Yoshiyuki Ohyama

Satoh and Taniguchi introduced the n-writhe [Formula: see text] for each non-zero integer n, which is an integer invariant for virtual knots. It is obvious that the virtualization of a real crossing is an unknotting operation for virtual knots. In our previous paper, we showed that if [Formula: see text] is a sequence of integers with [Formula: see text], then there exists a virtual knot K whose virtual unknotting number equals one and [Formula: see text] for any [Formula: see text]. In this paper, we show that there exist infinitely many virtual knots having such properties by using the vertex connected sum on Gauss diagrams.

Satoh and Taniguchi defined a virtual knot invariant [Formula: see text] called the [Formula: see text]-writhe for each non-zero integer [Formula: see text]. The [Formula: see text]-writhes give the coefficients of some polynomial invariants for virtual knots including the index polynomial, the odd writhe polynomial and the affine index polynomial. Higa, Nakamura, Nakanishi and Satoh introduce the polynomial invariant called the intersection polynomial. In this paper, we construct infinitely many virtual knots [Formula: see text] ([Formula: see text]) whose Delta unknotting numbers equal one. Delta unknotting number one knots have the same [Formula: see text]-writhes and the same intersection polynomials as the trivial knot.

Satoh and Taniguchi introduced the [Formula: see text]-writhe [Formula: see text] for each non-zero integer [Formula: see text], which is an invariant for virtual knots. The [Formula: see text]-writhes give the coefficients of some polynomial invariants for virtual knots including the index polynomial, the odd writhe polynomial and the affine index polynomial. It is obvious that the virtualization of a real crossing is an unknotting operation for virtual knots. The values of [Formula: see text]-writhes changed by some local moves are calculated. However for the virtualization, it is unknown. 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 by the virtualization is one and the value of the [Formula: see text]-writhe equals [Formula: see text]. Namely, the virtualization of a real crossing changes the value of [Formula: see text]-writhe by any given integer [Formula: see text].