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GENUS AND DEGREES OF TORUS KNOTS IN CP 2 MOHAMED AIT NOUH
 

Summary: GENUS AND DEGREES OF TORUS KNOTS IN CP 2
MOHAMED AIT NOUH
Department of Mathematics,
California State University Channel Islands
One University Drive, Camarillo, CA 93102
e-mail: mohamed.aitnouh@csuci.edu
ABSTRACT
We give the smooth slice genus and the possible degrees in CP 2 for all (2, q)-torus knots
for 3 q 11 . The proofs use gauge theory and twisting operations on knots.
1. Introduction
Throughout this paper, we work in the smooth category. All orientable manifolds will be as-
sumed to be oriented unless otherwise stated. A knot is a smooth embedding of S1 into the 3-sphere
S3 = R3
{}. All knots are oriented. Recall that the blow-up of a smooth oriented 4-manifold
X is the 4-manifold X#CP 2. We generalize the blow-up of smooth spheres defined in R. Gompf and
A. Stipticz [11] to the blow-up of any genus g smooth surface g X. If [g] H2(X, Z) satisfies
[g]2 = n, then blowing up g a number of times equal to n gives a new surface of the same genus
~g X#CP2 such that [~g]2 = 0, with [~] = [] -
i=n
i=1

  

Source: Ait Nouh, Mohamed - Mathematics Department, California State University Channel Islands

 

Collections: Mathematics