Parity and projection from virtual knots to classical knots (Q2850713)

The knot theory multiverse contains various generalizations of knots in \(S^3\): virtual knots, flat knots, free knots, pseudoknots, knots in thickened surfaces, homotopy classes of Gauss words, etc. An important aspect of the theory of these generalizations is that they are related by so-called functorial maps. A functorial map is map from a knot theory \(\mathcal{K}\) to another knot theory \(\mathcal{K}'\) such that if two knots are Reidemeister equivalent in \(\mathcal{K}\), then their images are Reidemeister equivalent in \(\mathcal{K}'\). In each case, the notion of Reidemeister equivalence is local to the particular generalized knot theory. An example of a functorial map is the inclusion of the classical knots into the virtual knots.\newline In the reviewed paper, the author proves the surprising result that there is a non-trivial functorial mapping from virtual knots to classical knots. Given a virtual knot diagram \(K\), it is shown that there is a classical knot \(\overline{K}\) which can be obtained from \(K\) by deleting some of its crossings (i.e. making them virtual). Moreover, \(\overline{K}=K\) if and only if \(K\) is a classical. The map (or projection) is functorial in the sense that if \(K_1\) and \(K_2\) are equivalent virtual knots, then \(\overline{K}_1\) and \(\overline{K}_2\) are equivalent classical knots. The projection is also surjective: every classical knot is in the image of the projection. \newline There is a very interesting corollary to this result. Suppose that you have a classical knot \(K\). If we consider \(K\) as a virtual knot, then \(K\) will have diagrams which have both classical and virtual crossings. Among all such diagrams, there is a smallest number \(\mu_{\text{virtual}}\) of classical crossings. On the other hand, if we consider only classical diagrams to which \(K\) is equivalent, then we also have a minimal number of classical crossings, \(\mu_{\text{classical}}\). Must it be the case that \(\mu_{\text{classical}}=\mu_{\text{virtual}}\)? The existence of the projection conclusively proves that \(\mu_{\text{classical}}=\mu_{\text{virtual}}\). A similar corollary about the bridge number of a classical knot is also established. \newline \newline The author also shows that the projection can be established in at least two distinct ways. Both methods rely on the following lemma/theorem: if the underlying surface of a virtual knot diagram \(K\) is not of minimal genus, then there is a knot diagram \(K' \leftrightharpoons K\) that can be obtained from \(K\) by deleting some of its crossings. Then beginning with a virtual knot diagram, the projection deletes classical crossings until a knot diagram is obtained such that its underlying surface is of minimal genus. The map may then be completed either by applying a projection due to I. Nikonov (a full proof is provided) or by using a projection derived from parity groups. The parity group method turns out to be the more discriminating of the two methods.\newline The author notes that it would be preferable for the projection to be computed directly from a Gauss diagram of the virtual knot. Indeed, both current methods require a reduction to a diagram whose underlying surface realizes the minimal genus. The author also asks whether there is a projection map satisfying the same properties as the given maps but which preserves as many classical crossings as possible.