Virtual Knot Theory

Homotopy invariants of Gauss words

“…It is shown in Figure 4. Kauffman first defined this move in [5] and he used the name virtual switch for it in [6]. In [5] he showed that the involutary quandle, a virtual knot invariant, is invariant even under virtual switches.…”

“…Kauffman first defined this move in [5] and he used the name virtual switch for it in [6]. In [5] he showed that the involutary quandle, a virtual knot invariant, is invariant even under virtual switches. Since there exist virtual knots with different involutary quandles, we may conclude that the virtual switch is not an unknotting operation for virtual knots.…”

Parity Bracket Polynomial via Some Parity of Virtual Links

“…Virtual knot theory is introduced by Kauffman as a generalization of classical knot theory so that if two classical link diagrams are equivalent as virtual links, then they are equivalent as classical links [4]. A virtual link diagram is a link diagram in R 2 possibly with some encircled crossings without over/under information, called virtual crossings.…”

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“…Furthermore, long virtual knots do not commute; that is, the connected sum of two long knots K 1 #K 2 is not in general equivalent to the connected sum K 2 #K 1 , and so on. In the papers [20,21,22,25,26,35,36,37] and in many other papers, numerous invariants of virtual knots have been constructed.…”

“…We construct certain variants of them enabling us, in particular, to show that some pairs of virtual knots do not commute (from which it follows that each of these virtual knots is non-classical). Kauffman's fundamental work [35] is devoted to a detailed discussion of the theory of virtual knots; that paper is a precursor to his book [15] and our work, as well as to the survey articles [16] and [5].…”