Satellite operators with distinct iterates in smooth concordance

Recall that given any knot $J$ with $\text{tb}(J)=2\text{g}(J)-1$, the untwisted Whitehead double of $K$ has the same property. Therefore, since any untwisted Whitehead double is topologically slice, we have several examples of topologically slice knots $K$ that we may use in our main theorem. For any of the operators $P$ as in the main theorem, each $P^{i}(K)$ will be topologically slice (since $P$ is unknotted as a knot in $S^{3}$) and $P^{i}(K)\neq P^{j}(K)$ for all $i\neq j\geq 0$ in smooth (as well as exotic) concordance. This gives us the following corollary.

Several examples of infinite families of smooth concordance classes of topologically slice knots exist in the literature, such as in [End95, Gom86, HK12, Hom11]. Our examples are novel only due to the ease with which they are obtained and the added property that they are iterated satellites.

We can also obtain an interesting corollary about $L$–space knots, which are defined as follows. A homology sphere $Y$ is an $L$–space if $\widehat{HF}(Y)$ is the same as that of a lens space (here $\widehat{HF}$ is the Heegaard–Floer invariant introduced in [OS04]). A knot $K$ is called an $L$–space knot if some positive integer surgery on $S^{3}$ along $K$ yields an $L$–space. All positive torus knots, i.e. the knots $T_{p,\,q}$ with $p,q>0$, are well-known $L$–space knots, since $pq\pm 1$ surgery on them yield lens spaces. $L$–space knots have received much interest lately since their knot Floer complexes may be computed directly from their Alexander polynomials [OS05]. It was also shown in [OS05] that there are strong restrictions on the Alexander polynomial of $L$–space knots. Since several $L$–space knots, we obtain the following corollary.

As a third application of the main theorem, we can construct infinitely many links which are not smoothly concordant to the Hopf link. Any 2–component link $L=(P,\eta)$ with $\eta$ unknotted gives a satellite operator by considering the knot $P$ in the solid torus $S^{3}-N(\eta)$, where $N(\eta)$ is a regular neighborhood of $\eta$. It is easy to see that if two such links are concordant they give identical functions on $\mathcal{C}^{\text{ex}}$ [CDR14, Proposition 2.3]. Given a satellite operator $P$, we may consider the associated 2–component link $(P,\eta(P))$ where $\eta(P)$ is the meridian of the solid torus containing $P$.

In addition, we know from [CFHH13, Corollary 2.2] that if a winding number one satellite operator $P$ is unknotted (such as the one in Figure 2), then the zero–surgery manifolds on $P(K)$ and $K$ are homology cobordant, for any knot $K$. Recall that the $n$–solvable, positive, negative, and bipolar filtrations of $\mathcal{C}$ (from [COT04] and [CHH13], denoted by $\{\mathcal{F}_{n}\}$, $\{\mathcal{P}_{n}\}$, $\{\mathcal{N}_{n}\}$, and $\{\mathcal{B}_{n}\}$ respectively) can be defined in terms of the zero–surgery manifolds of knots. Therefore, if we start with a knot $K$ in $\mathcal{F}_{n}/\mathcal{F}_{n-1}$ (resp. $\mathcal{P}_{n}/\mathcal{P}_{n-1}$ or $\mathcal{B}_{n}/\mathcal{B}_{n-1}$) with $\text{tb}(K)=2\text{g}(K)-1$ and an unknotted operator $P$ for which the main theorem applies, we obtain a family $\{P^{i}(K)\}$ of infinitely many classes of knots also in $\mathcal{F}_{n}/\mathcal{F}_{n-1}$ (resp. $\mathcal{P}_{n}/\mathcal{P}_{n-1}$ or $\mathcal{B}_{n}/\mathcal{B}_{n-1}$). Since if $K$ has $\text{tb}(K)=2\text{g}(K)-1$ it has $\tau(K)>0$, we cannot directly use the same construction for $\{\mathcal{N}_{n}\}$, but the mirror images of the examples for $\{\mathcal{P}_{n}\}$ suffice.

The author would like to thank Tim Cochran, Jung Hwan Park, and Christopher Davis for their time and insightful discussions. We are also indebted to the participants of the Heegaard Floer “computationar” held at Rice University in Spring 2014, particularly Allison Moore and Eamonn Tweedy, for their insights into $L$–space knots and Heegaard–Floer homology.

A satellite operator, or pattern, is a knot in the standard unknotted solid torus $V=S^{1}\times D^{2}$. The winding number of a satellite operator $P$, denoted by $w(P)$, is the signed count of the number of intersections of $P$ with a generic meridional disk of $V$.

The set of satellite operators is a monoid in the following natural way. Given a satellite operator $P$ in a solid torus $V$, we see the following curves:

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  $\mu(P)$, the meridian of $P$ within $V$,

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  $\lambda(P)$, the longitude of $P$ within $V$,

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  $m(P)$, the meridian of $V$, and

• •

  $\ell(P)$, the longitude of $V$.

Given operators $P$, $Q$ in solid tori $V(P)$, $V(Q)$, we construct the composed pattern $P\star\,Q$ as follows. Let $N(Q)$ be a regular neighborhood of $Q$ inside $V(Q)$. Glue $V(Q)-N(Q)$ and $V(P)$ by identifying $\mu(Q)\sim m(P)$ and $\lambda(Q)\sim\ell(P)$. The resulting 3–manifold is a solid torus. The image of $P$ inside this solid torus is the desired operator $P\star\,Q$. An example of this construction is shown in Figure 3.

The well-known action of satellite operators on knots is closely related to the above construction. Given a knot $K$ and a satellite operator $P$ in a solid torus $V$, we obtain the satellite knot $P(K)$ as follows. Denote the meridian of $K$ by $\mu(K)$ and the longitude by $\lambda(K)$. Let $N(K)$ be a tubular neighborhood of $K$. Glue $S^{3}-N(K)$ and $V$ by identifying $\mu(K)\sim m(P)$ and $\lambda(K)\sim\ell(P)$. The resulting 3–manifold is $S^{3}$ and the image of $P$ inside this manifold is the knot $P(K)$. An example of this construction is given in Figure 1. For a survey of the satellite construction see [Lic97, p. 10] or [Rol90, p. 111].

It is easily seen that $(P\star\,Q)(K)=P(Q(K))$, i.e.  the satellite construction gives a monoid action on $\mathcal{K}$. We denote the satellite operator $P\star P\star\cdots\star P$ by $P^{i}$. Therefore, $P^{i}(K)=P(P(\cdots(K)\cdots))=(P\star P\star\cdots\star P)K$, i.e. we get the same result whether we start with a knot $K$ and apply $P$ to it $i$ times or we apply the composed pattern $P\star P\star\cdots\star P$ to $K$ once.

Given a satellite operator $P$, we denote by $\widetilde{P}$ the knot $P(U)$, where $U$ is the unknot. $P$ is said to be strong winding number one [CDR14, Definition 1.1] if $\mu(P)$ normally generates $\pi_{1}(S^{3}-\widetilde{P})$. If $\widetilde{P}$ is unknotted, $H_{1}(S^{3}-\widetilde{P})$ and $\pi_{1}(S^{3}-\widetilde{P})$ are canonically isomorphic and thereofore $P$ is strong winding number one if and only if it winding number one [CDR14, Proposition 2.1].

If the knots $K_{0}$ and $K_{1}$ are concordant in any 4–manifold $M$ bounded by two disjoint copies of $S^{3}$, the satellites $P(K_{0})$ and $P(K_{1})$ are concordant in $M$ for every operator $P\subseteq V$. This is easily seen as follows. Let $C$ be the concordance between $K_{0}$ and $K_{1}$. We excise a neighborhood of $C$ and glue in $V(P)\times[0,1]$. The image of $P\times[0,1]$ in the resulting manifold (which is diffeomorphic to $M$) is a concordance between $P(K_{0})$ and $P(K_{1})$. As a result, the satellite construction is well-defined on $\mathcal{C}$ and $\mathcal{C}^{\text{ex}}$.

An embedding of a knot $K$ in $S^{3}$ is said to be Legendrian if at each point in $S^{3}$ it is tangent to the 2–planes of the standard contact structure on $S^{3}$. Legendrian knots can be studied concretely through their front projections as in Figure 4. Legendrian knots have two classical invariants, the Thurston–Bennequin number, $\text{tb}(\cdot)$, and the rotation number, $\text{rot}(\cdot)$, both of which can be easily calculated via front projections. See [Etn05] for an excellent review of these and related concepts.

Given a Legendrian knot with positive Thurston–Bennequin number, we may repeatedly stabilize at the cost of increasing the rotation number to get a Legendrian diagram with zero Thurston–Bennequin number; such a diagram is a different Legendrian knot, but has the same topological realization as the original knot. See Figure 4 for an example.

We will make use of the slice–Bennequin inequality [Rud95, Rud97][Etn05, pp. 133], which states that for any Legendrian knot $K$,

Here $\text{g}_{4}(\cdot)$ stands for the smooth 4–genus of a knot, i.e. the least genus of a connected, oriented, smooth, properly embedded surface bounded by $K$ in $B^{4}$. Since some of our work will be in the exotic category, we show that the slice–Bennequin inequality has an exotic analog.

Exotic 4–genus is clearly an invariant of exotic concordance of knots, and is bounded above by the (classical) smooth 4–genus.

The Legendrian satellite operation is discussed in [Ng01, NT04]. There, a Legendrian pattern $P$ in $S^{1}\times\mathbb{R}^{2}$ acts on a Legendrian knot $K$ in $\mathbb{R}^{3}$. A front diagram for a Legendrian pattern in shown in Figure 5. In order to construct the (Legendrian) satellite knot $P(K)$ we take an $n$–copy of $K$ ($n$ ‘vertical’ parallels of $K$) and insert $P$ in a strand of $K$, where $n$ is the number of strands of $P$. This process is described in Figure 6. The resulting knot is the $\text{tb}(K)$–twisted satellite of $K$ and $P$. If $\text{tb}(K)=0$, the resulting knot is a Legendrian realization of the classical untwisted satellite knot $P(K)$.

The same construction applies when a Legendrian pattern $P$ in $S^{1}\times\mathbb{R}^{2}$ acts on another pattern $Q$ also in $S^{1}\times\mathbb{R}^{2}$. This construction is described in Figure 7. The resulting operator $P\cdot Q$ is the $\text{tb}(Q)$–twisted $P$–satellite of $Q$. Therefore, if $\text{tb}(Q)=0$, $P\cdot Q$ corresponds to the operator $P\star Q$ described in Section 2.1.

The following lemmata describe how the Thurston–Bennequin number and rotation number of satellites are related.