A variant of the unknotting number of a knot with applications to the topology of biomolecules

Abstract

The creation of unwound or unknotted loops of DNA is facilitated by topoisomerases, which alter the conformation of DNA by cleaving the phosphate backbone on one or both the strands. A major unexplored question has been the possibility of topoisomerases failing to unpack supercoiled DNA on any one of the several knots, thereby failing to initiate the unwinding process. Computing the unknotting number enables biologists to estimate the smallest amount of times the topoisomerase must act to unpack DNA. In this paper, we define a new variant called the fault-tolerant unknotting number t(K) of the unknotting number u(K) for a knot K. Besides, t(K) is related to the unknotting and knotting phenomena of DNA. We determine t(D) for some families of knot diagrams and conclude that the difference between u(D) and t(D) for a knot diagram D can be arbitrarily large. We also conjectured the existence of fault-tolerant unknotting number for every knot diagram and every knot.