# Concordance

There are a lot of research talks this week at Stanford, so I’ll write about a few of them in the next few days.

Matt Hedden and Jen Hom both gave talks about knot concordance, so I’ll talk about what both of them said here. One goal in studying knot concordance is to try to understand something about 4-manifolds. Finding knots with certain concordance properties can be used to show the existence of exotic $\mathbb{R}^4$‘s. While knot concordance may not be able to tell us everything about smooth 4-manifolds, it can lead to interesting results. The goal then is to learn as much as possible about the size and structure of the concordance group.

Concordance definitions:

Two knots are smoothly concordant if they are the boundary of a proper embedding $(S^1\times[0,1], S^1\times\{0,1\})\to (S^3\times[0,1], S^3\times\{0,1\})$. This is an equivalence relation, and the space of knots modulo concordance equivalence forms an abelian group, (addition is connected sum, the unknot is the identity, and negation is the mirror image with reversed orientation). An equivalent definition for $K_1$ and $K_2$ to be concordant is that $K_1\#\overline{K_2}^r$ is smoothly slice (bounds a disk in the 4-ball). We can also consider the topological concordance equivalence, which instead of requiring the above mapping of the cylinder to be smooth, we only require that it is continuous and extends to a continuous map on a tubular neighborhood. Topological concordance is a weaker equivalence than smooth concordance. Let $\mathcal{C}$ denote the group of knots up to smooth concordance, and $\mathcal{C}^{top}$ denote the group of knots up to topological concordance.

There is good motivation to focus on topologically slice knots (topologically concordant to the unknot), up to the equivalence relation given by smooth concordance. Denote this space by $\mathcal{C}_{TS}$. Showing that $\mathcal{C}_{TS}$ is non-trivial implies the existence of exotic $\mathbb{R}^4$‘s.

History of results regarding $latex\mathcal{C}_{TS}$:

Casson showed in 82 using Donaldson’s diagonalization theorem that there are knots with trivial Alexander polynomial, which are not smoothly slice. Freedman showed around the same time that knots with trivial Alexander polynomial are necessarily topologically slice. Combining these results shows that $\mathcal{C}_{TS}$ is non-trivial.

In 95 Endo showed that this group is big, namely there is a copy of $\mathbb{Z}^{\infty}\subset \mathcal{C}_{TS}$. Livingston, and Manolescu-Owens more recently showed that $\mathcal{C}_{TS}$ contains a direct summand of $\mathbb{Z}^3$ distinguished using the $\tau$ and $s$ invariants from Heegaard Floer and Khovanov homologies, which are both concordance invariants.

Satellite operations and concordance:

One operation on knots that works compatibly with concordance is forming satellites. Given a pattern knot P embedded in a solid torus, one obtains a map from knots to knots sending a knot K, to its satellite with that pattern P(K) (embed the pattern solid torus in a neighborhood of the knot K). A particularly useful example is the Whitehead double whose pattern is:

It is a consequence of the Skein relation that the Whitehead double of a knot is trivial (resolve the crossing at the clasp), so by Freedman’s result above, all Whitehead doubles are topologically slice, and thus represent elements of $\mathcal{C}_{TS}$. One may want to understand how many elements in $\mathcal{C}_{TS}$ can be represented by Whitehead doubles. Hedden and Kirk prove that there there is a $\mathbb{Z}^{\infty}\subset Image(D) \subset \mathcal{C}_{TS}$. The knots are Whitehead doubles of torus knots, and the proof uses SO(3) gauge theory to show the knots are not smoothly concordant.

Another infinite family of independent topologically slice knots formed via satellite operations, which is independent of both Endo’s examples and the above examples, are (p,1) cables of the Whitehead double of the right-handed trefoil. Jen Hom distinguishes these in the concordance group using her concordance invariants from Heegaard Floer homology. She defines an invariant $\varepsilon\in \{-1,0,1\}$ which can be computed from the chain complex $CFK^{\infty}(K)$ through an algebraic process involving the $\tau$ invariant (which is also a concordance invariant), or more geometrically by looking at the triviality or nontriviality of cobordism maps on $\widehat{HF}$ induced by large integer surgeries on the knot in $S^3$. This is a concordance invariant of the knot, and it can be used to create a new equivalence relation on knots through their Heegaard Floer chain complexes. The idea is as follows. We can associate to a knot K, the complex $CFK^{\infty}(K)$ and to its inverse in the concordance class, we get $(CFK^{\infty}(K))^*$. The analog of addition in the knot concordance group is the tensor product of chain complexes by the following Kunneth formula: $CFK^{\infty}(K_1\#K_2) = CFK^{\infty}(K_1)\otimes CFK^{\infty}(K_2)$. If a knot K is smoothly slice, then $\varepsilon(CFK^{\infty}(K))=0$. With the concordance equivalence relation, we started with a monoid of knots under connected sum, and mod out by the concordance equivalence relation to get a group. Similarly, the $CFK^{\infty}$ complexes form a monoid under tensor product and we obtain a group if we mod out by the equivalence relation $CFK^{\infty}(K_1)\sim_{\varepsilon} CFK^{\infty}(K_2) \iff \varepsilon(CFK^{\infty}(K_1)\otimes CFK^{\infty}(K_2)^* = 0$. The resulting group $\mathcal{F} = \{CFK^{\infty}(K): K\subset S^3\}/\sim_{\varepsilon}$ has additional useful structure: a total ordering, a notion of much greater than, and a filtration. These structures can be used to show linear independence of knots in the concordance group.

Based on how $\varepsilon$ is defined, there is definitely a relation to the $\tau$ invariant, but $\varepsilon$ is a more powerful invariant. It turns out that the exact relation is related to the satellite operation. Jen proved that $CFK^{\infty}(K_1)\sim_{\varepsilon} CFK^{\infty}(K_2)$ if and only if $\tau(P(K_1)) = \tau(P(K_2))$ for every pattern P. Furthermore the satellite map descends to a well defined map on the group of knots up to $\varepsilon$ equivalence.

More pieces of the topologically slice concordance group:

Since we have a lot of examples of independent concordance classes with trivial Alexander polynomial obtained by Whitehead doubles, one may ask whether the smallest subgroup generated by knots with trivial Alexander polynomial gives all topologically slice concordance classes, i.e. does $\mathcal{C}_{\Delta} := \langle \{[K]: \Delta_K=1\}\rangle = \mathcal{C}_{TS}$? The answer to this question is strongly no. Hedden and Livingston prove that there is an infinitely generated free abelian subgroup in the quotient: $\mathbb{Z}^{\infty}\subset \mathcal{C}_{TS}/\mathcal{C}_{\Delta}$.

So now we know that there are lots of knot concordance classes with trivial Alexander polynomial, lots with nontrivial Alexander polynomial, but each of these constructions produce concordance classes of infinite order. We can also ask about torsion in the knot concordance group. The easiest kind of torsion to understand is 2-torsion. In this case $0=2[K]$ so $[K]=-[K]=[\overline{K}^r]$, i.e. the knot is isotopic to its reverse mirror image. Such knots have been studied for awhile, and are called amphichiral. There are lots of such knots, so it is reasonable to expect some of the topologically slice knots to have this property.

Indeed there are lots of amphichiral knots which are smoothly concordance independent, but also topologically slice. The theorem is due to Hedden, S.G. Kim, and Livingston:
$(\mathbb{Z}/2)^{\infty} \subset \mathcal{C}_{TS}$.

The knots is this family are constructed by starting with an amphichiral knot that is not topologically slice, and then performing satellite operations with different knots, and taking connected sums to obtain topologically slice knots that are amphichiral. Next one needs to show that these knots are not smoothly slice, and that they represent independent concordance classes. Here you need Heegaard Floer homology. The obstruction to K being slice comes from the d-invariant. The d-invariant, $d(Y,\mathfrak{s})$ is keeping track of the highest grading of the generator of the nontorsion elements in the (minus) Heegaard Floer homology of a $\mathbb{Z}/2$ homology 3-sphere. To use this to obstruct sliceness, first one notices that if K were smoothly slice, then its branched double cover $\Sigma(K)$ would be a $\mathbb{Z}/2$ homology 3-sphere and it would bound a $\mathbb{Z}/2$ homology 4-ball $Q^4$ (this comes from looking at the double branched cover of the 4-ball branched over the slice disk). This implies that the d-invariant $d(Y,\mathfrak{s})=0$ for all spin-c structures on Y which are a restriction of a spin-c structure on Q. Since we are trying to obtain a contradiction, and show that such a Q does not exist, we don’t know exactly which spin-c structures will show up on the boundary. However such spin-c structures will satisfy certain properties (e.g. they form a subgroup of a certain size in $H_1(Y)$. One can explicitly compute the d-invariants for the candidate 2-torsion knots, and look for possible spin-c subgroups satisfying the necessary conditions, and rule out the possibility that $d(Y,\mathfrak{s})$ vanishes for all required $\mathfrak{s}$.

Concordance genus:

The Seifert genus and 4-ball genus of a knot by definition satisfy the inequality $g_4(K)\leq g(K)$. We can define an intermediate genus, called the concordance genus $g_c(K) :=\min\{g(J): J\sim K\}$. One may ask what the possible size of the gaps can be in the inequality $g_4(K)\leq g_c(K) \leq g(K)$. The gap between concordance genus and slice genus can be made arbitrarily large by connect summing nontrivial slice knots, but it is more difficult to get gaps between $g_4(K)$ and $g_c(K)$. The first result regarding this problem is due to Nakanishi who found knots with concordance genus arbitrarily larger than 4-ball genus. Livingston improved this result and found algebraically slice (though not topologically slice knots) with $g_4(K)=1$ but $g_c(K)$ arbitrarily large. Jen improved this result even further with her $\varepsilon$ equivalence, finding examples of topologically slice knots that all have $g_4(K)=1$, but $g_c(K)=p$ for each $p\geq 1$.

The moral seems to be, invariants defined through Heegaard Floer homology have been very useful in mapping out more of the concordance group, and providing lots of example of topologically slice, concordance-independent knots.