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I am trying to proof that the unknotting number of prime knot $7_5$ is $2$.

For this purpose, I am studying the minimal number of crossings that need to be changed in the knot in order to get a diagram of the trivial knot.

However, I am stuck when I change the following crossing (circled in red):

knot with crossing especified

I know that I get a knot with five crossings, but I am not capable of distinguish which of the following is:

knots with five crossings

How do I proceed?

Thank you in advance!

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2 Answers 2

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We can cheat by using a computer tool. I'm using KnotFolio. I took your drawing of the knot and input it:

original knot

It recognizes this as $7_5$ by calculating some knot invariants and using the fact that no other knot with at most 7 crossings has the same invariants.

Flipping the crossing in question,

flipped crossing

by a similar process it recognizes it as $5_2$. Unfortunately, KnotFolio doesn't have any Reidemeister move tool yet, so you have to edit the diagram manually, but after applying Reidemeister II we get this:

flipped then Reidemeister II

It's still recognized as $5_2$. The "Beautify" tool smooths out the knot diagram (an isotopy of the diagram -- note that it does this isotopy on a sphere, so it might choose a different face to be the outer face). In this form, it's more obvious that it's $5_2$.

beautified

By the way, with two flipped crossings we can get the unknot:

two flipped crossings

I'm guessing you already know this and that you're systematically checking that flipping each individual crossing in your diagram of $7_5$ doesn't give the unknot. Beware that this alone is not sufficient in proving that the knot $7_5$ has unknotting number $2$ -- you'd only be showing that the diagram in question has unknotting number $2$. There might still be some diagram for $7_5$ that has unknotting number $1$!

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Do the obvious Reidemeister move. Now treat it as a planar graph (with the crossings as vertices of valency $4$). Label each region with the number of edges that surround it.

Which of the knots with $5$ crossings has the same "region structure" ?

enter image description here

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