Are there moves between Reidemeister moves? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T23:55:10Z http://mathoverflow.net/feeds/question/47636 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47636/are-there-moves-between-reidemeister-moves Are there moves between Reidemeister moves? Greg Muller 2010-11-29T03:45:11Z 2010-11-29T12:13:42Z <h3>Background</h3> <p>Knots are typically written in 2 dimensions as a loop in the plane with normal crossings. One then asks when two such diagrams describe the same knot. Two diagrams describe the same knot when one can be made into the other by a sequence of <strong><a href="http://en.wikipedia.org/wiki/Reidermeister_moves" rel="nofollow">Reidemeister moves</a></strong>. These are three simple manipulations of a diagram which obviously don't change the underlying knot.</p> <h3>The Reidemeister Graph</h3> <p>One may consider the 'Reidemeister graph' of a knot diagram (probably not an official term), which consists of every diagram which is equivalent to the original, and an edge for every Reidemeister move between them. Since every Reidemeister move can be undone by the same move again, this is an undirected graph.</p> <p>Two diagrams may be connected by a many different sequences of Reidemeister moves. It is not hard to find a sequence of moves which takes a knot to itself, and is not trivial in the sense that involves doing and immediately undoing the same move. As a consequence, the Reidemeister graph is infinite and not simply connected.</p> <h3>Question</h3> <p>I can think of a loop in the Reidemeister graph as a kind of 'relation between relations' (where the moves are the relations). I would like to find a finite list of relations between relations, such that every loop is built from these relations.</p> <p>I'll be more specific. Define a <strong>higher Reidemeister move</strong> to be a locally-defined sequence of Reidemeister moves which relate a given diagram to itself. I would like a finite list of higher Reidemeister moves, such that if one fills in the corresponding loops in the Reidemeister graph with a 2-cell, the resulting space is simply connected.</p> http://mathoverflow.net/questions/47636/are-there-moves-between-reidemeister-moves/47646#47646 Answer by Ryan Budney for Are there moves between Reidemeister moves? Ryan Budney 2010-11-29T05:47:01Z 2010-11-29T05:47:01Z <p>In some strict sense I think the answer to your question is <em>no</em>, there are likely no finite collection of 2-cells doing what you want. If you were to ask the more natural question where you're looking for a 2-complex whose inclusion into $Emb(S^1,\mathbb R^3)$ is an isomorphism on $\pi_1$ (component-by-component) the answer I'm near-certain is yes (via Thom-Mather singularity theory).</p> <p>For example, here is a non-trivial loop in the space of knots which you could imagine as a loop in your Reidemeister graph once you refine things suitably. This loop isn't a problem if you only want the map (2-complex) $\to Emb(S^1,\mathbb R^3)$ to be an isomorphism on $\pi_1$. But for the complex you want, these loops are a problem, as they're very much global things and can't be described readily in terms of local diagram moves. </p> <p><img src="http://rybu.org/photo/mathish/chalk/sum_move.png" alt="Pull-through move"></p> <p>The loop described in this picture can be done for any combination of summands -- as long as the summand knots are non-trivial this is a non-trivial loop. So how are you going to construct a finite collection of 2-cells that kill off all these loops? </p> http://mathoverflow.net/questions/47636/are-there-moves-between-reidemeister-moves/47667#47667 Answer by sleepless in beantown for Are there moves between Reidemeister moves? sleepless in beantown 2010-11-29T12:13:42Z 2010-11-29T12:13:42Z <p>Take a look at page 180 of <em>Low dimensional topology</em> by Tomasz Mrowka, Peter Steven Ozsvát, (following Ben Webster's comment about <strong>Movie Moves</strong> elucidated by Baez and Langford and their 30 basic movie moves, and by Carter and Saito who describe a 31st basic movie move.) A movie move is a sequence of frames of a braid (or subregion of a knot, I suppose). </p> <p>Carter and Saito have a theorem that</p> <blockquote> <p>two movies represent the same tangle cobordism iff they can be related by a sequence of <em>movie moves</em></p> </blockquote> <p>If you take the subset of Movie Moves where each movie is a composition of a sequence of Reidemeister moves, it seems like that would be similar or equivalent to what you are calling "Higher Reidemeister moves." Am I understanding you correctly?</p> <p>I would point you out to the appropriate page on that wiki o' info, but "Movie moves" does not even show up on their search page.</p>