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.
***Update***. Here is a figure from the Murakami reference kindly provided by Marco Golla:
(Murakami Fig.7, illustrating *#-unknotting operations*.)
> Murakami, Hitoshi. "Some metrics on classical knots." *Mathematische Annalen* **270**.1 (1985): 35-45. ([Göttinger Digitalisierungszentrum link to PDF](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0270&DMDID=DMDLOG_0010&IDDOC=160813).)