Affiliation:
1. Smith College, Northampton, MA 01063, USA
2. Max Planck Institute for Mathematics, Vivatsgasse 7, Bonn 53111, Germany
Abstract
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].
Publisher
World Scientific Pub Co Pte Lt
Subject
Applied Mathematics,General Mathematics
Cited by
4 articles.
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