Low Dimensional Topology

November 3, 2014

Can a knot be monotonically simplified using under moves?

Filed under: Knot theory — dmoskovich @ 12:55 am
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I would like to draw attention to a fascinating MO question by Dylan Thurston, originally asked, it seems, by John Conway:

Can a knot be monotonically simplified using under moves?

The question asks whether, rather than searching for Reidemeister moves to simplify a knot diagram, we could instead search for “big Reidemeister moves” in which we view a section which passes underneath the whole knot (only undercrossing) or over the whole knot (only overcrossing) as a single unit, and we replace it by another undersection (or oversection) which has the same endpoints.

This question (or more generally, the question of how to efficiently simplify knot diagrams in practice) loosely relates to a fantasy about being able to photograph a knot with a smartphone, and for the phone to be able to identify it and to tag it with the correct knot type. Incidentally, I’d like to also draw attention to a question by Ryan Budney on the topic of computer vision identification of knots, which is  topic I speculated about here:

Algorithm to go from a picture (or pictures) of a string in space, to a piecewise-linear representation of the curve.

A core question to which all of this relates is:

Are there any very hard unknots?

And perhaps more generally, are there any very hard ambient isotopies of knots?

May 3, 2012

Mysterious knot invariant on MathOverflow

Filed under: Knot theory — dmoskovich @ 5:14 am
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Andrew Rupinski asked a MathOverflow question about a knot invariant which he discovered, which distinguishes the left-hand trefoil from the right-hand trefoil. Nobody’s managed to identify this invariant yet, so maybe it really is new. The construction is by colouring vertices in a stick presentation of a knot by roots of unity, which is not something I’ve seen before, and is quite simple (you can read on MO exactly what it is- there are also pictures).

Finding a new knot invariant isn’t so exciting in as of itself, but a new perspective is a valuable thing to have, and, if nothing else, Rupinski’s question does seem to present a new perspective, at least to me. MathOverflow is becoming an ever more useful resource.

February 5, 2012

The 3-sphere has interesting triangulations

Filed under: 3-manifolds,Combinatorics,Triangulations — dmoskovich @ 1:06 am
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There is a recent interesting question on MO regarding a paper by Benedetti and Ziegler which I found most interesting.

Upon reading the question, I right away downloaded and printed Benedetti and Ziegler’s paper, after which I sat down to glance through it. My first impression is that it constitutes a really high-class piece of mathematics; the exposition is clear enough that a non-specialist can sit down and enjoy it, and the results are deep and interesting. There’s something really inspirational about a paper like that. (more…)

May 11, 2011

MO-problems: codimension zero embeddings

Filed under: 4-manifolds,Algebraic topology — Ryan Budney @ 7:38 pm
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Greetings,

Jesse recently recruited me as a special correspondent for the goings-on at Math Overflow. Perhaps he’ll eventually let me blog about other things! To begin I’d like to point out a lovely and easy-to-state but not-so-little problem that appeared on MO.

Is the universal covering of an open subset of Euclidean space diffeomorphic to an open subset of the same Euclidean space?

The above problem is perhaps a representative problem in a family of problems that have received little attention by the geometric topology community, which is the issue of low co-dimension embeddings. They are not well understood. This is because these can be rather difficult problems. More than that, there isn’t an edifice — there’s no standard machine to play with.

(more…)

April 27, 2011

Why do we study knots in S3?

Filed under: Knot theory — dmoskovich @ 5:18 pm
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Today, a question was posted on MathOverflow which asked why knot theorists choose to study knots in S^3 as opposed to knots in \mathbb{R}^3 (actually there’s another model also- we study knot (diagrams) in D^2). Mathematicians study mathematical models, and if we’re studying a certain model we should be able to explain why, especially because visualizing in \mathbb{R}^3 is easier than visualizing in S^3. So I found it rather embarassing that I had to think for a while before I came up with what I thought was a reasonable answer.
Knot theory is actually the theory of knot complements, and a knot complement in S^3 is compact, while a knot complement in \mathbb{R}^3 is open. What is the core reason that we choose to study compact knot complements in knot theory? I suppose that compactness must manifest itself in two essential ways: finiteness (for example, the knot complement is a finite number of simplexes glued together), and nice analytic properties (convergence of sequences, integrals, and so on).
After thinking for a while, I posted one answer. I really think that bordism is the most fundamental reason. Other people posted other answers (I encourage readers to post yet more- surely there’s still much more to be said). I think it was a useful exercize; it’s one of those questions that every knot theorist really should have a good answer to.

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