Virtual Crossing Numbers for Virtual Knots

Abstract

The aim of the present paper is to prove that the minimal number of virtual crossings for some families of virtual knots grows quadratically with respect to the minimal number of classical crossings.All previously known estimates for virtual crossing number ([1, 3, 18] etc.) were principally no more than linear in the number of classical crossings (or, what is the same, in the number of edges of a virtual knot diagram) and no virtual knot was found with virtual crossing number greater than the classical crossing number. MSC: 57M25, 57M27

Graphical constructions for the sl(3), C2 and G2 invariants for virtual knots, virtual braids and free knots

“…The above corollary easily allows one to reprove the theorem first proved in [32], that the number of virtual crossings of a virtual knot grows quadratically with respect to the number of classical knots for some families of graphs. In [32], it was done by using the parity bracket. Now, we can do the same by using With this invariant one can easily construct infinite series of trivalent bipartite graphs which serve as Kus for some sequence of knots Kn and such that the minimal crossing number for these graphs grows quadratically with respect to the number of crossings.…”

“…If this virtual crossing number is zero, then the link is classical. For some results about estimating virtual crossing number see [10,20,32] and see the results of Corollaries 3 and 4 in Section 3 of the present paper. In that section we not only count virtual crossings, we count combinatorial substructures of the diagram that may be unavoidable for a given invariant (in a sense that we specify later in the paper).…”

A graphical construction of the sl(3) invariant for virtual knots

“…If this virtual crossing number is zero, then the link is classical. For some results about estimating virtual crossing number see [8,13,17] and see the results of Corollaries 3 and 4 in Section 3 of the present paper.…”

Weak Parities and Functorial Maps

“…Additional information, which is introduced into knot diagrams with parity, allows one to strengthen knot invariants [1,[5][6][7][8][9][10][11][12]. The new invariants demonstrated the existence of nontrivial free knots as well as nontrivial cobordism classes of free knots [7].…”