Matrices in companion rings, Smith forms, and the homology of 3-dimensional Brieskorn manifolds

Gerald Williams; Vanni Noferini

REVIEW 1 major objections 4 minor 34 references

Matrices in companion rings, Smith forms, and the homology of 3-dimensional Brieskorn manifolds

T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves a quotient-polynomial reduction for Smith forms of companion-matrix polynomials and uses it to compute the first homology of every Brieskorn manifold $M(r,s,n)$ with $r,s$ coprime.

desk verdict Clean Smith-form reduction plus a plausible Brieskorn formula; the main theorem's proof skips a case that must be written out before acceptance. read the letter →

arxiv 1908.07331 v2 pith:Q72HZJTT submitted 2019-08-20 math.AT math.RA

Vanni Noferini , Gerald Williams This is my paper · ORCID

classification math.ATmath.RA MSC 11C2011C9915A1515A2115B3315B3620J0557M50

keywords SmithformelementarydivisordomaincirculantmatrixcompanioncyclicallypresentedgroupBrieskornmanifoldfirsthomologyAlexanderpolynomial

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes a general reduction for Smith forms of matrices $f(C_g)$ formed by evaluating a polynomial at the companion matrix of another monic polynomial $g$, over any elementary divisor domain. If $f=Fz$ and $g=Gz$ share a monic factor $z$, then $f(C_g)$ is equivalent to the smaller matrix $F(C_G)$ together with a block of $\deg z$ zeros; when $z$ is the $\gcd$, the last non-zero determinantal divisor is the resultant $\mathrm{Res}(F,G)$. This turns a matrix-computation problem into a polynomial-divisibility problem. The paper applies the reduction to the Alexander polynomial of the torus knot $K(r,s)$ with $g(t)=t^n-1$, giving the full Smith form of the resulting circulant matrix, and converts that into an explicit formula for the first homology of every 3-dimensional Brieskorn manifold $M(r,s,n)$ with $r,s$ coprime. The result matters because it unifies circulant, skew-circulant, and Toeplitz matrices under one mechanism and settles a whole family of knot-cover homologies rather than case by case.

What carries the argument

The companion ring $R_g=\{f(C_g):f\in R[t]\}$ is isomorphic to the quotient ring $Q_g=R[t]/\langle g\rangle$ through the map sending $[t]$ to $C_g$; this lets polynomial division in $Q_g$ predict matrix equivalence over $R$. The proof of the reduction uses a unimodular Toeplitz similarity $U_z$ that block-triangularizes $C_g$ with respect to the factorization $g=Gz$, and Lemma 5.3 shows the leftover block contributes only zero invariant factors. The resultant formula then follows from the eigenpair description $\theta\mapsto f(\theta)$ on the roots of $g$.

What would settle it

Work out the integer Smith normal form of the circulant $f(C_g)$ for $r=5$, $s=3$, $n=6$ (so $g(t)=t^6-1$ and $f$ is the Alexander polynomial of $K(5,3)$); Theorem C predicts invariant factors $1,1,1,1,5,5$, so a direct computation producing any other multiset would refute the paper's formula.

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Extended reading notes

Core claim

The central claim is that Smith forms of matrices in companion rings are controlled by the quotient ring $R[t]/\langle g\rangle$, with a sharp reduction theorem: when $g=Gz$ and $f=Fz$ for a monic $z$, $f(C_g)\sim F(C_G)\oplus 0_{(\deg z)\times(\deg z)}$, and the last non-zero determinantal divisor equals $\mathrm{Res}(F,G)$ when $z$ is the $\gcd$. Specializing to $R=\mathbb{Z}$, $g(t)=t^n-1$, and $f$ the Alexander polynomial of the torus knot $K(r,s)$, Theorem C states that for coprime $r,s$, with $x=(r,n)\le y=(s,n)$, the Smith form of the circulant $f(C_g)$ has non-unit invariant factors $r/x$ repeated $y-x$ times, $rs/(xy)$ repeated $x-1$ times, and $0$ repeated $(x-1)(y-1)$ times. Corollary D then identifies $H_1(M(r,s,n))$ with the corresponding direct sum of cyclic groups, with the roles of $r$ and $s$ swapped when $y<x$.

Load-bearing premise

The load-bearing premise of the topological application is that, for coprime $r,s$, the $n$-fold cyclic branched cover of $S^3$ over $K(r,s)$ has a cyclic presentation whose representer polynomial is the projection of the torus-knot Alexander polynomial into $\mathbb{Z}[t]/\langle t^n-1\rangle$; the homology conclusion collapses if that external bridge fails.

Editorial extensions

If this is right

Whenever $f$ and $g$ share a monic divisor $z$, the Smith form of $f(C_g)$ is the Smith form of the smaller matrix $F(C_G)$ plus a zero block of size $\deg z$; in particular the nonzero invariant factors can be read from a matrix of smaller dimension.
If $z$ is the full $\gcd$, the last non-zero determinantal divisor of $f(C_g)$ is the resultant $\mathrm{Res}(F,G)$, which over $\mathbb{Z}$ gives the order of the largest torsion subquotient when the matrix is singular.
For the Alexander polynomial of $K(r,s)$ and $g(t)=t^n-1$, the non-unit invariant factors are $r/x$ repeated $y-x$ times and $rs/(xy)$ repeated $x-1$ times, with $(x-1)(y-1)$ zero invariant factors, where $x=(r,n)\le y=(s,n)$.
Consequently $H_1(M(r,s,n))$ is the displayed direct sum of cyclic groups for every coprime pair $(r,s)$, reproducing the known $r=2$ family and giving trivial homology exactly when $r,s,n$ are pairwise coprime.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, the same quotient reduction should compute Smith forms over any elementary divisor domain, not just $\mathbb{Z}$; for example, over $\mathbb{F}[x]$ it would give the invariant-factor structure of polynomial circulant analogues.
A testable extension is to apply Theorem A to Alexander polynomials of torus links rather than knots; the reduction and cyclotomic technology should still constrain the homology of cyclic branched covers of links.
The resultant formula suggests a practical check: for random coprime $f,g\in\mathbb{Z}[t]$, the absolute value of $\mathrm{Res}(F,G)$ should equal the product of the nonzero invariant factors of $f(C_g)$, a computation that can be run without any of the paper's topological language.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

This paper deserves a serious look, but not a rubber stamp. The headline result is a genuinely useful reduction theorem: for an elementary divisor domain R, the Smith form of f(C_g) is the Smith form of F(C_G) plus a zero block, where F,G are quotients by a common divisor. That is clean, new, and broadly applicable; the proof via companion rings is elementary and convincing. The application to Brieskorn manifolds is the payoff: Theorem C gives a closed formula for the Smith form of the circulant matrix coming from the torus knot Alexander polynomial, and Corollary D turns it into H1 of M(r,s,n) for coprime r,s. This generalizes the known r=2 case and makes Randell's algorithm explicit. The external topological bridge (cyclic presentations from branched covers) is standard.

The soft spot is precisely where the stress-test lands. In the proof of Theorem C, the case x>1 is disposed of with 'we omit the details'. That is not a trivial case: it is the part of the theorem that goes beyond the r=2 setting, and the asserted invariant factors for f1(C_{h1}) and f2(C_{h2}) are exactly what produces the nonzero factors in the final answer. The analogy with Case 1 is plausible—the coprimality conditions that made the earlier argument work still hold after shifting the exponent range—but it is not immediate, and the interaction with the symmetric restriction on the other side is left unverified. As written, the theorem is not fully proven; a referee should require the details to be written out. If they check out, Theorem C stands and the paper is a solid contribution. There is also a minor wording issue: 'non-unit invariant factors' is not literal when r/x or rs/(xy) equals 1, though this doesn't affect the homology application.

My recommendation: send it to peer review. The main ideas are good, the reduction theorem is independently valuable, and the gap looks repairable. But the paper should not be accepted until the omitted case is fully proved, either in the text or in an appendix.

Referee Report

1 major / 4 minor

Summary. The paper develops a general theory of Smith forms of matrices of the form f(C_g), where C_g is the companion matrix of a monic polynomial g over an elementary divisor domain. Theorem A reduces f(C_g) to F(C_G) ⊕ 0_m when f and g share a monic common divisor z, with F=f/z and G=g/z; Corollary B identifies the last nonzero determinantal divisor as the resultant Res(F,G). After establishing auxiliary results on cyclotomic matrices (Theorem 7.4), the paper applies this machinery to the case where f(t) is the Alexander polynomial of the torus knot K(r,s) and g(t)=t^n−1. Theorem C claims a complete description of the non-unit invariant factors of the resulting circulant matrix, and Corollary D converts this into the first homology of the Brieskorn manifold M(r,s,n). The proof of Theorem A is elementary and detailed, and Theorem 7.4 is proved fully. The proof of Theorem C is complete for the case x:=gcd(r,n)=1, but the case x>1 contains a substantial omitted argument.

Significance. If the main results are correct, Theorem A is a useful and general reduction in the Smith-form theory of companion-matrix polynomials, and Theorem C together with Corollary D would settle the homology of all 3-dimensional Brieskorn manifolds M(r,s,n) with r,s coprime, generalizing the previously known r=2 case. The matrix-theoretic core is largely self-contained and rests on standard resultant and cyclotomic facts, and the proofs of Theorem A and Theorem 7.4 are written in sufficient detail to be checked. The paper's headline application, however, depends on an unproved step in Case 2 of the proof of Theorem C, so the central claim is not yet fully established as written.

major comments (1)

[Section 7, proof of Theorem C, Case 2 (page 17)] The assertions that the Smith form of f1(C_{h1}) has non-unit invariant factors s/y repeated x−1 times, and that the Smith form of f2(C_{h2}) has non-unit invariant factors r/x repeated y−1 times, are not proved. The manuscript states 'as apart from this subtlety the argument is completely analogous, we omit the details.' These two assertions are load-bearing: together with the (x−1)(y−1) zero factors they are exactly the nonzero non-unit invariant factors claimed in Theorem C, and Corollary D depends on them. The analogy to Case 1 is not immediate, because in Case 2 the exponent β runs from γ_i+1 to α_i rather than from 1 to α_i, and because the reduction modulo h1 involves quotients of cyclotomic polynomials whose numerator and denominator both vanish at roots of h1; establishing the divisibility of the first determinantal divisor requires a separate argument, for example via evaluation of the quotient using derivatives or a suitable variant of Lemma 7.2. The authors need to supply the complete proof of these two Smith-form statements before Theorem C can be accepted.

minor comments (4)

[Theorem C statement] The phrase 'non-unit invariant factors' is not literally correct in cases where r/x or rs/(xy) equals 1, for example r=2, s=3, n=6; in such cases some of the listed entries are units (1), and the Smith form also contains additional unit invariant factors whose number is not stated. The intended meaning is clear, but the wording should be adjusted, e.g., 'the invariant factors are: ..., together with the appropriate number of 1s.'
[Section 7, proof of Theorem C, Case 2 (notation)] The notation γ_i is used for the exponent of the prime p_i in the factorization of x, but γ_i already denotes the i-th determinantal divisor in Section 2.1. This reuse is confusing; a different symbol, such as τ_i or β_i, would be preferable.
[Section 7, proof of Theorem C, Case 2 (dimensions)] The sentence 'the sizes of the identity matrices is clear from the context' should be made explicit, since the identity blocks in I⊕f1(C_{h1}) and I⊕f2(C_{h2}) are essential for the final multiplication step; explicitly stating their dimensions would help the reader verify the product of the two Smith forms.
[Abstract and Introduction] The abstract states 'Brieskorn manifolds M(r,s,n) where r,s are coprime' without specifying n≥2, although n≥2 is used throughout and in Corollary D; this should be stated for accuracy.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Smith-form derivation is self-contained; the flagged gap in Case 2 of Theorem C is an omitted proof, not circularity.

full rationale

The paper's derivation chain is self-contained against standard external benchmarks. The central machinery (Theorem A, Lemma 5.1, Lemma 5.3, Theorem 6.1, Theorem 6.3) is proved in the paper from the Smith theorem and elementary matrix/ring facts, with no parameter fitted to the target result. Theorem C reduces the Smith form of f(C_g) to products of Smith forms of cyclotomic blocks, using external resultant identities ([1], [9], [26]) and the multiplicativity of Smith forms for coprime determinants ([29]); none of these inputs assumes Theorem C or Corollary D. The topological bridge to Brieskorn homology is cited to [25, Theorem 3.1] and [5, Proposition 7], which are independent of the present authors and are not merely self-citations. The authors' own previous work appears only as background on companion matrices and polynomial root-finding ([27], [28], [30]) and is not load-bearing. Per the review instruction to flag missing support: the proof of Theorem C, Case 2, asserts that the Smith forms of f1(C_h1) and f2(C_h2) follow by an argument 'completely analogous' to Case 1 and says 'we omit the details' (Section 7). This is a genuine gap in the write-up for the x > 1 case, and the asserted nonzero invariant factors are not derived there. That is a correctness and rigor concern, not circularity: no equation is being reused as its own input, and no result is being justified by a self-citation chain.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters or invented entities appear; the paper is a pure mathematical derivation. The central claim rests on standard Smith-form theory, cyclotomic resultant identities, and external topological results on cyclic presentations of branched covers.

assumptions (7)

domain assumption R is an elementary divisor domain, so the Smith normal form exists and is unique up to units for every matrix over R.
The paper builds all results on the Smith theorem over EDDs (Section 2.1, Theorem 2.1); Z is the main example and is a PID hence EDD.
standard math For an integral domain R and monic g, the determinant of f(C_g) equals the resultant Res(f,g).
Stated as equation (1.1) and proved in Section 4 via eigenpairs of the companion matrix in an algebraic closure; used throughout.
standard math Resultant identities for cyclotomic polynomials: Res(Φ_m,Φ_n) is 0, p^φ(n), or 1 according as m=n, m/n a prime power, or otherwise.
Taken from [1,9] and used in the proofs of Theorem 7.4 and Theorem C.
domain assumption The n-fold cyclic branched cover of S^3 over a (1,1)-knot has fundamental group with a cyclic presentation whose representer polynomial is the projection of the Alexander polynomial to Z[t]/<t^n−1>.
External topological result cited to [25, Theorem 3.1] and [5, Proposition 7] in Section 2.2; it connects the Smith form computations to the homology of Brieskorn manifolds.
standard math H_1(M) is isomorphic to the abelianization of π_1(M), and the abelianization of a cyclic presentation is obtained from the Smith form of its relation matrix.
Standard algebraic topology and group theory, cited to [15, Theorem 2A.1] and [23]/[19]; used to pass from matrices to manifolds.
standard math Every EDD is a Bezout domain, and R[t] is a GCD domain, so gcds of polynomials over R exist.
Used without further justification in Section 4 and in Theorem A to factor f and g and define the common divisor z.
standard math The Cayley-Hamilton theorem holds for matrices over any commutative ring, so g(C_g)=0.
Invoked in Section 3 and the proof of Theorem 3.1; standard linear algebra.

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Pith. "Pith review of Matrices in companion rings, Smith forms, and the homology of 3-dimensional Brieskorn manifolds." pith.science (2026). https://pith.science/paper/Q72HZJTT

@misc{pith2026190807331,
  author       = {Pith},
  title        = {Pith review of: Matrices in companion rings, Smith forms, and the homology of 3-dimensional Brieskorn manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q72HZJTT}},
  note         = {Machine review of arXiv:1908.07331}
}

abstract

We study the Smith forms of matrices of the form $f(C_g)$ where $f(t),g(t)\in R[t]$, $C_g$ is the companion matrix of the (monic) polynomial $g(t)$, and $R$ is an elementary divisor domain. Prominent examples of such matrices are circulant matrices, skew-circulant matrices, and triangular Toeplitz matrices. In particular, we reduce the calculation of the Smith form of the matrix $f(C_g)$ to that of the matrix $F(C_G)$, where $F,G$ are quotients of $f(t),g(t)$ by some common divisor. This allows us to express the last non-zero determinantal divisor of $f(C_g)$ as a resultant. A key tool is the observation that a matrix ring generated by $C_g$ -- the companion ring of $g(t)$ -- is isomorphic to the polynomial ring $Q_g=R[t]/<g(t)>$. We relate several features of the Smith form of $f(C_g)$ to the properties of the polynomial $g(t)$ and the equivalence classes $[f(t)]\in Q_g$. As an application we let $f(t)$ be the Alexander polynomial of a torus knot and $g(t)=t^n-1$, and calculate the Smith form of the circulant matrix $f(C_g)$. By appealing to results concerning cyclic branched covers of knots and cyclically presented groups, this provides the homology of all Brieskorn manifolds $M(r,s,n)$ where $r,s$ are coprime.

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Works this paper leans on

34 extracted references · 34 canonical work pages

[1]

T. M. Apostol. Resultants of cyclotomic polynomials. Proc. Am. Math. Soc., 24, 457-462 (1970)

work page 1970
[2]

D. A. Bini and V. Y. Pan. Polynomial and matrix computations. Fun- damental algorithms. Vol. 1. Boston, MA: Birkh¨ auser, 1994

work page 1994
[3]

Brieskorn

E. Brieskorn. Beispiele zur Diﬀerentialtopologie von Si ngularit¨ aten. Invent. Math. , 2:1–14, 1966

work page 1966
[4]

Cattabriga

A. Cattabriga. The Alexander polynomial of (1,1)-knots . J. Knot The- ory Ramiﬁcations, 15(9):1119–1129, 2006

work page 2006
[5]

Cattabriga and M

A. Cattabriga and M. Mulazzani. Strongly-cyclic branch ed coverings of (1 , 1)-knots and cyclic presentations of groups. Math. Proc. Camb. Philos. Soc. , 135(1):137–146, 2003

work page 2003
[6]

Cavicchioli

A. Cavicchioli. On some properties of the groups G(n, l). Ann. Mat. Pura Appl. (4) , 151:303–316, 1988

work page 1988
[7]

E. Y. S. Chan, R. M. Corless, L. Gonzalez-Vega, J. R. Sendr a, and J. Sendra. Algebraic linearizations of matrix polynomials. Linear Algebra Appl., 563:373–399, 2019

work page 2019
[8]

D. A. Cox. Galois theory. 2nd ed. Pure and Applied Mathema tics. A Wiley Series of Texts, Monographs, and Tracts. Hoboken, NJ : John Wiley & Sons

work page

Show all 34 references

[9]

J. E. Cremona, Unimodular integer circulants. Math. Comput. , 77, No. 263, 1639–1652, 2008. 18

2008
[10]

P. J. Davis. Circulant matrices. New York, NY: AMS Chelsea Publish- ing, 2nd edition, 1994

1994
[11]

M. J. Dunwoody. Cyclic presentations and 3-manifolds. In Groups - Korea ’94. Proceedings of the international conference, Pu san, Korea, August 18-25, 1994 , pages 47–55. Berlin: Walter de Gruyter, 1995

1994
[12]

Friedland

S. Friedland. Matrices. Algebra, analysis and applications. Hackensack, NJ: World Scientiﬁc, 2016

2016
[13]

Frobenius

G. Frobenius. Theorie der linearen formen mit ganzen co eﬃcienten. Journal f¨ ur die reine und angewandte Mathematik , 86:146–208, 1879
[14]

Gohberg, P

I. Gohberg, P. Lancaster, and L. Rodman. Matrix polynomials. , vol- ume 58. Philadelphia, PA: Society for Industrial and Applie d Mathe- matics (SIAM), reprint of the 1982 original edition, 2009

1982
[15]

A. Hatcher. Algebraic topology. Cambridge: Cambridge University Press, 2002

2002
[16]

O. Helmer. The elementary divisor theorem for certain r ings without chain condition. Bull. Am. Math. Soc. , 49:225–236, 1943

1943
[17]

N. J. Higham. Functions of matrices. Theory and computation. Philadelphia, PA: Society for Industrial and Applied Mathe matics (SIAM), 2008

2008
[18]

R. A. Horn and C. R. Johnson. Matrix analysis. Cambridge: Cambridge University Press, 2nd edition, 2013

2013
[19]

D. L. Johnson. Topics in the theory of group presentations. , volume 42. Cambridge University Press, Cambridge. London Mathematic al Soci- ety, London, 1980

1980
[20]

Kaplansky

I. Kaplansky. Elementary divisors and modules. Trans. Am. Math. Soc., 66:464–491, 1949

1949
[21]

Lorenzini

D. Lorenzini. Elementary divisor domains and B´ ezout d omains. J. Algebra, 371:609–619, 2012

2012
[22]

D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann. Skew-s ymmetric matrix polynomials and their Smith forms. Linear Algebra Appl. , 438(12):4625–4653, 2013

2013
[23]

Magnus, A

W. Magnus, A. Karrass, and D. Solitar. Combinatorial gr oup theory. Presentations of groups in terms of generators and relation s. 2nd rev. ed. Dover Books on Advanced Mathematics. New York: Dover Pub li- cations, 1976. 19

1976
[24]

J. W. Milnor. On the 3-dimensional Brieskorn manifolds M (p, q, r). Knots, Groups, 3-Manif.; Pap. dedic. Mem. R. H. Fox, 175-225 , 1975

1975
[25]

Mulazzani

M. Mulazzani. Cyclic presentations of groups and cycli c branched cov- erings of (1 , 1)-knots. Bull. Korean Math. Soc. , 40(1):101–108, 2003

2003
[26]

T. Nagell. Introduction to Number Theory. AMS Chelsea Publishing, Second Edition, reprint 2001

2001
[27]

Nakatsukasa and V

Y. Nakatsukasa and V. Noferini. On the stability of comp uting polyno- mial roots via confederate linearizations. Math. Comput., 85(301):2391– 2425, 2016

2016
[28]

Nakatsukasa, V

Y. Nakatsukasa, V. Noferini, and A. Townsend. Vector sp aces of lin- earizations for matrix polynomials: a bivariate polynomia l approach. SIAM J. Matrix Anal. Appl. , 38(1):1–29, 2017

2017
[29]

M. Newman. Integral matrices. Pure and Applied Mathema tics, 45. New York-London: Academic Press. XVII, 224 p., 1972

1972
[30]

Noferini and F

V. Noferini and F. Poloni. Duality of matrix pencils, Wo ng chains and linearizations. Linear Algebra Appl. , 471:730–767, 2015

2015
[31]

P. Orlik. On the homology of weighted homogeneous manif olds. Pro- ceedings of the Second Conference on Compact Transformatio n Groups (Univ. Massachusetts, Amherst, Mass., 1971), Part I, pp. 26 0–269. Lec- ture Notes in Math., Vol. 298, Springer, Berlin, 1972

1971
[32]

J. F. Queir´ o. Axioms for invariant factors. Port. Math. , 54(3):263–269, 1997

1997
[33]

R. C. Randell. The homology of generalized Brieskorn ma nifolds. Topol- ogy, 14:347–355, 1975

1975
[34]

H. J. S. Smith. On systems of linear indeterminate equat ions and con- gruences. Philosophical Transactions of the Royal Society of London, 151:293-326, 1861. 20

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

T. M. Apostol. Resultants of cyclotomic polynomials. Proc. Am. Math. Soc., 24, 457-462 (1970)

work page 1970

[2] [2]

D. A. Bini and V. Y. Pan. Polynomial and matrix computations. Fun- damental algorithms. Vol. 1. Boston, MA: Birkh¨ auser, 1994

work page 1994

[3] [3]

Brieskorn

E. Brieskorn. Beispiele zur Diﬀerentialtopologie von Si ngularit¨ aten. Invent. Math. , 2:1–14, 1966

work page 1966

[4] [4]

Cattabriga

A. Cattabriga. The Alexander polynomial of (1,1)-knots . J. Knot The- ory Ramiﬁcations, 15(9):1119–1129, 2006

work page 2006

[5] [5]

Cattabriga and M

A. Cattabriga and M. Mulazzani. Strongly-cyclic branch ed coverings of (1 , 1)-knots and cyclic presentations of groups. Math. Proc. Camb. Philos. Soc. , 135(1):137–146, 2003

work page 2003

[6] [6]

Cavicchioli

A. Cavicchioli. On some properties of the groups G(n, l). Ann. Mat. Pura Appl. (4) , 151:303–316, 1988

work page 1988

[7] [7]

E. Y. S. Chan, R. M. Corless, L. Gonzalez-Vega, J. R. Sendr a, and J. Sendra. Algebraic linearizations of matrix polynomials. Linear Algebra Appl., 563:373–399, 2019

work page 2019

[8] [8]

D. A. Cox. Galois theory. 2nd ed. Pure and Applied Mathema tics. A Wiley Series of Texts, Monographs, and Tracts. Hoboken, NJ : John Wiley & Sons

work page