[Math] 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.