Low Dimensional Topology

November 3, 2014

Can a knot be monotonically simplified using under moves?

Filed under: Knot theory — dmoskovich @ 12:55 am
Tags: ,

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?


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