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.
Objects are finite words of strand orientations. Morphisms are boundary-fixed ambient-isotopy classes of oriented tangles in a slab, composed by stacking compatible boundary words; horizontal juxtaposition is the tensor product. Cups, caps, identity strands and crossings generate the category. Ordinary unframed isotopy includes all three Reidemeister moves; framed variants impose different first-move relations.
A Conway mutation cuts a link along a sphere meeting it in four points, rotates the enclosed two-string tangle through , and reglues it.

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