PARAMETERIZATION OF KNOTTED SURFACES ARISING FROM CLASSICAL AND WELDED KNOTS

Abstract

In this thesis we discuss how to parameterize certain two dimensional knots in four space using simple polynomial functions. More specifically we have taken up 2 dimensional sphere S^2 and the torus T^2 embedded inside S^4 or R^4 as examples of knotted surfaces. We also touch upon knotted planes of special type, as non-compact 2 dimensional knots referred as the long 2 knots. It is difficult to provide a general scheme to parameterize all surface knots since a structural classification of surface knots is still not well understood. We have chosen those knotted surfaces which arise from some known 1 dimensional knot theories such as of the classical knots, long knots and the welded knots. For example, given a classical knot one can obtain knotted spheres using spun and twist spun constructions. Similarly using a result of S. Satoh, knotted tori can be obtained as a tube of some welded knot. Likewise, knotted planes can be obtained using the theory of long knots and also from slice knots. In this thesis, we have used a polynomial parameterization of a long knot K to parameterize the spun of K and the d twist spun of K for all d ≥ 1. To parameterize ribbon torus knots, trigonometric parameterizations of classical knots are utilized. Knotted planes are constructed using a parameterization of a long knot. In each case we have described a method to explicitly construct a parametrization. We have also demonstrated our results by examples and included the plots of their 3 dimensional projection obtained using Mathematica. In knot theory one of the main questions is to detect if a given knot is non-trivial. We have answered this for the ribbon torus knots by ensuring the associated welded knots have their fundamental groups not isomorphic to Z. In this thesis, we have discussed some invariants of welded knots that are un-knot detectors. We have examined for few families of classical knots (such as torus knots of type (2, 2n + 1) and twist knots) how to obtain different non-trivial ribbon torus knots by welding few of the crossings in their standard diagrams.