#### Title

A Generalization of Turaev’s Virtual String Cobracket and Self-Intersections of Virtual Strings

#### Document Type

Article

#### Publication Date

2016

#### Publication Title

Communications in Contemporary Mathematics

#### Abstract

Previously we defined an operation µ 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 µ 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 µ 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 α such that µ gives a formula for the minimal self-intersection number α. Finally, we construct an example that shows the bound on the minimal self-intersection number given by µ is always at least as good as, and sometimes stronger than, the bound ρ given by Turaev’s based matrix invariant.

#### Volume

19

#### Issue

4

#### Rights

© the author

#### Recommended Citation

Cahn, Patricia, "A Generalization of Turaev’s Virtual String Cobracket and Self-Intersections of Virtual Strings" (2016). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.

https://scholarworks.smith.edu/mth_facpubs/54

## Comments

Peer reviewed accepted manuscript.