THE THIRD TERM IN LENS SURGERY POLYNOMIALSMOTOO TANGE

Abstract. It is well-known that the second coeﬃcient of the Alexander polynomial of
any lens space knot is −1. We show that the non-zero third coeﬃcient condition ofthe Alexander polynomial of a lens space knot K in S 3 conﬁnes the surgery to the one
realized by the (2, 2g + 1)-torus knot, where g is the genus of K. In particular, such alens surgery polynomial coincides with ∆T (2,2g+1) (t).

1. Introduction
1.1. Lens space knots. If a knot K in a homology sphere Y yields a lens space byan integral Dehn surgery, then we call K a lens space knot in Y . The result obtainedby a Dehn surgery is written by Yp (K). Hence, the lens space surgery is presented as
Yp (K) = L(p, q). The homology class represented by the dual knot of the surgery isidentiﬁed with an element k in (Z/pZ)× . Precisely it is explained in Section 2. The pair(p, k) is called a lens surgery parameter.
We call a polynomial ∆(t) lens surgery polynomial (in Y ) if there exists a lens spaceknot K in Y such that ∆(t) = ∆K (t). It is well-known that any lens surgery polynomials
have interesting properties.
In [5], Ozsváth and Szabó proved that any lens surgery polynomials in S 3 are ﬂat andalternating. If the absolute values of all coeﬃcients of a polynomial are smaller than orequal to 1, we call the polynomial ﬂat. If the non-zero coeﬃcients of a polynomial arealternating sign in order, then we call the polynomial alternating. We call a polynomial∆ trivial, if ∆ = 1.
Any lens space knot with trivial Alexander polynomial in S 3 is isotopic to the unknot
due to [4]. Generally, if K is a lens space knot, then the degree of the Alexander polynomial
coincides with the Seifert genus g.
In this paper we use the following notations for coeﬃcients of any lens surgery polynomial:2gg∑∑∆(t) = t−g
αi ti =ai ti .
i=0i=−g

In other words, this equality implies αi = ai−g . By the symmetry of Alexander polynomial
we obtain ai = a−i and αi = α2g−i .
We consider non-trivial lens surgery polynomials from now. Then due to the author[7] and Hedden and Watson [3], any lens surgery polynomial in S 3 becomes the followingform around the top coeﬃcient tg :∆ = tg − tg−1 + · · · .
1991 Mathematics Subject Classiﬁcation. 57M25,57M27.
Key words and phrases. lens space surgery, Alexander polynomia, non-zero curve.
The author was partially supported by JSPS KAKENHI Grant Number 17K14180.
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In [3], it is shown that any L-space knot has the same form. Namely, the second coeﬃcientfrom the top has −1.
In [7], it is proven that the second top coeﬃcient of any lens space knot in any L-spacehomology sphere is −1.
1.2. Third top term of lens space polynomial. Let T (p, q) be the right-handed (p, q)torus knot. Teragaito asked an interesting question about the next coeﬃcient:
Question 1.1. If a non-trivial lens surgery polynomial in S 3 has the following form:∆ = tg − tg−1 + tg−2 + · · · ,then does ∆ coincide with ∆T (2,2g+1) ?
In other words, if a lens surgery polynomial is not a (2, 2g + 1)-torus knot polynomial
for some integer g, then α2 = 0?.
Here we give an aﬃrmative answer for this question.
Theorem 1.2. Teragaito’s question (Question 1.1) is true.
This theorem is true even if the lens space knot K lies in an L-space homology sphereand satisﬁes 2g(K) ≤ p by applying the same method.
Theorem 1.15 in [7] gave a criterion for a lens space knot K to satisfy ∆K (t) =∆T (2,2g+1) (t) for some positive integer g. On the other hand, we can also say that Theorem 1.2 gives a new criterion for a lens space knot to have the same Alexander polynomial
as that of T (2, 2g + 1).
1.3. Realization of lens surgery. We deﬁne the following terminology.
Deﬁnition 1.3. Let p, k be relatively prime positive integers. If a lens surgery Yp (K) =L(p, q) in a homology sphere Y has the lens surgery parameter (p, k), then we say that theparameter (p, k) is realized by a lens space knot K.
Corollary 1.4. Let K be a lens space knot in S 3 with the surgery parameter (p, k). The
Alexander polynomial ∆K (t) has the following form:∆K = tg − tg−1 + tg−2 + · · · ,if and only if (p, k) is realized by T (2, 2g + 1).
This condition in this corollary is equivalent to the condition of k = 2.
1.4. The cases of lens space knots Kp,k in Yp,k . Consider a simple (1, 1)-knot in alens space yielding a homology sphere by some integer slope. The ‘simple’ is deﬁned in [6]and [9]. If such a (1, 1)-simple knot generates the 1st homology of the lens space, we canalways ﬁnd such a slope. Hence any simple (1, 1)-knot is parameterized by a relatively
prime integers (p, k). The dual knot is a lens space knot in the homology sphere. The dual
knot is denoted by Kp,k and the homology sphere by Yp,k . The reader should probably
understand these facts by reading [6] and [9]. The main result in [2] gave a formula of theAlexander polynomial of Kp,k by using p, k. Here we give the following conjecture:Conjecture 1.5. If the third top term of the symmetrized Alexander polynomial ∆Kp,k (t)is non-zero, then ∆Kp,k (t) coincides with ∆T (2,2g+1) (t) for some integer g, in other words,
k = 2 holds.

THE THIRD TERM IN LENS SURGERY POLYNOMIALS

3This conjecture can be easily checked by a computer program (p ≤ 600) based on
the formula in [2]. Conjecture 1.5 is true under a little strong condition that Yp,k is
homeomorphic to S 3 , because of Theorem 1.2. The essential point is what if the third
top term of ∆Kp,k is non-zero, then Yp,k is homeomorphic to S 3 . Notice that in [7] theauthor proved that k = 2 holds if and only if Yp,k is homeomorphic to S 3 and Kp,k isisotopic to T (2, 2g + 1) for some integer g. This condition is also equivalent to the equality∆Kp,k (t) = ∆T (2,2g+1) (t).
Acknowledgements
Question 1.1 is presented by Masakazu Teragaito’s talk in the Mini-symposium “KnotTheory on Okinawa” at OIST on February the 17th to 21st, 2020. I appreciate him forteaching us this question.
2. Preliminaries and Proofs
2.1. Brief preliminaries. Here we deﬁne the lens surgery parameter (p, k).
Deﬁnition 2.1. Let K be a knot in a homology sphere Y . Suppose that Yp (K) = L(p, q)
and the dual knot K̃ has [K̃] = k[c] ∈ H1 (L(p, q), Z) for some orientation of K̃. Here thedual knot is the core knot in the solid torus obtained by the Dehn surgery. Furthermore,c is either of core circles of genus one Heegaard decomposition of L(p, q). Then we call(p, k) lens surgery parameter. The integer k is called a dual class.
If L(p, q) is a Dehn surgery of a homology sphere, the surgery parameter (p, k) is relatively prime and q = k 2 mod p. Note that we adopt the orientation of L(p, q) as thep/q-surgery of the unknot in S 3 .
The ambiguity of the orientation of K̃ and the choices of the core circles of genus one
Heegaard decomposition give (at most) four possibilities of the dual class k0 , −k0 , k0−1 , −k0−1(in Z/pZ), for some integer k0 . We always take the minimal integer k as a representative
satisfying 0 < k < p/2.
For any integer i we deﬁne the integer [i]p to be the integer with i ≡ [i]p mod p andp− 2 < [i]p ≤ p2 . Let k2 be the absolute value of the integer [k ′ ]p satisfying kk ′ ≡ 1 mod p.
We call k2 the second dual class. We set kk2 ≡ e mod p, e = ±1, m = kk2p−e , q = [k 2 ]p ,q2 = [(k2 )2 ]p . c = (k−1)(k+1−p)
and for some non-zero integer ℓ2{{1, 2, · · · , ℓ}
ℓ>0
Iℓ :=
{ℓ + 1, · · · , −1, 0} ℓ < 0.
From these data, we can compute the coeﬃcient ai due to [8].
Proposition 2.2 (Proposition 2.3 in [7]). Let K be a lens space knot in S 3 . For any
integer i with |i| ≤ p/2, the i-th coeﬃcient of the Alexander polynomial
ai = −e(m − #{j ∈ Ik |[q2 (j + ki + c)]p ∈ Iek2 }).
To prove Theorem 1.2, we use the non-zero curve deﬁned in [7]. First, we extend thecoeﬃcients ai of the Alexander polynomial periodically as āi = a[i]p . By the estimate in[1] proven in [4], āi is determined, where g(K) is the Seifert genus of K.
We deﬁne A-matrix and dA-matrix as follows:
Ai,j = āk2 (i+jek−c) , dAi,j = Ai,j − Ai−1,j ,4

MOTOO TANGE

where c = (k − 1)(k + 1 − p)/2. Due to the formula (9) in Lemma 2.6 in [7], we have[q2 i + k2 j]p ∈ I−k2
1(1)dAi,j = Eek2 (q2 i + k2 (j + e)) − Eek2 (q2 i + k2 j) = −1 [q2 i + k2 j]p ∈ Ik2
0otherwise.
We put Ai,j on each lattice point (i, j) in Z2 ⊂ R2 . For a non-zero coeﬃcient Ai,jwe draw a horizontal positive or negative arrow on (i, j) according to Ai,j = 1 or −1
respectively, where a positive (or negative) arrow means a horizontal arrow with positive(or negative) in the i-direction. After that, we connect the horizontally adjacent arrows
with the same orientation and compatibly connect arrows around the non-zero dAi,j as in[7]. Then we can obtain an inﬁnite family of simple curves on R2 with no ﬁnite ends (i.e.,they are properly embedded curves in R2 ). The arrows are not-increasing with respect tothe j-coordinate. We call the curves non-zero curves.
Proposition 2.3 ([7]). Any non-zero curve for any lens space knot in S 3 is included ina non-zero region N . In each non-zero region there is a single component non-zero curve.
Here a non-zero region N (introduced in [7]) is deﬁned as follows. First, we consider theunion of 2g+1 box-shaped neighborhoods of a vertical sequent lattice points corresponding
to α0 , α1 · · · , α2g . Two adjacent box neighborhoods are overlapped with a horizontal unit
segment. Next, we take the inﬁnite parallel copies moved by n · v where v is the vector(1, −k2 ) and n is any integer. We denote the union of the inﬁnite parallel copies of N
and call it non-zero region. Moving a non-zero region N by n · (0, p) for any integer n, weobtain inﬁnite non-zero regions on R2 .
The following lemma is important to prove the main theorem. This is also the case of
m = 0 in Lemma 4.4 in [7].
Lemma 2.4. If there exist integers i0 , j0 such that dAi0 ,j0 = −dAi0 ,j0 +1 = −1, then for
any integer i, there are no two adjacent zeros in the sequence {dAi,s |s ∈ Z}.
Proof. We assume the existence of integers i0 , j0 . Let x be i0 + (j0 − 1)ek. Usingthe formula (1), we have [q2 x]p ∈ I−k2 , [q2 x + k2 ]p ∈ Ik2 , and [q2 x + 2k2 ]p ∈ I−k2 . Hence,the sequence [q2 x + sk2 ]p starts at [q2 x]p and returns in I−k2 at s = 2. Therefore, we havep − k2 < (q2 x + 2k2 ) − q2 x < p + k2 and this means p < 3k2 .
We suppose dAi,j = dAi,j+1 = 0 for some integers i, j. Then [q2 (i + jek)]p , [q2 (i + jek) +k2 ]p ̸∈ I−k2 ∪ Ik2 . This implies p − k2 − k2 ≥ k2 . This contradicts the inequality above.
If for an integer I, the sequence {dAI,s |s ∈ Z} has no adjacent zeros, for any integer
i the same thing holds because the {dAi,s |s ∈ Z} is a parallel copies of {dAI,s |s ∈ Z}.
Hence, the desired condition is satisﬁed.
□Note that this lemma holds for any relatively prime positive integers (p, k). Actually,
to prove this lemma we do not require that the matrices A and dA come from a lens spaceknot in S 3 . In particular, if any Kp,k in Yp,k (deﬁned in Section 1.4) has non-zero thirdterm in the Alexander polynomial, then p < 3k2 holds. To prove Conjecture 1.5, ﬁrst weshould probably classify (Yp,k , Kp,k ) in the case of 3k2 < p.
2.2. Proof of Theorem 1.2. Let K be a lens space knot with lens surgery parameter(p, k) and with g = g(K). Suppose that α0 = 1, α1 = −1, and α2 = 1. Now we assume
that 2g − k2 ≥ 3. Since any non-zero curve has no ﬁnite ends, α4 = −1 holds naturally.

THE THIRD TERM IN LENS SURGERY POLYNOMIALSHence, we can assume thatα0 = 1α = −11α3 = 1
α4 = −1.

5(∗)

Let i, j be ﬁxed integers with k2 (i + jek − c) = −g mod p. Then Ai,j = α0 = 1.
Ai−1,j = Ai−1,j+1 = Ai−1,j+2 = Ai−1,j+3 = 0, because any non-zero curve is included in anon-zero region N due to Proposition 2.3.
We notice that the assumption of Lemma 2.4 is satisﬁed. Thus, we have dAi,j = 1,dAi,j+1 = −1, dAi,j+2 = 1, and dAi,j+3 = −1. The local values for matrices A and dA are
drawn in the top pictures in Figure 1.
Our situation falls into the following two cases (I), and (II) as in Figure 2.
(I): Ai+1,j+1 = −1, and Ai+1,j+2 = 1(II): Ai+1,j+1 = 0, and Ai+1,j+2 = 0.
In the case of (I), we obtain dAi+1,j+1 = dAi+1,j+2 = 0. This contradicts Lemma 2.4.
Next, consider the case of (II). We claim Ai+2,j+1 = Ai+2,j+2 = 0. If Ai+2,j+1 or
Ai+2,j+2 is non-zero, then the non-zero term is included in the non-zero region right next
to N , because there is only one non-zero curves in any non-zero region (Proposition 2.3).
This implies that by seeing the vertical coordinate in R2 , we havep − 2k2 ≤ 2.
Since 2k2 < p, we have p = 2k2 + 1 or 2k2 + 2. The equality p = 2k2 + 1 means it givesa (2, 2g + 1)-torus knot surgery. We consider the case of p = 2k2 + 2. Since p, k2 are
relatively prime, k2 is an odd number. The equality p = 2k2 + 2 can be deformed intok22 − 1 = k22−1 p ≡ 0 mod p. It is a lens space surgery yielding L(p, 1). This is the k2 = 1
case only due to [4]. Thus the claim above is true.
The remaining cases are −2 ≤ −2g + k2 ≤ 1. By using Theorem 1.15 and Theorem 4.20
in [7], the cases are realized by the (2, 2g + 1)-torus knot surgeries or the lens surgery on
T (3, 4) or P r(−2, 3, 7). The knots T (3, 4) and P r(−2, 3, 7) both do not satisfy (∗). Thus,our remaining cases satisfying this condition (∗) give the equality ∆K (t) = ∆T (2,2g+1) . □2.3. Proof of Corollary 1.4. We give a proof of Corollary 1.4. Let K be a lens space knot
in S 3 with parameter (p, k). If ∆K (t) has α0 = −α1 = α2 = 1, then ∆K (t) = ∆T (2,2g+1)
holds by Theorem 1.2. Using Theorem 1.15, the lens surgery parameter is (p, 2). The
parameter is realized by a (2, 2g + 1)-torus knot.
□References[1] J. Greene, The lens space realization problem, Ann. of Math. (2) 177 (2013), no. 2, 449–511.
[2] K. Ichihara, and T. Saito, and M. Teragaito, Alexander polynomials of doubly primitive knots, Proc.
Amer. Math. Soc. 135 (2007), 605-615.
[3] M. Hedden, and T. Watson, On the geography and botany of knot Floer homology, Selecta Mathematica
April 2018, Volume 24, Issue 2, pp 997–1037.
[4] P. Kronheimer, T. Mrowka, P. Ozsváth and Z. Szabó, Monopoles and lens space surgeries. Ann. ofMath. (2), 165(2):457-t-546, 2007.
[5] P. Ozsváth and Z. Szabó, On knot Floer homology and lens surgery, Topology, vol. 44, isuue 6, Nov
2005, 1281–1300.
[6] T. Saito, Knots in lens spaces with the 3-sphere surgery. Algebr. Geom. Topol. 8 (2008), no. 1, 53–79.
[7] M. Tange, On the Alexander polynomial of lens space knots, Topology Appl. 275 (2020), 107124.

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A-matrixdA-matrix

0 -1
0 10 -1
0 1-1
1-1
1A-matrix

dA-matrix

A-matrixdA-matrix-1
1 1-1 -1
100-1
1 0-1 01-11(I)(II)

Figure 1. Case (I) and (II)0 00 0Figure 2. An A-matrix of (II)[8] M. Tange, Ozsváth-Szabó’s correction term of lens surgery, Math. Proc. Cambridge Philos. Soc. 146(2009), no. 1, 119–134.
[9] M. Tange, Homology spheres yielding lens spaces, Proceedings of the Gökova Geometry-Topology Conference 2017, 73–121.
Email address: tange@math.tsukuba.ac.jp
Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 3058571, Japan