Basic Geometric Constructions

Basic geometric constructions, including tubing, boundary twisting, pushing down intersections, and contraction followed by push-off are presented. These moves are used repeatedly later in the proof. New, detailed pictures illustrating these constructions are provided. The Clifford torus at an intersection point between two surfaces in 4-dimensional space is described. The chapter closes with an important application of some of these moves called the Geometric Casson Lemma. This lemma upgrades algebraically dual spheres to geometrically dual spheres, at the cost of introducing more self-intersections. It is also shown that an immersed Whitney move is a regular homotopy of the associated surfaces.

Gamma-polynomial and its generalization to a 2-string tangle polynomial

The zeroth coefficient polynomial of the skein (HOMFLYPT) knot polynomial called the Γ-polynomial is studied from a viewpoint of regular homotopy of knot diagrams. In particular, an elementary existence proof of the knot invariance of the Γ-polynomial is given. After observing that there are three types for 2-string tangle diagrams, the Γ-polynomial is generalized to a polynomial invariant of a 2-string tangle. As an application, we have a new proof of the assertion that Kinoshita's θ-curve is not equivalent to the trivial θ-curve.

Approximate Analytical Solutions of the Regularized Long Wave Equation Using the Optimal Homotopy Perturbation Method

The paper presents the optimal homotopy perturbation method, which is a new method to find approximate analytical solutions for nonlinear partial differential equations. Based on the well-known homotopy perturbation method, the optimal homotopy perturbation method presents an accelerated convergence compared to the regular homotopy perturbation method. The applications presented emphasize the high accuracy of the method by means of a comparison with previous results.

REGULAR HOMOTOPIC DEFORMATION OF COMPACT SURFACE WITH BOUNDARY AND MAPPING CLASS GROUP

A necessary and sufficient algebraic condition for a diffeomorphism over a surface embedded in S3 to be induced by a regular homotopic deformation is discussed, and a formula for the number of signed pass moves needed for this regular homotopy is given.