Free knots and parity (Q2884341)

In this paper, the author applies the concept parity to free knots and links and virtual knots and links. Parity is used to construct new invariants of free and virtual knots and used to prove theorems about minimality and non-triviality. Section 2 contains an overview of virtual knot theory which is followed by an introduction to free knots and links -- equivalence classes of framed 4-graphs modulo the virtual Reidemeister moves.NEWLINENEWLINEIn Section 3, the author introduces the concept of parity and specifically defines Gaussian parity, in which each crossing in the diagram is designated as even or odd. The behavior of parity under the Reidemeister moves is discussed. A functorial map on the set of virtual knots is defined using parity. Under this map, crossings with odd parity are virtualized and the number of classical crossings in the image is less than or equal to the number of classical crossings in the pre-image. This functorial map gives rise to a parity hierachy and the applications of this are explored.NEWLINENEWLINEIn Section 4, for a framed 4-graph \( \Gamma\), parity is used to define a cohomology class on elements of \( H_1 ( \Gamma, \mathbb{Z}_2 ) \). The elements of the homology class are generated by the halves of the graph obtained by smoothing a vertex. In Section 5, the author defines knot invariants using parity. The invariant \( [ \Gamma ] \) is obtained by summing over states with 1 component; the states are obtained by smoothing even crossings. The author also defines an even Kauffman bracket. Applications of these invariants and references are provided in the article. The author also discusses results involving atoms and parity and an even analogue of Turaev's cobracket.NEWLINENEWLINEIn the concluding section, the author constructs a group invariant using parity. The group \(G\) is \( \{ a,b,b' | a{}^2 =b{}^2 = b'{}^2 =1, ab = b' a \} \). From a Gauss diagram \(D\) of a knot, a word \( \gamma (D) \) is constructed based on the parity of the crossings. As an element of \(G\), \( \gamma(D) \) is an invariant of free knots. The author concludes with an example of a free knot that is shown to be non-trivial using parity.NEWLINENEWLINEFor the entire collection see [Zbl 1237.57001].