A four-variable index polynomial invariant of long virtual knots (Q2890228)

A \textit{long virtual knot diagram} is a diagram \(D\) on \(\mathbb{R}^2\), which is the image of an immersion from \(\mathbb{R}\) to \(\mathbb{R}^2\) such that every sufficiently large circle intersects \(D\) at two points, and the crossings of \(D\) consist of classical crossings and virtual crossings. A \textit{long virtual knot} is a long virtual knot diagram up to generalized Reidemeister moves. Note that a long virtual knot is equivalent to a \textit{pointed virtual knot}. Throughout the paper, a long virtual knot is oriented.NEWLINENEWLINEFor a long virtual knot \(K\), the authors define a refinement of the \textit{index polynomial} due to \textit{Y.~H.~Im} and \textit{K.~Lee} [Eur. J. Comb. 30, No. 5, 1289--1296 (2009; Zbl 1178.57008)]. The new invariant is denoted by \(Q_K(s_1, s_2, t_1, t_2)\) (Definition 2.6). However the authors do not give a name for it. How about the \textit{parity index polynomial} of \(K\)? The reason why I suggest the name is that the refinement is according to over/under information and the parities of the classical crossings. By definition, \(Q_K(s_1, s_2, t_1, t_2)\) is an element of the Laurent polynomial ring \(\Lambda=\mathbb{Z}[s_1^{\pm 1}, s_2^{\pm 1}, t_1^{\pm 1}, t_2^{\pm 1}]\), and vanishes if \(K\) is classical. More precisely, \(Q_K(s_1, s_2, t_1, t_2)\in R\) where \(R\) is a \(\mathbb{Z}\)-submodule of \(\Lambda\) generated by \(s_j^k\) and \(t_j^k\)\ \((j=1, 2; k\in \mathbb{Z})\). The authors remark that \(Q_K(t, t, t, t)\) is the index polynomial of \(K\) (Remark 2.10 (2)).NEWLINENEWLINEIn Section 3, some important properties of the invariant are shown such that the invariant is in \(\mathbb{Z}[s_1^{\pm 2}, t_1^{\pm 2}]\) for a checkerboard colorable (\textit{normal}, in the paper) long virtual knot (Proposition 3.1), the invariant is additive under connected sum (Theorem 3.4), the mirror image formula and the inverting formula (Proposition 3.6) hold, and the invariant vanishes for an amphicheiral long virtual knot (Corollary 3.7).NEWLINENEWLINEIn Section 4, the effect of shifting the infinite point of a long virtual knot (the base point of a pointed virtual knot) to the invariant is studied (Lemma 4.4), a similar invariant of a long flat virtual knot is defined by using a functor from the set of long flat virtual knots to that of long virtual knots (Theorem 4.9), and if the invariant of a long flat virtual knot is not zero, then it is non-invertible (Corollary 4.12).NEWLINENEWLINEIn Section 3 and 4, suitable examples are raised, and in Section 5, some questions are proposed.