A Torres formula for twisted Reidemeister torsion

Peter Seokhee Seong

A Torres formula for twisted Reidemeister torsion

T0 review · 0 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Twisted Reidemeister torsion obeys a Torres-type formula: specializing one variable to 1 yields the sublink's torsion times det(Tρ'(K_μ)−I_n), up to units.

desk verdict Clean, careful re-proof of Morifuji's twisted Torres formula with an explicit determinant factor; the proof via Reidemeister torsion is new and the argument checks out. read the letter →

arxiv 2412.00080 v1 pith:323YGUF5 submitted 2024-11-26 math.GT

Peter Seokhee Seong This is my paper · ORCID

classification math.GT MSC 57K1057K1457Q10

keywords twistedReidemeistertorsionTorresformulaAlexanderpolynomiallinkMorifujitheoremdeterminant

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

This paper proves a twisted version of the classical Torres formula, which relates the Alexander polynomial of a link to the Alexander polynomial of a sublink obtained by deleting one component. The main theorem states that the twisted Reidemeister torsion of a μ-component link, with the variable of the last component set to 1, equals the twisted torsion of the sublink multiplied by the determinant det(Tρ'(K_μ)−I_n), where T=$t_1^{{ℓ_1}}$⋯t_{μ−1}^{ℓ_{μ−1}} records the linking numbers of the deleted component and ρ' is a representation of the sublink complement. The proof uses Reidemeister torsion directly, giving a second proof of Morifuji's twisted Torres condition that is closer in shape to Torres' original formula. When n=1, ρ' is trivial, and R=Z, the theorem recovers the classical Torres formula; when the representation takes values in SL(n,F), it recovers Morifuji's theorem.

What carries the argument

The proof is carried by the twisted homology of the CW pair formed by the two link exteriors X_{L'}=$S^{3}$∖νL' and X_L=$S^{3}$∖νL, with coefficients Γ^n where Γ=R[H_1(X_{L'})]. Lemma 3.2, obtained by excision and Poincaré duality with local coefficients, shows this homology is concentrated in degrees 2 and 3: H_2≅Γ^n/(Tρ'(K_μ)−I_n)Γ^n and H_3≅(Γ^n)^{π_1(K_μ)}. The multiplicativity of Reidemeister torsion for CW pairs then factorizes τ(X_{L'},ρ'⊗γ') as τ(X_L,(ρ'⊗γ')∘ι)·τ(X_{L'},X_L;ρ'⊗γ'), and the order–torsion relation from Proposition 2.7 identifies the pair torsion with 1/det(Tρ'(K_μ)−I_n). A localization property (Lemma 3.1) converts specializing t_μ to 1 into this pullback situation, so the determinant factor appears exactly where Torres' original formula has $t_1^{{ℓ_1}}$⋯t_{μ−1}^{ℓ_{μ−1}}−1.

What would settle it

Take the Hopf link as L with K_μ one component, so L' is the unknot, choose a non-abelian SL(2,C) representation ρ' of the unknot complement, and compute both sides of Theorem 1.1; alternatively, write down the cellular chain complex of (X_{L'},X_L) for this pair and check directly that H_2 is presented by Tρ'(K_μ)−I_n, resolving whether the determinant factor is correct up to units.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for a link L=K_1∪⋯∪K_μ⊂$S^{3}$ with μ≥2 components, sublink L'=L∖K_μ, and a representation ρ':π_1($S^{3}$∖νL')→GL(n,R) with R a Noetherian UFD, the twisted Reidemeister torsions satisfy τ^ρ_L(t_1,…,t_{μ−1},1)=det($t_1^{{ℓ_1}}$⋯t_{μ−1}^{ℓ_{μ−1}}ρ'([K_μ])−I_n)·$τ^{{ρ'}}$_{L'}(t_1,…,t_{μ−1}) up to multiplication by units of R[$t_1^{{±1}}$,…,t_{μ−1}^{±1}], where ℓ_i=lk(K_i,K_μ) and ρ is the pullback of ρ' along the inclusion-induced map. The equality is not merely formal: when the determinant factor vanishes or the sublink torsion vanishes, both sides vanish, and in the nondegenerate case the determinant is the characteristic polynomial of ρ'(K_μ) evaluated at T. This restores the shape of Torres' original formula in the twisted setting, with the classical formula appearing as the case n=1, ρ' trivial, R=Z.

Load-bearing premise

The load-bearing premise is Lemma 3.2's computation that the exterior pair's twisted homology is concentrated in degrees 2 and 3 with H_2 presented by the matrix Tρ'(K_μ)−I_n; if that presentation is wrong, the determinant factor never appears.

Editorial extensions

If this is right

Setting n=1, ρ' trivial and R=Z recovers the classical Torres formula for the Alexander polynomial of a link.
For SL(n,F)-representations, the theorem recovers Morifuji's twisted Torres condition: τ^ρ_L(t_1,…,t_{μ−1},1)=(T^n+ε_1T^{n−1}+⋯+ε_{n−1}T+(−1)^n)·τ^{ρ'}_{L'}(t_1,…,t_{μ−1}) with ε_i∈F.
The determinant factor det(Tρ'(K_μ)−I_n) is the characteristic polynomial of ρ'(K_μ) in the variable T, and is independent of the based homotopy class of K_μ.
If either det(Tρ'(K_μ)−I_n)=0 or the sublink torsion vanishes, both sides of the identity vanish, so the formula holds in the degenerate cases without separate treatment.

Reading between the lines

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

The same pair-homology mechanism might yield Torres-type identities for other torsion-based invariants, such as the Reidemeister torsion of sutured manifolds or higher-order Alexander invariants, where the analogous homology of the exterior pair would be presented by the same kind of matrix.
Because the theorem holds over any Noetherian UFD, the determinant factor is a purely algebraic object; a computational check over F_2 or F_q for a small link with a non-abelian representation could verify the equality exactly up to units and would isolate the role of the unit indeterminacy.
The μ=2 case deserves a careful comparison: the classical Torres formula contains an extra factor (t_1−1)^{-1} that must be absorbed into the unit indeterminacy between twisted torsion and twisted Alexander polynomial for n=1; tracking this absorption explicitly would clarify how the new theorem specializes to the classical one.

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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

Clear, careful paper. It proves Morifuji's twisted Torres formula by way of Reidemeister torsion, and the main new thing is the explicit determinant factor det(Tρ'(Kμ)−I_n) on the right-hand side, which Morifuji's theorem left as unnamed coefficients ε_k. The proof via torsion is genuinely different from Morifuji's Fox calculus approach, and it is done properly: Lemma 3.2 computes the relevant twisted homology of the pair (X_L', X_L) using excision and Poincaré duality, the rank argument is sound, and the multiplicativity chain in §3.2 leads to the stated formula without circularity. The paper is also honest about the price: for GL(n,R) the equality holds only up to units of the Laurent polynomial ring, which is a coarser indeterminacy than the ±t^nk units in Morifuji's original statement for SL(n,F). Remark 1.2 says exactly this, and Corollary 3.4 recovers Morifuji's theorem, albeit with that coarser indeterminacy. That is a real soft spot but not a fatal one; the main result is still the clean explicit determinant formulation.

The soft spots are minor. The result is logically equivalent to Morifuji's theorem for SL(n,F) representations, so the novelty is in the formulation and proof, not in the statement. The paper does not claim otherwise. The proof in Case 2.2 relies on the standard machinery of torsion multiplicativity and Proposition 2.7; I don't see any gap. The only thing that gave me pause is the H_3 identification in Lemma 3.2, but Poincaré duality gives invariants, and the stress-test note confirms the typesetting ambiguity is just that. So the central computation holds.

This is a paper for people who work with twisted Alexander polynomials and Reidemeister torsion: it gives them a better-formulated Torres formula and a proof that fits the torsion framework. It is not a major advance on the level of new theorems, but it is a solid, honest contribution. I would send it to a serious referee; it deserves to be published in a good knot theory journal.

Recommendation: accept after minor revision, mainly checking the indeterminacy discussion and perhaps expanding Remark 1.2.

Referee Report

0 major / 5 minor

Summary. The paper proves a Torres-type formula for twisted Reidemeister torsion: for a mu-component link L in S^3 and a representation rho' of the complement of L'=L\K_mu into GL(n,R) with R a Noetherian UFD, the torsion of L specialized at t_mu=1 equals det(T rho'(K_mu) - I_n) times the torsion of L', up to units. The proof computes the twisted homology of the pair (X_{L'},X_L) via excision and Poincare duality, uses multiplicativity of Reidemeister torsion, and derives the determinant factor from the order of the H_2 module. Corollary 3.4 recovers Morifuji's formula for SL(n,F) with a coarser indeterminacy.

Significance. The result is a clean torsion-theoretic proof of the twisted Torres formula, valid for GL(n,R) over a Noetherian UFD and having the same shape as the classical Torres formula. The proof is careful and relies on standard theorems; Lemma 3.2 is the key computation and is correct. The paper explicitly identifies the indeterminacy and notes that the n=1, trivial-representation case recovers the classical Torres formula. The comparison with Morifuji's theorem is appropriate, and the coarser indeterminacy is honestly acknowledged. I found no unsupported assumptions, circularity, or invented entities.

minor comments (5)

[Abstract] Please fix the typos 'Alexan-der polyomial' and 'th e twisted' in the abstract.
[Section 2.4, Definition 2.10] The tensor product in the definition of twisted Reidemeister torsion should be over Z[pi_1(X)]; as written, the expression ~S^n \otimes C_*(~X,~Y) omits the module structure of C_*(~X,~Y), which could confuse readers.
[Lemma 3.2] In the proof of Lemma 3.2, the exact sequence should use Gamma^n rather than R^n, and the phrase 'The Gamma n-homology' in the statement should be 'The Gamma^n-homology'.
[Lemma 3.2, proof] The sentence 'As K_mu is a 1-manifold, H_i=0 for i <= 1' is terse; it follows from the preceding identification H_i \cong H^{3-i}(K_mu), and stating this explicitly would improve readability.
[Theorem 1.1] The notation tau^rho_L(t_1,...,t_{mu-1},1) is defined informally; a brief reminder that (rho'\otimes gamma')\circ iota_* is the representation obtained by setting t_mu=1 would help the reader connect the statement to Definition 2.13.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity identified: the main theorem is derived from standard torsion machinery and external results, with no fitted parameters or self-citation chains.

full rationale

The paper's derivation chain is self-contained with respect to its inputs. Theorem 1.1 is proved by computing the twisted homology of the pair (XL', XL) in Lemma 3.2 using excision, Poincaré duality, and the infinite-cyclic fundamental group of the knot K_mu. The determinant factor det(T rho'(K_mu) - I_n) arises as the order of H_2, which is presented by the matrix T rho'(K_mu) - I_n; this is a genuine algebraic computation, not a definition or a fitted parameter. The subsequent localization and multiplicativity step (Section 3.2) uses standard Reidemeister torsion multiplicativity from Turaev and Proposition 2.7, and the cited external results (Turaev, Friedl-Vidussi, Kawauchi, Morifuji) are independent of the paper's own claim. The paper does not cite the author's own prior work, and no central premise is justified by a self-citation. Morifuji's theorem is recovered as a corollary rather than assumed, and the paper explicitly notes a less refined indeterminacy, confirming that the result is not identical to its input by construction. There is no fitted input renamed as a prediction, no uniqueness theorem imported from the authors, and no ansatz smuggled in via self-citation. The only close dependence is on standard torsion theory and on Lemma 3.2, which is proved in the paper. Therefore the analysis yields no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof uses standard tools from the Reidemeister torsion literature. No free parameters are fitted; no new entities are postulated. The main non-trivial input is the computation of the twisted homology of the pair (XL',XL) via Lemma 3.2.

assumptions (5)

standard math Twisted Reidemeister torsion is multiplicative under composition of based chain complexes and well-defined up to ±det(ρ(g)) for 3-manifolds with toroidal boundary.
Invoked in the proof of Theorem 1.1 via equation (4) and Proposition 2.11, citing Turaev [Tur01] and Friedl-Vidussi [FV11].
standard math Excision and Poincaré duality hold for twisted (co)homology with local coefficients in Γ = R[H1(XL')].
Used in Lemma 3.2 to identify H_i(XL',XL) with H^{3-i}(Kμ) with local coefficients.
standard math The order of the cokernel of an injective square matrix A over a Noetherian UFD is det(A) up to units.
Used to compute Δ(H2) = det(Tρ'(Kμ)-I_n) in Case 2.2 of the proof.
domain assumption The link exterior XL' has the homotopy type of a finite 2-dimensional CW complex.
Used to conclude H3(XL')=0 in Case 1; standard for link complements in S^3.
domain assumption The coefficient ring R is a Noetherian UFD.
Explicit assumption in Theorem 1.1; needed for Proposition 2.7 and the order function.

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Cite this review

Pith. "Pith review of A Torres formula for twisted Reidemeister torsion." pith.science (2026). https://pith.science/paper/323YGUF5

@misc{pith2026241200080,
  author       = {Pith},
  title        = {Pith review of: A Torres formula for twisted Reidemeister torsion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/323YGUF5}},
  note         = {Machine review of arXiv:2412.00080}
}

read the original abstract

The Torres formula, which relates the Alexander polynomial of a link to the Alexander polyomial of its sublinks, admits a generalization to the twisted setting due to Morifuji. This paper uses twisted Reidemeister torsion to obtain a second proof of Morifuji's result that is closer in appearance to Torres' original formula.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 15 canonical work pages

[1]

T. A. Chapman. Topological invariance of Whitehead torsion. Am. J. Math. , 96:488--497, 1974

work page 1974
[2]

Studying the multivariable A lexander polynomial by means of S eifert surfaces

David Cimasoni. Studying the multivariable A lexander polynomial by means of S eifert surfaces. Bol. Soc. Mat. Mexicana (3) , 10(Special Issue):107--115, 2004

work page 2004
[3]

Invariants of colored links and generalizations of the B urau representation

Anthony Conway. Invariants of colored links and generalizations of the B urau representation . PhD thesis, Universit\'e de Gen\`eve, 2017. 10.13097/archive-ouverte/unige:99352 http://dx.doi.org/10.13097/archive-ouverte/unige:99352

work page doi:10.13097/archive-ouverte/unige:99352 2017
[4]

Thurston norm, fibered manifolds and twisted alexander polynomials, 2006

Stefan Friedl and Taehee Kim. Thurston norm, fibered manifolds and twisted alexander polynomials, 2006

work page 2006
[5]

A survey of the foundations of four-manifold theory in the topological category, 2024

Stefan Friedl, Matthias Nagel, Patrick Orson, and Mark Powell. A survey of the foundations of four-manifold theory in the topological category, 2024

work page 2024
[6]

Topology

Stefan Friedl. Topology . AMS open notes \# OMN:202307.111368. , 2023. https://www.ams.org/open-math-notes/omn-view-listing?listingId=111368

arXiv 2023
[7]

A survey of twisted A lexander polynomials

Stefan Friedl and Stefano Vidussi. A survey of twisted A lexander polynomials. In The mathematics of knots , volume 1 of Contrib. Math. Comput. Sci. , pages 45--94. Springer, Heidelberg, 2011

work page 2011
[8]

A survey of knot theory

Akio Kawauchi. A survey of knot theory . Birkh\" a user Verlag, Basel, 1996. Translated and revised from the 1990 Japanese original by the author

work page 1996

Show all 16 references

[9]

Twisted A lexander polynomial and R eidemeister torsion

Teruaki Kitano. Twisted A lexander polynomial and R eidemeister torsion. Pacific J. Math. , 174(2):431--442, 1996

1996
[10]

Twisted A lexander invariants, R eidemeister torsion, and C asson- G ordon invariants

Paul Kirk and Charles Livingston. Twisted A lexander invariants, R eidemeister torsion, and C asson- G ordon invariants. Topology , 38(3):635--661, 1999

1999
[11]

Representations of knot groups and twisted A lexander polynomials

Xiao Song Lin. Representations of knot groups and twisted A lexander polynomials. Acta Math. Sin. (Engl. Ser.) , 17(3):361--380, 2001

2001
[12]

A T orres condition for twisted A lexander polynomials

Takayuki Morifuji. A T orres condition for twisted A lexander polynomials. Publ. Res. Inst. Math. Sci. , 43(1):143--153, 2007

2007
[13]

On the A lexander polynomial

Guillermo Torres. On the A lexander polynomial. Ann. of Math. (2) , 57:57--89, 1953

1953
[14]

V. G. Turaev. Reidemeister torsion in knot theory. Uspekhi Mat. Nauk , 41(1(247)):97--147, 240, 1986

1986
[15]

u rich. Birkh\

Vladimir Turaev. Introduction to combinatorial torsions . Lectures in Mathematics ETH Z\" u rich. Birkh\" a user Verlag, Basel, 2001. Notes taken by Felix Schlenk

2001
[16]

Twisted A lexander polynomial for finitely presentable groups

Masaaki Wada. Twisted A lexander polynomial for finitely presentable groups. Topology , 33(2):241--256, 1994

1994

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

T. A. Chapman. Topological invariance of Whitehead torsion. Am. J. Math. , 96:488--497, 1974

work page 1974

[2] [2]

Studying the multivariable A lexander polynomial by means of S eifert surfaces

David Cimasoni. Studying the multivariable A lexander polynomial by means of S eifert surfaces. Bol. Soc. Mat. Mexicana (3) , 10(Special Issue):107--115, 2004

work page 2004

[3] [3]

Invariants of colored links and generalizations of the B urau representation

Anthony Conway. Invariants of colored links and generalizations of the B urau representation . PhD thesis, Universit\'e de Gen\`eve, 2017. 10.13097/archive-ouverte/unige:99352 http://dx.doi.org/10.13097/archive-ouverte/unige:99352

work page doi:10.13097/archive-ouverte/unige:99352 2017

[4] [4]

Thurston norm, fibered manifolds and twisted alexander polynomials, 2006

Stefan Friedl and Taehee Kim. Thurston norm, fibered manifolds and twisted alexander polynomials, 2006

work page 2006

[5] [5]

A survey of the foundations of four-manifold theory in the topological category, 2024

Stefan Friedl, Matthias Nagel, Patrick Orson, and Mark Powell. A survey of the foundations of four-manifold theory in the topological category, 2024

work page 2024

[6] [6]

Topology

Stefan Friedl. Topology . AMS open notes \# OMN:202307.111368. , 2023. https://www.ams.org/open-math-notes/omn-view-listing?listingId=111368

arXiv 2023

[7] [7]

A survey of twisted A lexander polynomials

Stefan Friedl and Stefano Vidussi. A survey of twisted A lexander polynomials. In The mathematics of knots , volume 1 of Contrib. Math. Comput. Sci. , pages 45--94. Springer, Heidelberg, 2011

work page 2011

[8] [8]

A survey of knot theory

Akio Kawauchi. A survey of knot theory . Birkh\" a user Verlag, Basel, 1996. Translated and revised from the 1990 Japanese original by the author

work page 1996