ON COLORED QUANDLE LONGITUDES AND ITS APPLICATIONS TO TANGLE EMBEDDINGS AND VIRTUAL KNOTS

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 $\Phi_{Q}^{q}(K)$ of quandle automorphisms which is a knot invariant. For every element x ∈ Q one can consider the formal sum $S_{\Phi}^{x}(K) = \sum_{\phi}\phi(x)$, taken over all $\phi \in \Phi_{Q}^{q}$. Such formal sums can be applied to a tangle embedding problem and recognizing non-classical virtual knots.