 30, 2009 @ 4:15 pm |
Ryan-
A free knot (Manturov’s term) is a virtual knot on which you are allowed to perform crossing changes on real (non-virtual) crossings (of course).
I’m not thinking of a specific next step, but rather just reflecting on life, knottiness, and everything. I suppose I would challenge you to interest me in knotted handlebodies and handcuff knots, which seem so popular among Japanese knot theory youth…
I haven’t yet managed to find virtual knots exciting, to tell you the truth…
Comment by Daniel Moskovich — November 1, 2009 @ 7:48 am |
I’m not sure if I’m up to the task Daniel! :)
A lot of things Japanese youth do get taken up by the rest of the world, eventually. I’ve seen a few in my life: pet rocks, nintendo… pokeman.
More seriously, it really depends on what it takes to get you motivated. Knotted handlebodies are vaguely on my mind but they’re fairly peripheral objects for me. I have a set of problems that I consider significant and important for moving topology and mathematics forward. My interest in an issue pretty closely correlates to how that issue relates to my internal list of major problems. One of these is the smooth Schoenflies problem in the 4-sphere. The relation is pretty simple: look at the proof of the 3-dimensional Schoenflies theorem. You take a 2-sphere in R^3, find a height function which restricts to a Morse function on the sphere, and cut across level sets disjoint from the critical points. From this you construct local models for how the compact 3-manifold bounded by the 2-sphere is constructed. This is doable in R^3 largely because the slices consist of nested circles in a level-plane, so you have an innermost circle argument working for you. But for an S^3 in R^4, the slices are nested surfaces, and in general they’re knotted surfaces. So Richard Kent’s comments apply. So knotted graphs come into the picture. Does the story die here? Perhaps a subtle-enough understanding of knotted graphs could help you push through a proof of the Schoenflies problem. I haven’t been able to imagine one. But it’s a relation.
Comment by Ryan Budney — November 1, 2009 @ 11:19 pm
Out of curiosity, what’s the motivation for regarding knotted trivalent graphs as natural? I spent some time working on knotted graphs, but unfortunately I never found a clear motivation for the topic.
Comment by Gilbert Bernstein — October 31, 2009 @ 5:37 pm |
IIRC, knotted theta graphs (and higher) have some significant role to play in loop quantum gravity theories.
Comment by John Armstrong — October 31, 2009 @ 6:24 pm |
Why do you find trivalent graphs natural, Daniel?
I’ll take a stab at that question. A trivalent graph is a “generic” graph from the point of view of cell-complexes and attaching maps. Build a graph by attaching edges to a graph with less edges. Provided your attaching map is as transverse as it can be to 0-skeleton of the previously-constructed graph, you construct a trivalent graph.
In that regard, there’s nothing really special about trivalent graphs. They’re just graphs where you minimize the valency of the vertices, provided it’s considered okay to perturb the attaching maps a little. So trivalent graphs appear all over the place because graphs do.
The analogue for 2-complexes is called a “shadow” in the literature nowadays. IMO it should be called something more along the lines of “generic 2-complex”.
Comment by Ryan Budney — October 31, 2009 @ 8:47 pm |
Gilbert: Understanding knotted trivalent graphs is often useful in constructing 3-manifolds with high genus boundary that have certain properties. For example, by the Fox reembedding theorem, every (tame) 3-manifold that embeds in the 3-sphere may be reembedded so that it is the complement of a graph. As the complement of a graph is always the complement of a 3-valent graph, understanding how the latter can be embedded in the sphere is of some use.
For example, using a theorem Myers, and Thurston’s geometrization theorem, you can construct lots of hyperbolic manifolds with high genus boundary this way.
Comment by Richard Kent — October 31, 2009 @ 8:51 pm |
There are two things which make knotted trivalent graphs natural in my mind. First, the fixed point locus of a 3-dimensional orbifolds is usually a knotted trivalent graph. The vertices correspond to global fixed points, and the arcs to fixed points on 0-, 1-, and 2-cells of a complex fixed by the action. An example would be quotients of the 3-sphere by symmetries of a regular polygon or of a Platonic solid. See Chapter 2 of http://www.gbv.de/dms/goettingen/388326085.pdf (Three Dimensional Orbifolds and their Geometric Structures by Boileau et.al.)
Secondly, as John Armstrong pointed out, they are natural extensions of knots when one considers quantum invariants of knots and 3-manifolds. If one expands the space of knots to the space of knotted trivalent graphs with knots, with an unzip operation, universal finite type invariants seem better behaved, so much so that people like Bar-Natan and Dylan Thurston think there might exist a 3-dimensional algebraic topology based on these ideas. This is the “Algebraic Knot Theory” project. See http://www.math.toronto.edu/~drorbn/papers/AlgebraicStructures/AlgebraicStructures.html
The first of these reasons one might perhaps associate with Bill Thurston (the father), and the second perhaps with Dylan Thurston (the son) :)
Comment by Daniel Moskovich — November 1, 2009 @ 7:43 am |
@Daniel Moskovich: have you seen the interpretation of virtual knots as knot diagrams on higher genus surfaces? That is, if you draw a knot diagram on the torus (for instance) you can get a virtual crossing by passing two strands around the loop in two different ways so they never actually cross on the surface itself.
Comment by John Armstrong — November 1, 2009 @ 12:06 pm |
Thanks for the response.
… I don’t find that compelling though. Why should a knot in a thickenned surface be more important and worthy of study than a knot in a general 3-manifold? Especially because one doesn’t really care about the surface (stabilize at will), but about the virtual knot itself.
One thing about virtual knots which I do find compelling is that they give geometric realizations to Gauss diagrams which cannot be realized by knots. But I find it hard to get excited about that, except if one is already excited about Gauss diagrams.
Another thing about virtual knots which I find compelling is the connection with Etingof-Kazhdan. But there, it’s the quantum algebra which I find fascinating, and virtual knots are merely the “excuse”.
Comment by Daniel Moskovich — November 1, 2009 @ 10:38 pm |
Knotted graphs also arise when studying tunnel number one knots. For instance, Goda, Scharlemann, and Thompson use thin position for trivalent graphs to study the bridge position of tunnel number one knots and their unknotting tunnels. (See Jesse’s latest post :) ) You can also consider the 1-skeleton of a triangulation to be an (un?)knotted graph in a 3-manifold.
A question for those who know something about virtual knots: I was under the impression that if you think about a virtual knot as living in surface x I, you can’t actually keep the genus of the surface fixed. That is, that some of the Reidemeister moves for virtual knots imply that you have to allow handles to be attached and removed. Is that correct? I suppose I could look it up, but it’s easier to ask :)
Comment by Scott Taylor — November 1, 2009 @ 7:38 pm |
That’s correct… more than that, the stable homeomorphism class only depends on the germ of the ambient surface near the curve. Therefore, what happens outside a regular neighbourhood does not matter, and the surface itself is not really a part of the structure.
Comment by Daniel Moskovich — November 9, 2009 @ 11:28 pm |
I also think knotted graphs are interesting. (… because I have been thinking about it recently.) Someone nearby me talked about tunnel number one, theta curve. At first, it was an unfamiliar terminology.
It can be regarded as part of Heegaard splitting theory with just genus increased by one. But, there was an exmaple called Kinoshita’s theta curve that attracted my attention.
As I think, whether a knotted object is natural or not depends on individual taste to some extent.
Comment by JungHoon Lee — November 2, 2009 @ 2:12 am |
“To them it would seem to specific, too specialized, not really interesting; but we know that they are wrong, right?” You bet we do. Mathematics is all about being specific. The value is; we can take a physical model which
is a mixture of practical rules of thumb, incorrect inferences and misuse of language, strip it down to its core, and then ask “What does it really predict?”, if it predicts anything at all. Most published math papers have about the
same intellectual value as a worked crossword puzzle. On the other hand all that fussing around with details leads to a clarity of language which in turn leads to scientific and technological revolutions. The aggregate sum of our behavior is much bigger than any single part.
Comment by topologicalperspectives — October 30, 2011 @ 5:54 am |