TY - JOUR

T1 - Virtualization and n -writhes for virtual knots

AU - Ohyama, Yoshiyuki

AU - Sakurai, Migiwa

PY - 2019/10/1

Y1 - 2019/10/1

N2 - Satoh and Taniguchi introduced the n-writhe Jn for each non-zero integer n, which is an invariant for virtual knots. The n-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 n-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 n and any given integer N, there exists a virtual knot whose unknotting number by the virtualization is one and the value of the n-writhe equals N. Namely, the virtualization of a real crossing changes the value of n-writhe by any given integer N.

AB - Satoh and Taniguchi introduced the n-writhe Jn for each non-zero integer n, which is an invariant for virtual knots. The n-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 n-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 n and any given integer N, there exists a virtual knot whose unknotting number by the virtualization is one and the value of the n-writhe equals N. Namely, the virtualization of a real crossing changes the value of n-writhe by any given integer N.

KW - Gauss diagram

KW - n -writhe

KW - Virtual knot

KW - virtualization

UR - http://www.scopus.com/inward/record.url?scp=85078491680&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85078491680&partnerID=8YFLogxK

U2 - 10.1142/S0218216519500743

DO - 10.1142/S0218216519500743

M3 - Article

AN - SCOPUS:85078491680

VL - 28

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

SN - 0218-2165

IS - 12

M1 - 1950074

ER -