Census L-space knots are braid positive, except for one that is not

We exhibit braid positive presentations for all L-space knots in the SnapPy census except one, which is not braid positive. The normalized HOMFLY polynomial of o9_30634, when suitably normalized is not positive, failing a condition of Ito for braid positive knots. We generalize this knot to a 1-parameter family of hyperbolic L-space knots that might not be braid positive. Nevertheless, as pointed out by Teragaito, this family yields the first examples of hyperbolic L-space knots whose formal semigroups are actual semigroups, answering a question of Wang. Furthermore, the roots of the Alexander polynomials of these knots are all roots of unity, disproving a conjecture of Li-Ni.


Introduction
Based on observation, most L-space knots are braid positive.Here L-space knots are knots in S 3 with a positive Dehn surgery to an L-space (see Ozsváth and Szabó [26]), and a knot that is the closure of a positive braid is braid positive.The L-space torus knots are the positive torus knots, and hence they are braid positive.Notably however, the .2;3/-cable of the .2;3/-torus knot is an L-space knot (see Hedden [16]) that is not braid positive; see eg Dunfield [12,Table 8] and Anderson, Baker, Gao, Kegel, Le, Miller, Onaran, Sangston, Tripp, Wood, and Wright [1, Example 1].It stands to reason that there probably are other cable L-space knots which are not braid positive.Nevertheless, it was questioned if every hyperbolic L-space knot is braid positive; see eg Hom, Lipschitz, and Ruberman [19,Problem 31(2)].
Dunfield showed that there are exactly 1267 complements of knots in S 3 in the SnapPy census of 1-cusped hyperbolic manifolds that can be triangulated with at most nine ideal tetrahedra [11].He further determined that (up to mirroring) 635 are not L-space knots, 630 are L-space knots, and left two as undetermined [12].These last two have been shown to have quasialternating surgeries (see Baker, Kegel, and McCoy [3]) and hence they are L-space knots as well.Thus there are exactly 632 L-space knots in the SnapPy census.Theorem 1.1 Every L-space knot in the SnapPy census of up to nine tetrahedra is braid positive except for o9_30634, which is not.
The knot o9_30634 is nearly braid positive in the sense that it has a braid presentation that is braid positive except for one strongly quasipositive crossing that jumps over only one strand.We do not know if o9_30634 admits a positive diagram.Question 1.2 Is every hyperbolic L-space knot nearly braid positive?
Proof of Theorem 1.1 In [3] we obtained braid words for every census L-space knot by automating the process from [1].(An alternative approach is taken by Dunfield, Obeidin, and Rudd [13].)Here, utilizing the braid and simplification methods in SnapPy [10] and Sage [27], we managed to cajole braid positive presentations for all of the knots except for one, o9_30634.The L-space census knots and positive braids with them as closures are detailed in the online supplement and verified in [2].
As one may check, the knot K D o9_30634 is the closure of the 4-braid ˇD OE2; 1; 3; 2; 2; 1; 3; 2; 2; 1; 3; 2; 1; 2; 1; 1; 2: Here the list of nonzero integers represents a braid word by letting the integer k stand for the standard generator k or its inverse 1  k , depending on whether k is positive or negative.Ito gives new constraints on a suitably normalized version of the HOMFLY polynomial for positive braids [20].The Ito-normalized HOMFLY polynomial z 13 69 133 121 55 12 1 17 66 83 45 11 1 0 4 10 6 1 0 0 0 1 1 0 0 0 0 0 where the indexing starts at 00, so that h 00 D 13.One may calculate this with Sage (or the knot theory package [21] for Mathematica) from the braid word, using the built-in HOMFLY polynomial and adjusting it to achieve Ito's normalization.The computations can be found at [2].
According to [20,Theorem 2], if a link K is braid positive then the Ito-normalized HOMFLY polynomial should only have nonnegative coefficients.As one observes, the coefficients h 30 and h 31 are negative.Hence o9_30634 is not braid positive.
In Section 2, we generalize the knot o9_30634 to an infinite family of hyperbolic L-space knots that are nearly braid positive but for which Ito's constraints fail to obstruct braid positivity, at least for the examples we managed to calculate.In Section 3, we further extend this family to a doubly infinite family of knots K n;m in hopes of providing more potential examples.While that doesn't quite work out, we highlight several properties of these knots in Proposition 3.1.Notably, we show that all but K 1;m and six other exceptional cases of these knots are hyperbolic, identify a small Seifert fibered space surgery for each, determine that when n 0 they are L-space knots if and only if m Ä 0, compute their Alexander polynomials, and examine their structures as positive braids and strongly quasipositive braids.
Algebraic & Geometric Topology, Volume 24 (2024) Census L-space knots are braid positive, except for one that is not 571 Lastly, in Section 4 we observe that our infinite family of hyperbolic L-space knots of Section 2 have Alexander polynomials that induce formal semigroups that are actually semigroups (which Teragaito pointed out to us), and have all their roots on the unit circle, disproving Li and Ni's Conjecture 1.3 in [22].
2 A family of hyperbolic L-space knots that might not be braid positive Let fK n g be the family of knots that are the closures of the braids ˇn D OE.2; 1; 3; 2/ 2nC1 ; 1; 2; 1; 1; 2 and includes our knot o9_30634 as K 1 ; see Figure 1, bottom right.Observe that ˇn gives a strongly quasipositive braid presentation for these knots that is almost braid positive -it is braid positive except for one negative crossing.
Proposition 2.1 For n 1, the knots K n are hyperbolic L-space knots.
Proof This follows from Lemmas 2.2 and 2.3.
Proof Figure 2 shows how a strongly invertible surgery description of the knot K n along with its .8nC6/surgerymay be obtained.Figure 3 demonstrates how one may take the quotient and perform rational tangle replacements associated to the surgeries to produce a link whose double branched cover is .8nC6/-surgeryˇ2n C 1 ˇn Figure 1: Top left: the braid ˇis positive except for one strongly quasipositive crossing.Its closure O ˇis the hyperbolic L-space knot o9_30634, which we show is not braid positive.Bottom left: dragging the base of the strongly quasipositive band of ˇinto the position shown exhibits O ˇas a positive Hopf basket.Top right: this braid has the .2;3/-cable of the .2;3/-torus knot as its closure.Bottom right: the closures of the braids ˇn are L-space knots that may also fail to be braid positive.The surgery coefficient on the closure knot is adjusted accordingly.Bottom right: after closure and isotopy, we obtain a surgery description for .8nC6/-surgery on K n .on K n .We observe this link to be the Montesinos link M 2=.4n C 5/; 1 2 ; .2nC 3/=.4n C 4/ .Hence its double branched cover is the Seifert fibered space M n 0I 2=.4n C 5/; 1 2 ; .2nC 3/=.4n C 4/ .Here we use the notation of Lisca and Stipsicz [24] where the Seifert fibered space M.e 0 I r 1 ; r 2 ; : : : ; r k / is obtained by e 0 -surgery on an unknot with k meridians having .1=r i /-surgery on the i th one.
These Seifert fibered spaces are determined to be L-spaces via [24,Theorem 1].More specifically, Lisca and Stipsicz [24, Theorem 1] show that the Seifert fibered space M D M.e 0 I r 1 ; r 2 ; r 3 / -with 1 r 1 r 2 r 3 0 -is an L-space if and only if either M or M does not carry a positive transverse contact structure.Then by Lisca and Matić [23], such a Seifert fibered space M carries no positive transverse contact structure if and only if either e 0 0 or e 0 D 1 and there exists no coprime integers a and m such that mr 1 < a < m.1 r 2 / and mr 3 < 1.
Rewriting to apply [24, Theorem 1], we obtain that M n D M 1I 1 2 ; .2nC 1/=.4n C 4/; 2=.4n C 5/ .Then, since 1 r 2 D .2nC 3/=.4n C 4/, we assume for contradiction that there are coprime integers a and m such that m 1 2 < a < m.2n C 3/=.4n C 4/ and m2=.4nC 5/ < 1.The first gives The second implies m < 2n C 2 C 1 2 , so that m Ä 2n C 2 and m 2nC2 Ä 1: Together they yield 0 < 2a m < 1.However, since 2a m is an integer, there are no pairs of integers .a;m/ that satisfy this equation.This is a contradiction.The quotient of the surgery description followed by some isotopy to straighten the arcs.(d) Rational tangle replacements along the arcs produce a link whose double branched cover is .8nC4/surgeryon K n .(e)-(h) A sequence of isotopies shows that this link is the Montesinos link M.OE0; 2n 3; 2; OE0; 2; OE0; Therefore M n does not carry a positive transverse contact structure, and thus it is an L-space.Hence K n is an L-space knot for each n 1.

Lemma 2.3
For n 1, the knots K n are hyperbolic.
Proof We check that L12n1739.Thus for n > 1 we fill with slopes longer than 2 and therefore directly get hyperbolic manifolds by Gromov and Thurston's 2 theorem; see for example [7,Theorem 9].
Teragaito (personal communication, 2022) suggested an alternative approach to this lemma that does not use SnapPy or any computer calculation.The referee also proposed a similar approach.Since it is more "hands-on", we include a proof along the lines of their suggestions here: Another proof of Lemma 2.3 As knots in S 3 are either torus knots, satellite knots, or hyperbolic knots by [29], we must show that K n is neither a torus knot nor a satellite knot.
In the proof of Theorem 4.4 the Alexander polynomial of K n D K n;0 is presented as As this is not equivalent to the Alexander polynomial of a torus knot, K n cannot be a torus knot.(Also, the formal semigroup of K n has rank 3 as noted in Remark 4.3, whereas the formal semigroup of a torus knot has rank 2.) So now suppose K n is a satellite knot.Observe that an unknotting tunnel put at the unique negative crossing for K n D O ˇn in Figure 1, bottom right, shows that K n has tunnel number 1. Since the bridge index of K n is at most 4, Morimoto and Sakuma's classification of tunnel number 1 satellite knots [25] tells us that K n has the 2-bridge torus knot T .2;q/ as a companion knot for some odd q and a pattern of wrapping number 2. As K n is an L-space knot by Lemma 2.2, this pattern must be a braided pattern by [4, Lemma 1.17].Hence the pattern must be a 2-cable.Thus if K n is a satellite knot, then it is a 2-cable knot of T .2;q/.Indeed, the Alexander polynomial of K n shown above implies that K n must be the .2;4nC5/-cable of the T .2;2nC1/ torus knot.However, the distance of the cabling slope 8n C 10 and the slope 8n C 6 of the Seifert fibered surgery on K n is .8nC 10; 8n C 6/ D 4 > 1.Thus the cabling torus remains incompressible after surgery; see eg [15,Lemma 7.2].This contradicts that .8nC6/-surgery on K n produces a small Seifert fibered space.Thus K n cannot be a satellite knot.
However, the constraints of Ito on HOMFLY polynomials appear to not obstruct K n from being braid positive when n 2. Using Sage for computations, we see that Ito's constraints on the HOMFLY polynomials of K n for n D 2; : : : ; 10 do not obstruct braid positivity for these knots.Furthermore, we have been unsuccessful in finding a braid positive presentation for these knots.
Question 2.4 Are the knots K n for n 2 braid positive?

A doubly infinite family of knots
From our description of the family of knots K n in Figure 2, one finds a natural 2-parameter family generalization.While one may initially hope this family yields further examples of hyperbolic L-space knots that fail to be braid positive, we show this is not the case.
(2) .8nC64m/-surgery on K n;m gives the Seifert fibered space (4) Assume n 0. Then K n;m is an L-space knot if and only if m Ä 0.
(5) If n 0 and m < 0, then ˇn;m is a positive braid and K n;m is a braid positive knot of genus jmj C 4n C 3 (6) If 2n C 1 m 0, then ˇn;m is conjugate to a strongly quasipositive braid and K n;m is a strongly quasipositive knot of genus 4n m C 2.
(a) If 2n m 0, then K n;m is a fibered strongly quasipositive knot.Moreover it is a Hopf plumbing basket.
(b) If 2n C 1 D m > 0, then K n;m is a nonfibered strongly quasipositive knot.
Proof (1) Since the surgery description of K n;m given in Figure 4(e) is on a hyperbolic link, using the 2 theorem a couple of times yields a finite list of pairs .n;m/ for which K n;m might not be hyperbolic.A further check in SnapPy confirms that all but five of them are hyperbolic.These remaining five are readily confirmed to be torus knots.The computations are displayed at [2].
(5) When n 0 and m < 0, the braid ˇn;m as described in Figure 4 (6) When 0 Ä m Ä 2n C 1, through braid isotopy and braid conjugacy, we may isotope in pairs 2m of the 2m C 1 negative crossings over to m of the 2n C 1 copies of the "2-cabled" positive crossing that appear in ˇn;m so that they appear as in the left-hand side of Figure 6, top.Hence by a further braid isotopy as indicated by Figure 6, each of these 2m negative crossings contributes to an SQP band.The final negative crossing also contributes to an SQP band towards the end of the braid, ultimately giving us the strongly quasipositive braid, shown in Figure 6, middle, to which ˇn;m is conjugate.One counts that the braid index is Furthermore, when 0 Ä m Ä 2n so that 2n m 0, we may instead perform braid isotopy and conjugation to arrive at the strongly quasipositive braid shown in Figure 6, bottom.This braid however contains the "dual Garside element" ı D 3 2 1 .Hence, as Banfield points out [5], the closure of such an SQP braid is fibered and a Hopf basket.
When m D 2n C 1, the braid ˇn;2nC1 is conjugate to an SQP braid but its closure K n;2nC1 might not be fibered.Indeed, we find that the Alexander polynomial of K n;2nC1 is not monic, so the closure is not fibered.Explicitly, from our computations of K n;m for (3) below, we have  m with exterior E n;m obtained from K and the core curves of .1=.nC1//-surgery on c and .1=.mC2nC3//-surgery on c 0 .Thus E n;m Š E where x n;m D x; y n D y 1 x 4.nC1/ and z m D zx where the : D indicates that we have divided out the unit t 4nC3 m .
(4) Using our Alexander polynomial calculations provides obstructions to the knots K n;m for n > 0 being L-space knots when m > 0. As an example, taking n > 0 and m D 1 gives One may observe that the constant coefficient is 3. Hence the knots K n;1 cannot be L-space knots.Indeed, one may further observe that, when n > 0 and m > 0, the central terms will have overlap with the end terms to give coefficients ˙2 or ˙3 for terms with degree of small magnitude.Thus none of the knots K n;m with n > 0 and m > 0 are L-space knots.
In the other direction, where n > 0 and m Ä 0, we may observe via [23; 24], as in Lemma 2.2, that the Seifert fibered space M resulting from .8nC64m/-surgery on K n;m is an L-space.For that we need to distinguish several cases.We continue with the notation of Lisca and Stipsicz [24] as in Lemma 2.2.
So we must reckon with the coefficient

2nCmC1/C3
: > 0: If we now assume that there exist coprime integers a and b such that we conclude from the first inequality that 0 < 2a b < b=.2n C 2/ and the second inequality implies that Putting both together yields the contradiction Thus M carries no positive transverse contact structure and is therefore an L-space.
If 2n C m C 1 D 0 we get the Seifert fibered space M 1I 2 3 ; 1 2 ; .2nC 1/=.4n C 4/ .We assume that there exist coprime integers a and b such that 2  3 b < a < 1 2 b and ..2n C 1/=.4n C 4//b < 1, from which we conclude 4b < 6a < 3b and b < 2 C 2=.2n C 1/ Ä 4, which is a contradiction.Therefore M does not carry a positive transverse contact structure and is thus an L-space.
which is a lens space and hence an L-space.
which is a lens space and hence an L-space.
and thus the correctly normalized Seifert fibered space is Á ; which admits a positive contact structure.Next, we consider then the correct ordering of the Seifert invariants is M 1I 2 3 ; .2nC 3/=.4n C 4/; 1 2 .We readily see that there exist no coprime integers a and b such that But putting them together yields the contradiction Thus M does not admit a positive transverse contact structure and is therefore an L-space.
Remark 3.2 In the cases of the above proof when 2n C m C 1 D 1 or 2, the knots K n;m have lens space surgeries.These knots can be seen to be Berge knots as follows.With m 2n 3 D 1 or 0, Figure 5(d) can be seen to divide along a horizontal line into two rational tangles.A vertical arc in the middle would be the arc dual to the rational tangle replacement on the 0-framed arc from Figure 5(c).
In the double branched cover, this vertical arc will lift to a knot in the lens space with an S 3 -surgery.Furthermore, one may observe that this arc lifts to a .1;1/-knot in the lens space.Hence the knot K n;m must be a Berge knot [6].

Curiosities about the Alexander polynomial of o9_3063and its generalizations
Like the failure of braid positivity, the hyperbolic L-space knot o9_30634 exhibits two more curious properties that had previously only been observed for L-space knots among iterated cables of torus knots.The first, regarding formal semigroups, Teragaito communicated to us near the completion of the initial preprint.The second, regarding the roots of its Alexander polynomial, came after that.Both actually generalize to the infinite family fK n g n 1 as well.

An infinite family of hyperbolic L-space knots whose formal semigroups are semigroups
Teragaito informed us about the work of Wang [31] on formal semigroups of L-space knots, and that there are only two L-space knots in the SnapPy census whose formal semigroups were actual semigroups.He had also observed that one of these knots appeared to fail to be braid positive.It turns out that this is the knot o9_30634, which we had confirmed to not be braid positive.Upon seeing an early draft of this article, Teragaito further showed that all of our hyperbolic L-space knots K n have formal semigroups that are semigroups.Below we overview the formal semigroup and then record Teragaito's results in Theorem 4.1.
An algebraic link is defined to be the link of an isolated singularity of a complex curve in C 2 .Algebraic knots are known to be iterated cables of torus knots [14] and they are all L-space knots; see [17].Moreover, one can assign to any algebraic knot K an additive semigroup S K < N 0 which determines the Heegaard Floer chain complex and is computable from the Alexander polynomial of K; see [8].
In [31] Wang has generalized this definition, but now S K is not necessarily a semigroup anymore.Let K be an L-space knot with (symmetrized) Alexander polynomial K .Then the formal semigroup S K N 0 is defined by where g.K/ denotes the genus of K. (Note that t g.K / K .t/ is now an ordinary polynomial of degree 2g.K/.)The set S K still determines the Heegaard Floer chain complex of K but is not necessarily a semigroup.This is used by Wang to construct an infinite family of L-space knots which are iterated cables of torus knots but not algebraic [31].On the other hand, it remained open if there exists an L-space knot which is not an iterated cable of torus knots but whose formal semigroup is a semigroup [31,Question 2.8].
Theorem 4.1 (Teragaito, personal communication, 2022) There exists an infinite family of hyperbolic L-space knots whose formal semigroups are semigroups.More concretely: (1) o9_30634 and t09847 are hyperbolic L-space knots whose formal semigroups are semigroups.The formal semigroup of every other L-space knot in the SnapPy census is not a semigroup.
(2) The formal semigroups S K n of the infinite family of hyperbolic L-space knots fK n g from Section 2 are all semigroups.
Consequently, the knots fK n g provide an infinite family of knots answering [31, Question 2.8] negatively.
Proof (1) The formal semigroup S K of an L-space knot is computable from the Alexander polynomial of K; in particular, S K always contains all natural numbers larger than g.K/ and the finitely many other elements of S K can be read off from the Alexander polynomial.In [2] we present code that computes the formal semigroups of all SnapPy census L-space knots and determines that o9_30634 and t 09847 are the only ones whose formal semigroups are semigroups.
( From these presentations of their Alexander polynomials, one sees that all of their roots are roots of unity.(2) As one may check, the hyperbolic L-space knots fK n; 2 g n 1 have Alexander polynomials with roots that are not roots of unity.
Remark 4.6 In light of Theorem 4.4 and [9, Corollary 1.2], one may hope that at least one of the hyperbolic L-space knots among fK n g n 1 and fK n; 1 g n 1 has a double branched cover that is an L-space.This would answer a question of Moore in the negative; see [9,Question 1.3].However, as one may check, these knots are not definite.Indeed, j .K n /j D g.K n / C 2 < 2g.K n / while j .K n; 1 /j D g.K n; 1 / C 3 < 2g.K n; 1 /.

2 Figure 2 :
Figure 2: Top left: the braid ˇn with a surgery coefficient of 8n C 6 for its closure knot K n .Bottom left and top right: twists in the braid are expressed and collected into surgeries on unknots.The surgery coefficient on the closure knot is adjusted accordingly.Bottom right: after closure and isotopy, we obtain a surgery description for .8nC6/-surgery on K n .

Figure 4 :
Figure 4: (a) The braid ˇn;m with a surgery coefficient of 4m C 8n C 6 for its closure knot K n;m .(b)-(d) Twists in the braid are expressed and collected into surgeries on unknots.The surgery coefficient on the closure knot is adjusted accordingly.(e) After closure and isotopy, we obtain a surgery description for .4mC8nC6/-surgery on K n;m .

4
and there are 2m C 1 SQP bands and 4.2n C 1 m/ C 2 regular crossings.Hence .K n;m / D .8n2m C 3/ and g.K n;m / D 4n m C 2.

( 3 )
View the surgery description for K n;m as the link L D K [ c [ c 0 where we do .1=.nC1//-surgery on c and .1=.mC2nC3//-surgery on c 0 .Observe that c [ c 0 is the trivial 2-component link, and we may orient the link so that lk.K; c/ D 4 and lk.K; c 0 / D 2. Let E be the exterior of L D K [ c [ c 0 .Then H 1 .E/ D hOE K ; OE c ; OE c 0 i Š Z 3 where K , c , and c 0 are oriented meridians of K, c, and c 0 .Let K , c , and c 0 be their preferred longitudes.Observe that OE c D 4OE K and OE c 0 D 2OE K in H 1 .E/.Now consider the family of links L n;m D K n;m [ c n [ c 0

Remark 4 . 5 ( 1 )
While we do not yet know if any of the knots in fK n g n 1 are braid positive, all of the knots fK n; 1 g n 1 are braid positive by Proposition 3.1(5).
of nonorientable hyperbolic 3-manifolds JUAN LUIS DURÁN BATALLA and JOAN PORTI 141 Realization of Lie algebras and classifying spaces of crossed modules YVES FÉLIX and DANIEL TANRÉ 159 Knot Floer homology, link Floer homology and link detection FRASER BINNS and GAGE MARTIN 183 Models for knot spaces and Atiyah duality SYUNJI MORIYA 251 Automorphismes du groupe des automorphismes d'un groupe de Coxeter universel YASSINE GUERCH 277 The RO.C 4 / cohomology of the infinite real projective space NICK GEORGAKOPOULOS 325 Annular Khovanov homology and augmented links HONGJIAN YANG 341 Smith ideals of operadic algebras in monoidal model categories DAVID WHITE and DONALD YAU 393 The persistent topology of optimal transport based metric thickenings HENRY ADAMS, FACUNDO MÉMOLI, MICHAEL MOY and QINGSONG WANG 449 A generalization of moment-angle manifolds with noncontractible orbit spaces LI YU 493 Equivariant Seiberg-Witten-Floer cohomology DAVID BARAGLIA and PEDRAM HEKMATI 555 Constructions stemming from nonseparating planar graphs and their Colin de Verdière invariant ANDREI PAVELESCU and ELENA PAVELESCU 569 Census L-space knots are braid positive, except for one that is not KENNETH L BAKER and MARC KEGEL 587 Branched covers and rational homology balls CHARLES LIVINGSTON (3)(4) usingand(3)(4)where we set x n;m D t, y n D 1, and z m D 1, we obtain K n;m .t/D t 1 .t4 1/.t 2 1/ L .t; t 4.nC1/ ; t 2.mC2nC3/ /: D .x 2 1/.x 3 y 2 z C x 2 y 3 z x 2 y 2 z C x 2 y C xy 2 z xy C x C y/: Ct m t 1 m Ct 2 m Ct mC1 t mC2 Ct mC3 Ct mC4nC5 / .t 4 1/ : D .t1/..t m 4n 2 t mC2 /C.t m Ct 2 m Ct mC1 Ct mC3 /C.t 4nC5 m t 1 m // .t 4 1/ [30]m .xn;m;yn;zm / D L .x n;m ; y 1 n x 4.nC1/ n;m ; z m x 2.mC2nC3/ n;m/:Using the Torres formulae[30], one obtains that(3-4) K n;m .xn;m/ D x n;m 1 x 4 n;m 1 K n;m [c n .xn;m; 1/ D x n;m 1 K n;m [c n [c 0 m .x n;m ; 1; 1/: L .x; y; z/ [18, Proposition 3.1(3)we have computed the Alexander polynomials of K n , from which we read off the formal semigroup S K n to be Remark 4.3 The semigroups from Theorem 4.1 and the preceding remark all have rank 3, ie the minimal number of a generating set is 3. On the other hand, Teragaito constructs in[28]an infinite family of hyperbolic L-space knots whose formal semigroups are semigroups of rank 5.From this one may observe that all of its roots are roots of unity.Since o9_30634 is a hyperbolic L-space knot, it provides a counterexample to[22, Conjecture 1.3]; see also the discussion surrounding its reference as[18, Conjecture 6.10].Indeed, we have infinite families of hyperbolic L-space knots that are counterexamples to this conjecture: Theorem 4.4 The two infinite families of hyperbolic L-space knots fK n g n 1 and fK n; 1 g n 1 consist of knots whose Alexander polynomials have all of their roots on the unit circle.8nC7Ct 4nC5 C t 4nC2 C 1 .tC1/.t 2 C 1/ D .t4nC5C 1/.t 4nC2 C 1/ .tC1/.t 2 C 1/ ; while setting m D 1 yields K n; 1 .t/: D .t8nC9 C t 4nC6 t 4nC5 C t 4nC4 / C .t 4nC3 t 4nC4 C t 4nC5 C 1/ .tC 1/.t 2 C 1/ D t 8nC9 C t 4nC6 C t 4nC3 C 1 .tC 1/.t 2 C 1/ D .t4nC6 C 1/.t 4nC3 C 1/ .tC 1/.t 2 C 1/ : which is very close to our braid word for o9_30634.One can similarly show that t 09847 fits into an infinite family of hyperbolic L-space knots with braid words OE.2; 1; 3; 2/ 2nC1 ; 1; 2; 1; 1; 2 whose formal semigroups are semigroups.Algebraic & Geometric Topology, Volume 24 (2024) Proof Proposition 3.1(1) and (4) show that the knots of fK n g n 1 and fK n; 1 g n 1 are hyperbolic L-space knots.Proposition 3.1(3) gives a general formula for K n;m .t/.In the course of that proof, we obtained the first equality below.Dividing out the unit t and rearranging gives the second:K n;m .t/Dt 4nC3 m .t1/.t m 4n 2 C t m t 1 m C t 2 m C t mC1 t mC2 C t mC3 C t mC4nC5 / .t41/ : D .t8nC7 C t 4nC4 t 4nC3 C t 4nC2 /t 2m C .t 4nC3 t 4nC4 C t 4nC5 C 1/ .tC 1/.t 2 C 1/ : Setting m D 0 yields K n;0 .t/: D .t8nC7 C t 4nC4 t 4nC3 C t 4nC2 / C .t 4nC3 t 4nC4 C t 4nC5 C 1/ .tC 1/.t 2 C 1/ D t