Knot and Link Tricolorability

Abstract

Knot Theory, a field of Topology, can be used to model and understand how enzymes (called topoisomerases) work in DNA processes to untangle or repair strands of DNA. In a human cell nucleus, the DNA is linear, so the knots can slip off the end, and it is difficult to recognize what the enzymes do. However, the DNA in mitochondria is circular, along with prokaryotic cells (bacteria), so the enzyme processes are more noticeable in knots in this type of DNA. A strand of DNA as the 41 knot. Image from paper by De Witt Sumners. Invariants prove to be a useful tool in studying when two knots are different. Tricolorability is an easily understood invariant we will use to distinguish doubles (replications) of certain prime knots. Our team studied knots and links which have been observed in DNA. Specifically, we considered what happens to the colorability after performing a doubling operation.