Clasp pass moves and arrow polynomials of virtual knots (Q6611937)

Two knots have the same values for any Vassiliev invariant of degree up to 3 if and only if they can be transformed into each other through a sequence of clasp pass moves. \textit{T. Tsukamoto} calculated the difference in the values of the third Vassiliev invariant derived from the JonesKauffman polynomial for two knots differing by a single clasp pass move [Proc. Am. Math. Soc. 128, No. 6, 1859--1867 (2000; Zbl 0946.57016)]. This result was later extended by \textit{M.-J. Jeong} and \textit{D.-G. Kim} to virtual knots by using the Kauffman polynomial [J. Knot Theory Ramifications 25, No. 8, Article ID 1650045, 19 p. (2016; Zbl 1354.57014)].\N\NIn this paper, the author computes the third derivative of the arrow polynomial, after a change of variables, for two virtual knots related by a single clasp pass move. Let \(h_L(A)=f_L\left(A^{n_0}, A^{n_1}, \ldots\right)\), where \(f_L(A)\) is the arrow polynomial of a polar link \(L\) and \(n_0,n_1,\ldots\) are integers. The third derivative of \(h_L(A)\) at \(A=1\) is a Vassiliev invariant of degree \(3\). It is shown that\N\[\Nh_{K_1}^{(3)}(1)-h_{K_2}^{(3)}(1)=0 \quad \text { or } \quad \pm 2304 \N\]\Nif the virtual knots \(K_1\) and \(K_2\) are related by a single clasp pass move. This result provides a lower bound for the number of clasp pass moves required to transform one virtual knot into another when they are related by a finite sequence of such moves.