Abstract
We identify a subcategory of biracks which define counting invariants of unoriented links, which we call involutory biracks. In particular, involutory biracks of birack rank N = 1 are biquandles, which we call bikei or 双圭. We define counting invariants of unoriented classical and virtual links using finite involutory biracks, and we give an example of a non-involutory birack whose counting invariant detects the non-invertibility of a virtual knot.
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