HELICITY AS ISOTOPY INVARIANT

Abstract- In this paper we will investigate whether or not the writhe of knot is an isotopy invariant? By adding a parallel strand to an oriented knot/link, we will justify that writhe is a useful isotopy invariant. Since helicity has its origin in knot theory, consequently, we will eatablish that helicity is an isotopy invariant which will be useful to study plasma physics, solar physics and astrophysics. Keywords- Writh, Isotopy Invariant, Magnetic Field, Reidemeister Moves, Helicity. I. INTRODUCTION Mathematicians were perplexed at the seemingly unending number of ways a knot could be shaped and turned. Consequently, these give rise to the central problem of knot theory i.e., whether two knots (links) are equivalent or not. This was the motivation for much of the recent work in knot theory, which is devoted to search for invariants of knots. The study of invariants underwent in a kind of phase transition, which has linked knot theory to chemistry, molecular chemistry, mathematical physics, particles physics, polymer physics, statistical mechanics, fluid mechanics, kinematics, C*-algebra, conformal field theory, crystallography, cryptography, graph theory, computer systems and networks, etc. In the recent past, biologists and chemists studying genetics discovered an exciting link of knot theory with DNA (genetic material of all cells, containing coded information about cellular molecules and processes) and synthetic chemistry. DNA is just one application of knot theory, which presently is an area of intense mathematical activities worldwide. The relation between knots (links) and graphs has provided a new way of visualizing, redefining and deriving many concepts in knot theory. One can enjoy transferring some qualitative information between knot theory and graph theory. Plasma Physicists study helicity to solve complex problems using useful skills from knot theory. In this study we will investigate an analogue property of helicity from knot theory perspective. We will investigate whether or not the writhe of knot is an isotopy invariant? By adding a parallel strand to an oriented knot/link, we will justify that writhe is a useful isotopy invariant. Since helicity has its origin in knot theory, consequently, we will eatablish that helicity is an isotopy invariant which will be useful to study plasma physics, solar physics and astrophysics.