Birman-Ko-Lee left canonical form and its applications & improving students confidence in mathematics and problem solving skills

In the first part, we investigate Birman, Ko, and Lee's left canonical form of a braid. Topology is an area of mathematics which studies objects in space and their properties. Knots and braids are common objects that are studied in Topology. A braid is comprised of n-strands where the strands are twisted together. Figure 1 shows an example of a 4-braid.

In Topology, we are interested in differentiating between objects. When are two braids the same (or different)? The left canonical form of a braid solves this problem in an algebraic way. We used Birman, Ko, and Lee's left canonical form algorithm to create a diagrammatic version. This is a more geometric approach that can be simpler to compute for larger braids.

We are interested in a special class of braids called almost strongly quasipositive braids, which are braids with all positive crossings and at most one negative crossing. Figure 1 shows an almost strongly quasipositive braid, with the single negative crossing highlighted. We investigate this class of braids in terms of the left canonical form to find new and exciting properties and applications.

In the second part, we investigate students’ confidence in mathematics and problem solving skills. Every math instructor has heard students utter the phrase, “I’m bad at math.” This attitude toward mathematics causes students to give up easily when working through a tough problem. With hard work, a positive attitude, and the right supports in place, every student can be successful at mathematics. This project implements new teaching methods to improve students’ confidence, attitude, and motivation toward mathematics.

In particular, we look at five different factors that influence students’ attitudes toward mathematics. We use new teaching strategies to positively influence students confidence in mathematics, such as implementing group work, being available outside of class for one-on-one help, and creating a supportive and safe class environment, among others.