Arunima Ray

Injectivity of satellite operators in knot concordance

Let P be a knot in a solid torus, K a knot in 3-space and P(K) the satellite knot of K with pattern P. This defines an operator on the set of knot types and induces a satellite operator P:C--> C on the set of smooth concordance classes of knots. There has been considerable interest in whether certain such functions are injective. For example, it is a famous open problem whether the Whitehead double operator is weakly injective (an operator is called weakly injective if P(K)=P(0) implies K=0 where 0 is the class of the trivial knot). We prove that, modulo the smooth 4-dimensional Poincare Conjecture, any strong winding number one satellite operator is injective on C. More precisely, if P has strong winding number one and P(K)=P(J), then K is smoothly concordant to J in S^3 x [0,1] equipped with a possibly exotic smooth structure. We also prove that any strong winding number one operator is injective on the topological knot concordance group. If P(0) is unknotted then strong winding number one is the same as (ordinary) winding number one. More generally we show that any satellite operator with non-zero winding number n induces an injective function on the set of Z[1/n]-concordance classes of knots. We extend some of our results to links.

Satellite operators as group actions on knot concordance

introduce a generalization of patterns that form a group (unlike traditional patterns), modulo a generalization of concordance. Generalized patterns induce functions, called generalized satellite operators, on concordance classes of knots in homology spheres; using this we recover the recent result of Cochran and the authors that patterns with strong winding number 1 induce injective satellite operators on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth 4‐dimensional Poincare conjecture. We also obtain a characterization of patterns inducing surjective satellite operators, as well as a sufficient condition for a generalized pattern to have an inverse. As a consequence, we are able to construct infinitely many nontrivial patterns P such that there is a pattern P for which P.P.K// is concordant to K (topologically as well as smoothly in a potentially exotic S 3 a0;1c) for all knots K ; we show that these patterns are distinct from all connected-sum patterns, even up to concordance, and that they induce bijective satellite operators on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth 4‐dimensional Poincare conjecture. 57M25

Satellite operators with distinct iterates in smooth concordance

Let P be a knot in an unknotted solid torus (i.e. a satellite operator or pattern), K a knot in S 3 and P (K) the satellite of K with pattern P. For any satellite operator P , this correspondence gives a function P : C ! C on the set of smooth concordance classes of knots. We give examples of winding number one satellite operators P and a class of knots K, such that the iterated satellites P i (K) are distinct as smooth concordance classes, i.e. if i 6 j 0, P i (K) 6 P j (K), where each P i is unknotted when considered as a knot in S 3 . This implies that the operators P i give distinct functions onC, providing further evidence for the fractal nature ofC. There are several other applications of our result, as follows. By using topologically slice knots K, we obtain innite families fP i (K)g of topologically slice knots that are distinct in smooth concordance. We can also obtain innite families of 2{component links (with unknotted components and linking number one) which are not smoothly concordant to the positive Hopf link. For a large class of L{space knots K (including the positive torus knots), we obtain innitely many prime knots fP i (K)g which have the same Alexander polynomial as K but are not themselves L{space knots.

On the Upsilon invariant and satellite knots

We study the effect of satellite operations on the Upsilon invariant of Ozsváth–Stipsicz–Szabó. We obtain results concerning when a knot and its satellites are independent; for example, we show that the set

Shake slice and shake concordant knots

\{D_{2^i,1}\}_{i=1}^\infty

Concordance of knots in S1×S2: CONCORDANCE OF KNOTS IN S1×S2

{D2i,1}i=1∞ is a basis for an infinite rank summand of the group of smooth concordance classes of topologically slice knots, for D the positive clasped untwisted Whitehead double of any knot with positive

Casson towers and filtrations of the smooth knot concordance group

\tau