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https://youtu.be/co78AEqsv3s?t=1901

Within this video at 31:41, Professor Ronald Brown handles knots algebraically, I am confused to how he deals with the crossing indices. He substitutes and simplifies a way of interlacing a separate rope in the trefoil and then joining the ends. Algebraically it equals "1", which means that the trefoil is able to be untangled from the other rope, resulting in two separate knots; a trefoil and an unknot. What is this type of knot theory algebra called so that I can look it up and study it further?

For example, is the difference between x and x^-1 the direction, or is it the difference between overpass and underpass?

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  • $\begingroup$ I have finally cracked the code to what the difference between x and x^-1 is, however I still do not know the name of this theory/technique. If anyone knows please drop an answer. Thank you! $\endgroup$
    – Alexander Stroborg
    Commented Aug 18, 2016 at 18:30

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He is computing the fundamental group of the complement of the knot. This is also known as the knot group and is a key invariant in knot theory.

More specifically, he is computing the Wirtinger presentation of this group.

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