| Well-known concordance invariants for a satellite knot R (eta, J) tend to be functions of R and J but depend only weakly on the axis eta. The Alexander polynomial, the Blanchfield linking form, and Casson-Gordon invariants all fail to distinguish concordance classes of satellites obtained by slightly varying the axes. By applying higher-order invariants and using filtrations of the knot concordance group, satellite concordance may be distinguished by determining the term of the derived series of pi1( S3 R ) in which 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 (eta1, J) and R (eta2, J) are non-concordant when eta 1 and eta2 have distinct orders in the classical Alexander module of R . In the second, we show that even when eta1 and eta 2 have the same order, R (eta1, J) and R (eta2, J) may be distinguished when the classical Blanchfield form of eta1 with itself differs from that of eta2 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. |
