A generalization of Turaev’s virtual string cobracket and self-intersections of virtual strings

Previously we defined an operation [Formula: see text] that generalizes Turaev’s cobracket for loops on a surface. We showed that, in contrast to the cobracket, this operation gives a formula for the minimum number of self-intersections of a loop in a given free homotopy class. In this paper, we consider the corresponding question for virtual strings, and conjecture that [Formula: see text] gives a formula for the minimum number of self-intersection points of a virtual string in a given virtual homotopy class. To support the conjecture, we show that [Formula: see text] gives a bound on the minimal self-intersection number of a virtual string which is stronger than a bound given by Turaev’s virtual string cobracket. We also use Turaev’s based matrices to describe a large set of strings [Formula: see text] such that [Formula: see text] gives a formula for the minimal self-intersection number [Formula: see text]. Finally, we compare the bound given by [Formula: see text] to a bound given by Turaev’s based matrix invariant [Formula: see text], and construct an example that shows the bound on the minimal self-intersection number given by [Formula: see text] is sometimes stronger than the bound [Formula: see text].

COVERINGS, COMPOSITES AND CABLES OF VIRTUAL STRINGS

A virtual string can be defined as an equivalence class of planar diagrams under certain kinds of diagrammatic moves. Virtual strings are related to virtual knots in that a simple operation on a virtual knot diagram produces a diagram for a virtual string. In this paper we consider three operations on a virtual string or virtual strings which produce another virtual string, namely covering, composition and cabling. In particular we study virtual strings unchanged by the covering operation. We also show how the based matrix of a composite virtual string is related to the based matrices of its components, correcting a result by Turaev. Finally we investigate what happens under cabling to some invariants defined by Turaev.

A SEQUENCE OF DEGREE ONE VASSILIEV INVARIANTS FOR VIRTUAL KNOTS

For ordinary knots in R3, there are no degree one Vassiliev invariants. For virtual knots, however, the space of degree one Vassiliev invariants is infinite-dimensional. We introduce a sequence of three degree one Vassiliev invariants of virtual knots of increasing strength. We demonstrate that the strongest invariant is a universal Vassiliev invariant of degree one for virtual knots in the sense that any other degree one Vassiliev invariant can be recovered from it by a certain natural construction. To prove these results, we extend the based matrix invariant introduced by Turaev for virtual strings to the class of singular flat virtual knots with one double-point.

AN INVARIANT FOR OPEN VIRTUAL STRINGS

Extended Alexander groups are used to define an invariant for open virtual strings. Examples of non-commuting open strings and a ribbon-concordance obstruction are given. An example is given of a slice open virtual string that is not ribbon. Definitions are extended to open n-strings.