An Algorithm to Generate Two-Dimensional Drawings of Conway Algebraic Knots

• Main Author: Tung, Jen-Fu
• Format: Others
• Published: TopSCHOLAR® 2010
• Subjects: knot theory; Conway algebraic knots; discrete mathematics; Discrete Mathematics and Combinatorics; Mathematics; Numerical Analysis and Computation
• Online Access: http://digitalcommons.wku.edu/theses/163; http://digitalcommons.wku.edu/cgi/viewcontent.cgi?article=1164&context=theses

The problem of finding an efficient algorithm to create a two-dimensional embedding of a knot diagram is not an easy one. Typically, knots with a large number of crossings will not nicely generate two-dimensional drawings. This thesis presents an efficient algorithm to generate a knot and to create a nice two-dimensional embedding of the knot. For the purpose of this thesis a drawing is “nice” if the number of tangles in the diagram consisting of half-twists is minimal. More specifically, the algorithm generates prime, alternating Conway algebraic knots in O(n) time where n is the number of crossings in the knot, and it derives a precise representation of the knot’s nice drawing in O(n) time (The rendering of the drawing is not O(n).). Central to the algorithm is a special type of rooted binary tree which represents a distinct prime, alternating Conway algebraic knot. Each leaf in the tree represents a crossing in the knot. The algorithm first generates the tree and then modifies such a tree repeatedly to reduce the number of its leaves while ensuring that the knot type associated with the tree is not modified. The result of the algorithm is a tree (for the knot) with a minimum number of leaves. This minimum tree is the basis of deriving a 4-regular plane map which represents the knot embedding and to finally draw the knot’s diagram.