Kristin Kooiman April 2, 2016 Math 101 Knot Theory Physicist s first became interested in studying knots in the mid 1800 s when scientists were looking for a way to model the atom. Although this idea was later proven false, mathematicians became interested in the concept of knot theory and today it can be applied to areas in biology, chemistry, and physics. Carl Friedrich Gauss was the first to make an advance in the study of Knot Theory in mathematics. He created a way to tabulate knots by drawing the knot, labeling the crossings, and recording the order of letters you pass when starting at one point on the knot and finishing in the same spot. For example, this trefoil knot is recorded as ABCABC (Colberg). A C B Guass went on to study knots in his work in electrodynamics. There are several more mathematicians that contributed to the area of Knot Theory.
Johann Benedict Listing was interested in whether knots were equivalent when looking at the mirror image and discovered the right and left trefoil knots are not. Peter Guthrie Tait made the first table of knots, classifying each knot up to seven crossings O Connor, 2016). Thomas Kirkman made tables with diagrams classifying each knot up to eleven crossings. Tait joined Charles Newton Little to continue working on Kirkman s classification of different knot types. They put together tables that consisted of 800 different knots, from the simplest to the most complicated (Murasugi). Tait and Little eventually published the first official table of alternating knots up to ten crossings (Colberg). James Waddell Alexander created the Alexander polynomial in 1928. This polynomial is an invariant, which means it never changes. This was the first discovered polynomial invariant and helped tell the difference between knots from other knots (Colberg). First Alexander began with the Alexander matrix by labeling a knot diagram. To achieve the Alexander matrix, label the universe of the link as shown: 1-1 -t t -1 t 1 -t A knot with n crossings has (n+2) in the region and a diagram with n crossings will be
an n x (n+2) matrix. The last two rows are removed and the reduced matrix of can be used to determine the Alexander polynomial of the knot (Colberg). The Alexander polynomial is defined as follows: =det ( ) The Alexander polynomial for the trefoil knot can be used by first labeling the regions and crossings as shown: C3 R4 R3 R5 C1 C2 R1 R2 The C s represent the crossings and the R s represent the regions. Next the Alexander matrix of this trefoil knot can be represented as: R1 R2 R3 R4 R5 C1 -t 1 0 t -1 C2 -t t 1 1-1 C3 -t 0 t t -1 The last two rows are removed and the reduced Alexander matrix is:
-t 1 0 (t) = -t t 1 -t 0 t Next, to get the Alexander polynomial, determine the determinant of the reduced matrix: = -t 3 +t 2 -t Finally, a t can be factored out of the determinant to get the final polynomial: = -t 2 +t-1 Much more work has gone into the study of Knot Theory over the years and many applications of Knot Theory are used today in other areas. Knot Theory is used in biology for topology of DNA. Topoisomerases operate the DNA strands, and biochemist look at the effects this has on closed or circular DNA molecules (Colberg). Chemistry also uses Knot Theory to help determine the chirality or asymmetry of molecules, because some molecules are knotted. Physics uses actual statistical models to obtain certain knot invariants (Colberg). There are still so many questions about Knot Theory, and mathematicians will no doubt continue in the ongoing research to find the answers.
References Colberg, E. A Brief History of Knot Theory. Retrieved from http://www.math.ucla.edu/ ~radko/191.1.05w/erin.pdf Murasugi, K. (1996). Knot Theory and its Applications. Retrieved from http://www.maths.ed.ac.uk/~aar/papers/murasug3.pdf O Connor, J. and Robertson, E. (2016). MacTutor History of Mathematics Archive. Retrieved from http://www-history.mcs.st-andrews.ac.uk/index.html