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4:00 pm Monday, August 27, 2012 Stulken Topology Seminar: Injectivity of satellite operators in knot concordanceby Arunima Ray (Rice University) in HB 427- (Joint work with Tim Cochran and Christopher Davis) Any knot P in a solid torus gives a well-defined map from the (smooth or topological) knot concordance group to itself, by mapping any knot K in S^3 to P(K) - the satellite knot with pattern P and companion K. We are interested in the question of when such an operator is injective, i.e. if P(K)=P(J), is K necessarily concordant to J? This family of questions has received much interest; for example, it is a famous open problem whether the Whitehead double operator is 'weakly injective', i.e. if Wh(K)=Wh(unknot), is K slice? We will show that any strong winding number one operator is injective on the smooth knot concordance group, modulo the 4-dimensional Poincare conjecture. We also show that such operators are injective on the topological knot concordance group. We also show that any winding number n operator is injective on the group of Z[1/n]-concordance classes of knots.
Submitted by cochran@rice.edu |