The writhe is the sum of the signs of all crossings in an oriented link diagram. It changes under the first Reidemeister move and is unchanged under the second and third moves.
A tangle is a proper embedding of arcs and circles in a three-ball with prescribed arc endpoints on its boundary. A two-string tangle has four boundary endpoints.
A Conway mutation cuts a link along a sphere meeting it in four points, rotates the enclosed two-string tangle through , and reglues it.
The Kauffman bracket is the unoriented diagram invariant determined by resolving each crossing into its two smoothings and replacing every additional disjoint circle by the loop factor . It is invariant under the second and third Reidemeister moves.
The Jones polynomial is obtained from the Kauffman bracket of an oriented diagram by the writhe normalization
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