 
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
email: 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 3sphere
S3 = R3
{±}. All knots are oriented. Recall that the blowup of a smooth oriented 4manifold
X is the 4manifold X#CP 2. We generalize the blowup of smooth spheres defined in R. Gompf and
A. Stipticz [11] to the blowup 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
