FEATURED AUTHOR

# Heather Ann Dye

Associate Professor of Mathematics
McKendree University

I am an Associate Professor of Mathematics at McKendree University. McKendree is a small, primarily undergraduate school in Illinois near St. Louis, Missouri. My area of research is virtual knot theory and I have published multiple articles in this area. I enjoy working with undergraduates on research projects and teaching them knot theory. I also enjoy organizing conference sessions for the American Mathematical Society. I am a member of the AMS and MAA.

Subjects: Mathematics

Knot theory

Knot theory
Probability
Quilting

Featured Title

## Articles

#### Minimal surface representations of virtual knots and links.

Published: Feb 21, 2016 by Algebr. Geom. Topol. 5 (2005), 509–535.
Authors: Dye, H. A.; Kauffman, Louis H.

Kuperberg [Algebr. Geom. Topol. 3 (2003) 587-591] has shown that a virtual knot corresponds (up to generalized Reidemeister moves) to a unique embedding in a thichened surface of minimal genus. If a virtual knot diagram is equivalent to a classical knot diagram then this minimal surface is a sphere. Using this result and a generalised bracket polynomial, we develop methods that may determine whether a virtual knot diagram is non-classical (and hence non-trivial).

#### The three loop isotopy and framed isotopy invariants of virtual knots.

Published: Feb 21, 2016 by Topology Appl. 173 (2014), 107–134.
Authors: Chrisman, Micah W.; Dye, Heather A.

This paper introduces two virtual knot theory analogues'' of a well-known family of invariants for knots in thickened surfaces: the Grishanov-Vassiliev finite-type invariants of order two.

#### Vassiliev invariants from parity mappings.

Published: Feb 21, 2016 by J. Knot Theory Ramifications 22 (2013), no. 4, 1340008,
Authors: Dye, H. A.

Parity mappings from the chords of a Gauss diagram to the integers is defined. The parity of the chords is used to construct families of invariants of Gauss diagrams and virtual knots. One family consists of degree n Vassiliev invariants.

#### Smoothed invariants.

Published: Feb 21, 2016 by J. Knot Theory Ramifications 21 (2012), no. 13, 1240003,
Authors: Dye, H. A.

We construct two knot invariants. The first knot invariant is a sum constructed using linking numbers. The second is an invariant of flat knots and is a formal sum of flat knots obtained by smoothing pairs of crossings. This invariant can be used in conjunction with other flat invariants, forming a family of invariants. Both invariants are constructed using the parity of a crossing.

#### On two categorifications of the arrow polynomial for virtual knots.

Published: Feb 21, 2016 by The mathematics of knots, 95–124, Contrib. Math. Comput. Sci., 1, Springer, Heidelberg, 2011.
Authors: Dye, Heather Ann; Kauffman, Louis Hirsch; Manturov, Vassily Olegovich

Two categorifications are given for the arrow polynomial, an extension of the Kauffman bracket polynomial for virtual knots. The arrow polynomial extends the bracket polynomial to infinitely many variables, each variable corresponding to an integer {\it arrow number} calculated from each loop in an oriented state summation for the bracket.

#### Lower bounds on virtual crossing number and minimal surface genus.

Published: Feb 21, 2016 by The mathematics of knots, 31–43, Contrib. Math. Comput. Sci., 1, Springer, Heidelberg, 2011.
Authors: Bhandari, Kumud; Dye, H. A.; Kauffman, Louis H.

We compute lower bounds on the virtual crossing number and minimal surface genus of virtual knot diagrams from the arrow polynomial. In particular, we focus on several interesting examples.

#### Anyonic topological quantum computation and the virtual braid group.

Published: Feb 21, 2016 by J. Knot Theory Ramifications 20 (2011), no. 1, 91–102.
Authors: Dye, H. A.; Kauffman, Louis H.

We introduce a recoupling theory for virtual braided trees. This recoupling theory can be utilized to incorporate swap gates into anyonic models of quantum computation.

#### Virtual crossing number and the arrow polynomial.

Published: Feb 21, 2016 by J. Knot Theory Ramifications 18 (2009), no. 10, 1335–1357.
Authors: Dye, H. A.; Kauffman, Louis H.

We introduce a new polynomial invariant of virtual knots and links and use this invariant to compute a lower bound on the virtual crossing number and the minimal surface genus.

#### Virtual knots undetected by 1- and 2-strand bracket polynomials.

Published: Feb 21, 2016 by Topology Appl. 153 (2005), no. 1, 141–160.
Authors: Dye, H. A.

Kishino's knot is not detected by the fundamental group or the bracket polynomial; these invariants cannot differentiate between Kishino's knot and the unknot. However, we can show that Kishino's knot is not equivalent to unknot by applying either the 3-strand bracket polynomial or the surface bracket polynomial. In this paper, we construct two non-trivial virtual knot diagrams, KD and Km, that are not not detected by the bracket polynomial or the 2-strand bracket polynomial.

#### Promoting REU participation from students in underrepresented groups.

Published: Jan 01, 2014 by Involve 7 (2014), no. 3, 403–411.
Authors: Russell, Heather M.; Dye, Heather A.

Promoting REU participation from students in underrepresented groups.