1 Introduction

A closed oriented 3-manifold Y is called an L-space if it is a rational homology sphere and its Heegaard–Floer homology has minimal possible rank i.e. \(\widehat{HF}(Y)=|H_1(Y;\mathbb {Z})|\). A knot \(K\subset S^3\) is a called (positive) L-space knot if a surgery with a sufficiently large positive coefficient is an L-space.

L-space knots were introduced by Ozsváth and Szabó in [10] in their attempt to classify knots such that a surgery on them gives a lens space. In particular, they proved the following result.

Theorem 1.1

[10, Corollary 1.3] If \(\Delta \) is the Alexander polynomial of an L-space knot, then \(\exists \)\(a_1,\ldots ,a_n\) such that \(a_0<a_1\ldots <a_n\) and

$$\begin{aligned} \Delta =t^{a_0}- t^{a_1}+t^{a_2}\cdots +t^{a_n} \end{aligned}$$
(1.1)

The class of L-space knots includes all torus knots. More generally, all algebraic knots are L-space knots; see [6, 7]. Moreover, in [8] Hedden proves the following result:

Theorem 1.2

An L-space knot is strongly quasipositive and fibred.

We refer to [12] for the definition and the properties of strongly quasipositive knots.

We have the following result of Borodzik and Feller.

Theorem 1.3

(see [1]) Every link L is topologically concordant to a strongly quasipositive link \(L'\).

In this light, the main theorem of the paper seems a bit surprising.

Theorem 1.4

There are knots that are not topologically concordant to any combinations of L-space knots and their mirrors.

2 Proof of Theorem 1.4

Let us recall the following fact, which can be found e.g. in [11, Section 8.1].

Theorem 2.1

(Cauchy’s bound) Suppose \(P=\alpha _0+\alpha _1 t+\dots +\alpha _m t^m\) is a complex polynomial with \(\alpha _m\ne 0\).

For any root z of P we have \(|z|<1+\max _{j<m}\frac{|\alpha _j|}{|\alpha _m|}\).

From Cauchy’s bound we obtain the following proposition.

Proposition 2.2

The Alexander polynomial of an L-space knot has roots with modulus at most 2.

Remark 2.3

From the multiplicativity of the Alexander polynomial under connected sums we infer also that if K is a connected sum of L-space knots and their mirrors, then \(\Delta _K(t)\) has all roots inside of the disk of radius 2. Note that by [9], a non-trivial connected sum of L-space knots is not an L-space knot anymore.

Following [4], for \(n=1,2,\dots \) we define

$$\begin{aligned} P_n=1+nt-(2n+1)t^2+nt^3+t^4. \end{aligned}$$
(2.1)

We need some properties of roots of this polynomial.

Theorem 2.4

(see [4, Lemma 4]) The polynomial \(P_n\) is irreducible over \(\mathbb {Q}[t]\) and it has two roots on the unit circle.

Denote these two roots by \(\theta _n^+\) and \(\theta _n^-\). We will use a more specific control for one of the other roots of \(P_n\).

Lemma 2.5

For \(n\ge 1\), the polynomial \(P_n\) has a root with modulus greater than 2.

Proof

We have \(P_n(-n-1)=1-2n-3n^2-n^3<0\). On the other hand, \(P(-(n+2))=13+10n+2n^2>0\). Therefore \(P_n\) has a real root in the interval \((-n-2,-n-1)\). \(\square \)

Theorem 2.6

Let K be connected sum of L-space knots and some mirrors of L-space knots, then \(\Delta _K \) cannot vanish at \(\theta _n^+\).

Proof

Since the Alexander polynomial of a connected sum of knots is the product of the Alexander polynomial of each knot, it is enough to prove the theorem for K being an L-space knot.

Suppose \(\Delta _K(\theta _n^+)=0\). Then \(\gcd (\Delta _K,P_n)\) has positive degree and it divides \(P_n\). As \(P_n\) is irreducible over \(\mathbb {Q}[t]\), it follows that \(P_n|\Delta _K\). But \(P_n\) has a root outside a disk of radius 2 and all the roots of \(\Delta _K\) are inside this disk. \(\square \)

Theorem 2.7

Suppose K is a knot that is concordant to a knot \(K'\) which is a connected sum of L-space knots and mirrors of L-space knots. Then the order of the root of \(\Delta _K\) at \(\theta _n^+\) is an even number.

Proof

As K and \(K'\) are concordant, an easy corollary of Fox-Milnor theorem [5] implies that there exist polynomials \(f,g\in \mathbb {Z}[t,t^{-1}]\) such that

$$\begin{aligned} \Delta _K(t)f(t)f(t^{-1})= \Delta _{K'}(t)g(t)g(t^{-1}). \end{aligned}$$
(2.2)

Claim. If \(\xi (t)\in \mathbb {Z}[t,t^{-1}]\) vanishes at \(\theta _n^+\), then \(\xi ^{-1}(t)\) vanishes at \(\theta _n^+\) with the same order.

To prove the claim, note that if \(\xi \) vanishes at \(\theta _n^+\) up to order s, then it is divisible by \((t-\theta _n^+)^s\) and also by \((t-\theta _n^-)^s\) (because \(\xi \) has real coefficients and \(\overline{\theta _n^+}=\theta _n^-\)). As \(\theta _n^+\theta _n^-=1\) we have

$$\begin{aligned} (t^{-1}-\theta _n^+)(t^{-1}-\theta _n^{-})=t^{-2}(t-\theta _n^+)(t-\theta _n^-). \end{aligned}$$

From the above identity the claim follows readily.

From the claim we conclude that the order of the root of \(\Delta _{K'}g(t)g(t^{-1})\) at \(\theta _n^+\) is an even number. Using the claim once again, this time for f(t), we see that (2.2) implies that \(\Delta _K(t)\) vanishes at \(\theta _n^+\) up to an even power (maybe zero). \(\square \)

To conclude the proof of Theorem 1.4 we will show that there exist knots such that their Alexander polynomial vanishes at \(\theta _n^+\) with an odd order. As \(P_n\) is a symmetric polynomial and \(P_n(1)=1\), for any n there exist a knot \(K_n\) such that \(\Delta _{K_n}=P_n\), see [13]. Furthermore, the knot \(K_n\) from [4, Figure 1] has Alexander polynomial \(P_n\).

Example 2.8

A notable example of a knot that is not concordant to a combination of L-space knots is the knot 12n642. According to KnotInfo web page [2], its Alexander polynomial is \(P_7\). On the other hand, 12n642 is strongly quasipositive and fibered. It is also almost positive in the sense of [3].