#### Event Title

Session 1F: Brunnian Links and Tricolorability

#### Session Number

Session 1F: 2nd Presentation

#### Advisor(s)

Dr. Louis Kauffman, University of Illinois Chicago

#### Location

Room A113

#### Start Date

26-4-2018 9:40 AM

#### End Date

26-4-2018 10:25 AM

#### Abstract

As ropes and other one dimensional extended objects, knots and links can be found in everyday life. In the study of Knot Theory, a knot is an embedding of a circle in three dimensional space and is represented by a diagram in the plane. A link is an embedding of number of disjoint circles and is correspondingly represented by a diagram. Tricolorability is a classifying trait of such graphs. For a knot to be tricolorable, each arc in the diagram is assigned a color according to the following rules (1) each arc in the diagram receives a color, (2) at a crossing either all 3 colors appear, or only one color appears. Non tricolorable links cannot be unlinked! We use this theorem to prove that an infinite class of Brunnian Links are linked. This demonstrates an intriguing inverse method for proving that links are linked via uncolorability. We shall denote a specific Brunnian link of n components as an n-Brunnian Link. Such links have the property that the removal of any single component results in a trivial link of n-1 components.

Session 1F: Brunnian Links and Tricolorability

Room A113

As ropes and other one dimensional extended objects, knots and links can be found in everyday life. In the study of Knot Theory, a knot is an embedding of a circle in three dimensional space and is represented by a diagram in the plane. A link is an embedding of number of disjoint circles and is correspondingly represented by a diagram. Tricolorability is a classifying trait of such graphs. For a knot to be tricolorable, each arc in the diagram is assigned a color according to the following rules (1) each arc in the diagram receives a color, (2) at a crossing either all 3 colors appear, or only one color appears. Non tricolorable links cannot be unlinked! We use this theorem to prove that an infinite class of Brunnian Links are linked. This demonstrates an intriguing inverse method for proving that links are linked via uncolorability. We shall denote a specific Brunnian link of n components as an n-Brunnian Link. Such links have the property that the removal of any single component results in a trivial link of n-1 components.