#### Event Title

Bi-leveled Trees and Quotient of Twist2 Map

#### Session Number

Project ID: MATH 6

#### Advisor(s)

Dr. Aaron Lauve; Loyola University Chicago

#### Discipline

Mathematics

#### Start Date

22-4-2020 8:30 AM

#### End Date

22-4-2020 8:45 AM

#### Abstract

We analyze the structure of the bi-leveled trees labeling the vertices of the Multiplihedra, a family of polytopes placed between the permutahedra and associahedra. For both permutahedra and associahedra it is understood that there exist Hopf algebras on their vertices. In 2010, Lauve and others attempted to provide the vertices of the Multiplihedra with a similar Hopf algebra structure but did not achieve the establishment of a full Hopf algebra on bi-leveled trees.

Using the notion of the Twistk Hopf algebra introduced by Pilaud (2018), we establish a Hopf map Φ from Twist2 to its dual. From combinatorial evidence it appears that the image of Φ has the same graded dimensions as the span of the bi-leveled trees. Using this map and analyzing the Twist2 Hopf algebra using SageMath, we attempt to find a bijection between the bi-leveled trees and a basis for Twist2 / ker(Φ) to form a Hopf algebra on the vertices of the Multiplihedra.

In addition, we aim to prove a general statement for positive integers k. In particular, we desire to show that a Hopf map Φk exists from Twistk to its dual whose image has equal graded dimensions to the vector space on k-leveled trees.

Bi-leveled Trees and Quotient of Twist2 Map

We analyze the structure of the bi-leveled trees labeling the vertices of the Multiplihedra, a family of polytopes placed between the permutahedra and associahedra. For both permutahedra and associahedra it is understood that there exist Hopf algebras on their vertices. In 2010, Lauve and others attempted to provide the vertices of the Multiplihedra with a similar Hopf algebra structure but did not achieve the establishment of a full Hopf algebra on bi-leveled trees.

Using the notion of the Twistk Hopf algebra introduced by Pilaud (2018), we establish a Hopf map Φ from Twist2 to its dual. From combinatorial evidence it appears that the image of Φ has the same graded dimensions as the span of the bi-leveled trees. Using this map and analyzing the Twist2 Hopf algebra using SageMath, we attempt to find a bijection between the bi-leveled trees and a basis for Twist2 / ker(Φ) to form a Hopf algebra on the vertices of the Multiplihedra.

In addition, we aim to prove a general statement for positive integers k. In particular, we desire to show that a Hopf map Φk exists from Twistk to its dual whose image has equal graded dimensions to the vector space on k-leveled trees.