Distance between two knots

Question

Are there some well-studied functions defining natural distance measures between two knots? One can imagine a function that counts, say, the minimum number of moves, each of which passes one strand of a knot through a crossing strand, in order to convert one knot to another. Or perhaps there are functions that rely on knot polynomial similarity.

Any references would be appreciated.

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Update. Here is a figure from the Murakami reference kindly provided by Marco Golla:

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          ^{(Murakami Fig.7, illustrating #-unknotting operations.)}

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Murakami, Hitoshi. "Some metrics on classical knots." Mathematische Annalen 270.1 (1985): 35-45. (Göttinger Digitalisierungszentrum link to PDF.)

Answer 1

The Gordian distance measures precisely the number of crossings you need to change to turn a knot into another.

MathWorld gives the reference:

Murakami, H. Some Metrics on Classical Knots, Math. Ann. 270, 35--45, 1985.

Answer 2

You might be interested in this paper of Tim Cochran and Shelly Harvey, which studies the large-scale geometry of various metrics on the set of concordance classes of knots.

Answer 3

If you want to define some distance on knots, you should have a (local) transformation/move on knots or or knot diagrams such that any two knots can be connected by finitely many of them. As mentioned by Ryan, the most important local move is crossing change, and the corresponding distance is called the Gordian distance.

Besides of this, $\sharp$-operation (H. Murakami, Some metrics on classical knots. Math. Ann. 270, 35-45, 1985), $\triangle$-operation (H. Murakami, Y. Nakanishi, On a certain move generating link-homology. Math. Ann. 284, 75-89, 1989) and $n$-gon move (H. Aida, Unknotting operation for Polygonal type. Tokyo J. Math. Vol. 15, No. 1, 111-121, 1992) all are unknotting operations, hence can be used to define a distance between knots. Recently, Ayaka Shimizu proved that region crossing change is also an unknotting operation for knots (A. Shimizu, Region crossing change is an unknotting operation. J. Math. Soc. Japan. Volume 66, Number 3 (2014), 693-708). While this move depends on the choice of the diagram, hence the distance defined by region crossing change are restricted on minimal diagrams.

On the other hand, there also exist some local operations that we do not know whether it can unknot every knot or not, for example the 4-move (see problem 1.59 on Kirby's list for more details and Dabkowski and Przytycki's paper: Unexpected connections between Burnside groups and knot theory. Proc Natl Acad Sci U S A. 2004 Dec 14;101(50):17357-60 for some update).

Answer 4

I tried Googling 'Knot distances', but you get a whole load of nautical stuff, which I don't think you meant!

Instead I went for the broader 'Solving knot Polynomials' and found:

Calculating knot distances and solving tangle equations involving Montesinos links by Hyeyoung Moon, University of Iowa.

Which comes with a free 35 page suite of programs for you to type in!