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

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          ^{(Murakami Fig.7.)}

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> Murakami, Hitoshi. "Some metrics on classical knots." *Mathematische Annalen* **270**.1 (1985): 35-45. [GDZ site](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0270&DMDID=DMDLOG_0010&IDDOC=160813)

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](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0270&DMDID=DMDLOG_0010&IDDOC=160813).)

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.

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.

---

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

---

         
          ^{(Murakami Fig.7.)}

---

> Murakami, Hitoshi. "Some metrics on classical knots." *Mathematische Annalen* **270**.1 (1985): 35-45. [GDZ site](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0270&DMDID=DMDLOG_0010&IDDOC=160813)