Abstract
In this short note we use methods of Friedl, Livingston and Zentner to show that there are knots that are not algebraically concordant to a connected sum of positive and negative L-space knots.
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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
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
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
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
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].
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Acknowledgements
The author is very grateful to his advisor, Maciej Borodzik, for his help during preparation of the manuscript. He also expresses his gratitude towards Chuck Livingston, Stefan Friedl and András Stipsicz for their comments on the preliminary version of the manuscript.
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The author was supported by the National Science Center Grant 2016/22/E/ST1/00040.
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Yozgyur, R. Knots not concordant to L-space knots. Period Math Hung 80, 269–272 (2020). https://doi.org/10.1007/s10998-020-00329-y
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DOI: https://doi.org/10.1007/s10998-020-00329-y