The Miyazawa polynomial for long virtual knots
In this article, a Miyazawa polynomial is defined for long virtual knots. A long virtual knot is essentially a single component one-one tangle that possibly contains virtual crossings. From any long virtual knot, we can recover a virtual knot by taking the closure, i.e. identifying the endpoints of the tangle. This defines a map from the equivalence classes of long virtual knots to the equivalence classes of virtual knots. The closure of a long virtual knot \(K\) is denoted as \( \hat{K} \). Unlike in the classical case, there is no inverse map. That is, selecting a point on a virtual knot and simply cutting produces a long virtual knot but the location of the point matters; different cut points can produce different long virtual knots. The Miyazawa polynomial \(M_K (A,t) \) of a knot \(K\) is an oriented version of the Jones polynomial and its states consist of bivalent graphs that are augmented by weight maps on the edges of the graphs. The contribution of the variable \(t\) is computed by evaluating the virtual writhe which is obtained by summing the Miyazawa index of each virtual crossing. The Miyazawa index assigns a weight to each crossing based on the values of edges in the state from the weight map. Here the authors define a related invariant, \( \Lambda_K (A,t) \), for a long virtual knot \(K\). This invariant has several properties which are detailed in the main result of the paper. The main results are as follows. Let \(K\) be a long virtual knot and let \( \hat{K}\) denote its closure. If \(K\) is a product (concatenation) of the long virtual knots \(K_1\) and \(K_2\) then \( \Lambda_K (A,t) = \Lambda_{K_1} (A,t) \Lambda_{K_2} (A,t) \). If \(M_{ \hat{K} } (A,t) \) is the Miyazawa polynomial of \( \hat{K} \) then \(M_{ \hat{K} } (A,t) = \frac{1}{2} ( \Lambda_K (A,t) + \Lambda_K (A, t^{-1}) )\). The Kauffman \(f\)-polynomial of \( \hat{K}\) can be recovered by evaluating \( \Lambda_K (A,1)\). Finally, the real crossing number and the virtual crossing number are bounded below by the maximum of the absolute value of the highest and lowest degree of \( \Lambda_K (A,t) \) in the variable \(t\). The paper concludes with numerous interesting examples and corollaries.
scientific article; zbMATH DE number 5641772
scientific article