Combinatorics of Twist Boxes of Knots
Arts and Sciences
Mathematics and Statistics
In the study of knots, we can look at twist boxes--a set of consecutive half-twists in a knot diagram. Our project considers generalizations of twist boxes to higher-order twist boxes. We seek what combinations of first- and second-degree twist boxes exist in knots and how to construct any knot with those arbitrary combinations. We carefully define higher-order twist boxes in terms of checkerboard graphs of knots, and then confirm that the number of higher-order twist boxes depends only on the underlying knot chosen, not the graph or diagram used to represent it. We found several general cases showing general relations of number of degree one and degree two twist boxes of a knot that can coexist. Past results have shown the number of twist boxes corresponds to coefficients in the Jones polynomial, and for the future, we will seek a connection between degree two twist boxes and the Jones polynomial.
Smith, Nichole S., "Combinatorics of Twist Boxes of Knots" (2016). Symposium on Undergraduate Research and Creative Expression (SOURCE). 506.