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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 of quandle automorphisms which is a knot invariant. For every element x ∈ Q one can consider the formal sum , taken over all . Such formal sums can be applied to a tangle embedding problem and recognizing non-classical virtual knots.