The M ±3 2-move and n -writhes for virtual knots

Satoh and Taniguchi introduced the [Formula: see text]-writhe [Formula: see text] for each non-zero integer [Formula: see text], which is an invariant of virtual knots. The [Formula: see text]-writhes provide the coefficients of several polynomial invariants for virtual knots, including the index polynomial, the odd writhe polynomial, and the affine index polynomial. Miyazawa, Wada, and Yasuhara defined a local move for virtual knots, called a [Formula: see text]-move. The [Formula: see text]-move is equivalent to the [Formula: see text]-move and serves as an unknotting operation for virtual knots. We have shown that, for any given non-zero integer [Formula: see text] and any given integer [Formula: see text], there exists a virtual knot whose unknotting number with respect to the virtualization equals one and whose value of the [Formula: see text]-writhe is equal to [Formula: see text]. In this paper, we show that, for any given non-zero integer [Formula: see text] and any given integer [Formula: see text], there exists a virtual knot whose unknotting number with respect to the [Formula: see text]-move equals one and whose value of the [Formula: see text]-writhe is equal to [Formula: see text]. In other words, the [Formula: see text]-move can change the value of the [Formula: see text]-writhe by an arbitrary integer.