Abstract
Given a long knot diagram D and a finite quandle Q, we consider the set of all quandle colorings of D with a fixed color q of its initial arc. Using this set we define the family [Formula: see text] of quandle automorphisms which is a knot invariant. For every element x ∈ Q one can consider the formal sum [Formula: see text], taken over all [Formula: see text]. Such formal sums can be applied to a tangle embedding problem and recognizing non-classical virtual knots.
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