Obstructions to the Concordance of Satellite Knots
Cochran, Tim D.
Doctor of Philosophy thesis
Formulas which derive common concordance invariants for satellite knots tend to lose information regarding the axis a of the satellite operation R(a,J). The Alexander polynomial, the Blanchfield linking form, and Casson-Gordon invariants all fail to distinguish concordance classes of satellites obtained by slightly varying the axis. By applying higher-order invariants and using filtrations of the knot concordance group, satellite concordance may be distinguished by determining which term of the derived series of the fundamental group of the knot complement the axes lie. There is less hope when the axes lie in the same term. We introduce new conditions to distinguish these latter classes by considering the axes in higher-order Alexander modules in three situations. In the first case, we find that R(a,J) and R(b,J) are non-concordant when a and b have distinct orders viewed as elements of the classical Alexander module of R. In the second, we show that R(a,J) and R(b,J) may be distinguished when the classical Blanchfield form of a with itself differs from that of b with itself. Ultimately, this allows us to find infinitely many concordance classes of R(-,J) whenever R has nontrivial Alexander polynomial. Finally, we find sufficient conditions to distinguish these satellites when the axes represent equivalent elements of the classical Alexander module by analyzing higher-order Alexander modules and localizations thereof.
Geometric topology; Mathematics; Non commutative algebra; Knot theory