A lower bound for the number of forbidden moves to unknot a long virtual knot. (English) Zbl 1310.57013
As a generalization of classical knot theory, virtual knot theory was first proposed by L. H. Kauffman in [Eur. J. Comb. 20, No. 7, 663–690 (1999; Zbl 0938.57006)]. It was independently proved by Nelson and Kanenobu that the forbidden move is an unknotting operation for virtual knots. In this paper the author introduces a sequence of local moves \(F_n\) on a virtual knot diagram such that \(F_1\) is equivalent to the forbidden move. In general \(F_n\) is not an unknotting operation, but we can consider the Gordian distance between two \(F_n\)-equivalent virtual knots \(K_1\) and \(K_2\). For two long virtual knots with distance one, some constrains on degree two finite type invariants are given. As a corollary the author gives a lower bound for the number of \(F_n\) needed to connect \(K_1\) and \(K_2\).
Some related results can be found in [M. Sakurai, J. Knot Theory Ramifications 22, No. 3, 1350009, 10 p. (2013; Zbl 1271.57028)].
Some related results can be found in [M. Sakurai, J. Knot Theory Ramifications 22, No. 3, 1350009, 10 p. (2013; Zbl 1271.57028)].
Reviewer: Zhiyun Cheng (Beijing)
MSC:
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
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