Adjoint Hyperbolic Torsion Polynomial

• The adjoint hyperbolic torsion polynomial is a normalized twisted Alexander polynomial derived from composing a discrete faithful lift of hyperbolic holonomy with the adjoint action.
• It encapsulates nonabelian Reidemeister torsion, serving as a bridge between hyperbolic geometry, knot invariants, and representation-theoretic moduli spaces.
• Explicit formulas for torus and twist knots demonstrate its computability and its role in detecting genus, fibering, and linking to quantum 1-loop invariants.

The adjoint hyperbolic torsion polynomial is the normalized twisted Alexander polynomial associated with the $\mathrm{SL}_3(\mathbb{C})$-representation $\mathrm{Ad}\circ \rho_0$, where $\rho_0:G(K)\to \mathrm{SL}_2(\mathbb{C})$ is a discrete faithful lift of the hyperbolic holonomy of a knot exterior and $\mathrm{Ad}$ is the adjoint action of $\mathrm{SL}_2(\mathbb{C})$ on $\mathfrak{sl}_2(\mathbb{C})$. In this form it is a Laurent polynomial defined up to $\pm t^i$, and it is simultaneously the adjoint twisted Alexander polynomial and the nonabelian torsion polynomial attached to the hyperbolic structure; one normalization imposes the antisymmetry $$\mathcal{T}^{\mathrm{Ad}}_K(t^{-1})=-\,\mathcal{T}^{\mathrm{Ad}}_K(t)$$ up to multiplication by $t^i$ (Tran, 20 Aug 2025, Tran, 2013).

1. Definition and algebraic construction

Let $K\subset S^3$ be a hyperbolic knot, let $\mathrm{Ad}\circ \rho_0$0 be its exterior, and let

$\mathrm{Ad}\circ \rho_0$1

Choose a Wirtinger presentation

$\mathrm{Ad}\circ \rho_0$2

and let

$\mathrm{Ad}\circ \rho_0$3

be abelianization. For a representation $\mathrm{Ad}\circ \rho_0$4, the induced maps on group rings produce a homomorphism

$\mathrm{Ad}\circ \rho_0$5

and Wada’s twisted Alexander polynomial is

$\mathrm{Ad}\circ \rho_0$6

where $\mathrm{Ad}\circ \rho_0$7 is obtained from the Fox-derivative matrix by deleting the $\mathrm{Ad}\circ \rho_0$8-th column (Tran, 20 Aug 2025).

For the adjoint hyperbolic torsion polynomial, one takes $\mathrm{Ad}\circ \rho_0$9, a discrete faithful lift of the hyperbolic holonomy

$\rho_0:G(K)\to \mathrm{SL}_2(\mathbb{C})$0

and composes it with

$\rho_0:G(K)\to \mathrm{SL}_2(\mathbb{C})$1

With respect to a basis of $\rho_0:G(K)\to \mathrm{SL}_2(\mathbb{C})$2, the adjoint action is explicit: if

$\rho_0:G(K)\to \mathrm{SL}_2(\mathbb{C})$3

then

$\rho_0:G(K)\to \mathrm{SL}_2(\mathbb{C})$4

Applying Wada’s construction to $\rho_0:G(K)\to \mathrm{SL}_2(\mathbb{C})$5 gives

$\rho_0:G(K)\to \mathrm{SL}_2(\mathbb{C})$6

up to $\rho_0:G(K)\to \mathrm{SL}_2(\mathbb{C})$7; Dubois–Yamaguchi showed that this is a Laurent polynomial rather than merely a rational function. Its normalized form is denoted

$\rho_0:G(K)\to \mathrm{SL}_2(\mathbb{C})$8

and the normalization is chosen so that

$\rho_0:G(K)\to \mathrm{SL}_2(\mathbb{C})$9

up to multiplication by $\mathrm{Ad}$0 (Tran, 20 Aug 2025).

The general degree estimate for twisted Alexander polynomials yields

$\mathrm{Ad}$1

reflecting that the adjoint coefficient system has dimension $\mathrm{Ad}$2 (Tran, 20 Aug 2025).

2. Torsion interpretation and character-variety formulation

A central structural fact is that the adjoint twisted Alexander polynomial is the nonabelian Reidemeister torsion polynomial. In Tran’s formulation,

$\mathrm{Ad}$3

coincides with the nonabelian Reidemeister torsion polynomial attached to an irreducible $\mathrm{Ad}$4 representation $\mathrm{Ad}$5, and Yamaguchi’s formula recovers the non-acyclic torsion by the limit

$\mathrm{Ad}$6

This makes the adjoint twisted Alexander polynomial a one-parameter torsion package whose specialization at $\mathrm{Ad}$7 recovers the nonabelian Reidemeister torsion (Tran, 2013).

For finite-volume hyperbolic 3-manifolds with torus boundary, this torsion also appears as a function on a character variety. In Porti’s framework, for a one-cusped manifold $\mathrm{Ad}$8 and a peripheral curve $\mathrm{Ad}$9, the adjoint torsion

$\mathrm{SL}_2(\mathbb{C})$0

is a regular function on an open subset of the distinguished component of the $\mathrm{SL}_2(\mathbb{C})$1-character variety. The tangent space at a character is identified with twisted cohomology,

$\mathrm{SL}_2(\mathbb{C})$2

so adjoint torsion is naturally interpreted as a volume form on the deformation space. Changing the peripheral curve changes the torsion by a Jacobian determinant,

$\mathrm{SL}_2(\mathbb{C})$3

which makes its dependence on boundary coordinates completely explicit (Porti, 2015).

This viewpoint explains why the adjoint hyperbolic torsion polynomial is simultaneously a knot invariant, a torsion invariant, and a function on a representation-theoretic moduli space. A plausible implication is that the polynomial is best understood not only as a single Laurent polynomial for one holonomy representation, but also as a distinguished specialization of a broader torsion function on a geometric component of the character variety.

3. Explicit formulas for torus knots and twist knots

Tran computed the adjoint twisted Alexander polynomial explicitly for torus knots and twist knots. Although torus knots are not hyperbolic, they supply a model family of irreducible $\mathrm{SL}_2(\mathbb{C})$4 representations; twist knots include many hyperbolic examples, so the same formulas give genuine adjoint hyperbolic torsion polynomials in that setting (Tran, 2013).

For the $\mathrm{SL}_2(\mathbb{C})$5-torus knot $\mathrm{SL}_2(\mathbb{C})$6, with $\mathrm{SL}_2(\mathbb{C})$7 an irreducible representation lying in the component indexed by integers $\mathrm{SL}_2(\mathbb{C})$8, the adjoint twisted Alexander polynomial is

$\mathrm{SL}_2(\mathbb{C})$9

Applying Yamaguchi’s limit recovers the nonabelian Reidemeister torsion

$\mathfrak{sl}_2(\mathbb{C})$0

For the twist knot $\mathfrak{sl}_2(\mathbb{C})$1, let $\mathfrak{sl}_2(\mathbb{C})$2 be a nonabelian representation, written in Riley form, and set

$\mathfrak{sl}_2(\mathbb{C})$3

Then

$\mathfrak{sl}_2(\mathbb{C})$4

For longitude-regular $\mathfrak{sl}_2(\mathbb{C})$5, the associated nonabelian Reidemeister torsion is

$\mathfrak{sl}_2(\mathbb{C})$6

These formulas show that, in concrete families, the adjoint hyperbolic torsion polynomial is often an explicitly computable rational expression in standard character coordinates. For twist knots this is particularly effective because many $\mathfrak{sl}_2(\mathbb{C})$7 are hyperbolic for $\mathfrak{sl}_2(\mathbb{C})$8, so the formulas apply directly to hyperbolic holonomy-type representations.

4. Ideal triangulations, gluing equations, and the twisted 1-loop invariant

For hyperbolic once-punctured torus bundles, the adjoint hyperbolic torsion polynomial admits a second, purely triangulational realization. Let $\mathfrak{sl}_2(\mathbb{C})$9 be a hyperbolic once-punctured torus bundle with monodromy triangulation $\pm t^i$0, and let

$\pm t^i$1

be the epimorphism induced by the bundle projection. Garoufalidis–Yoon’s twisted 1-loop invariant is defined from twisted gluing equation matrices

$\pm t^i$2

and the logarithmic derivatives

$\pm t^i$3

by

$\pm t^i$4

For all hyperbolic once-punctured torus bundles,

$\pm t^i$5

where $\pm t^i$6 is the adjoint twisted Alexander polynomial associated to the geometric representation and the bundle epimorphism. The same paper also gives a Jacobian formula

$\pm t^i$7

and a monodromy formula

$\pm t^i$8

showing that the polynomial is the characteristic polynomial of monodromy acting on twisted cohomology of the fiber (Yoon, 2021).

A complementary formulation uses infinitesimal gluing equations. For a geometric ideal triangulation with shape parameters $\pm t^i$9, one forms

$$\mathcal{T}^{\mathrm{Ad}}_K(t^{-1})=-\,\mathcal{T}^{\mathrm{Ad}}_K(t)$$0

where the hatted matrices incorporate edge and completeness equations. Siejakowski identified the torsion of the tangential gluing complex with

$$\mathcal{T}^{\mathrm{Ad}}_K(t^{-1})=-\,\mathcal{T}^{\mathrm{Ad}}_K(t)$$1

and formulated a generalized 1-loop conjecture expressing Porti’s adjoint hyperbolic Reidemeister torsion by

$$\mathcal{T}^{\mathrm{Ad}}_K(t^{-1})=-\,\mathcal{T}^{\mathrm{Ad}}_K(t)$$2

verifying the reduced form for the sister manifold of the figure-eight knot complement (Siejakowski, 2017).

Taken together, these results identify the adjoint hyperbolic torsion polynomial with determinants of Jacobians arising from gluing equations, Ptolemy coordinates, and monodromy actions. This places the invariant at the intersection of Reidemeister torsion, ideal triangulations, and the 1-loop sector of complex Chern–Simons theory.

5. Higher-rank generalizations and geometric formulas

The adjoint viewpoint extends beyond $$\mathcal{T}^{\mathrm{Ad}}_K(t^{-1})=-\,\mathcal{T}^{\mathrm{Ad}}_K(t)$$3. For a compact oriented 3-manifold $$\mathcal{T}^{\mathrm{Ad}}_K(t^{-1})=-\,\mathcal{T}^{\mathrm{Ad}}_K(t)$$4 with torus boundary and a connected semisimple algebraic group $$\mathcal{T}^{\mathrm{Ad}}_K(t^{-1})=-\,\mathcal{T}^{\mathrm{Ad}}_K(t)$$5, the adjoint torsion function is defined on the moduli stack of $$\mathcal{T}^{\mathrm{Ad}}_K(t^{-1})=-\,\mathcal{T}^{\mathrm{Ad}}_K(t)$$6-local systems satisfying a boundary-adjoint-regularity condition: $$\mathcal{T}^{\mathrm{Ad}}_K(t^{-1})=-\,\mathcal{T}^{\mathrm{Ad}}_K(t)$$7 If $$\mathcal{T}^{\mathrm{Ad}}_K(t^{-1})=-\,\mathcal{T}^{\mathrm{Ad}}_K(t)$$8 is cusped hyperbolic and

$$\mathcal{T}^{\mathrm{Ad}}_K(t^{-1})=-\,\mathcal{T}^{\mathrm{Ad}}_K(t)$$9

is a principal embedding, then the image of the complete hyperbolic structure is boundary-adjoint-regular. Writing the exponents of $t^i$0 as $t^i$1, the adjoint torsion satisfies the product formula

$t^i$2

For $t^i$3 and the figure-eight knot complement, the principal-embedded adjoint torsion is

$t^i$4

while a second boundary-unipotent $t^i$5-local system, not arising from a principal embedding, yields an explicit algebraic number over a degree-$t^i$6 field (Ishibashi et al., 28 Feb 2026).

A different higher-dimensional approach expresses adjoint twisted Reidemeister torsion directly in geometric holonomy variables. For fundamental shadow link complements, Wong and Yang obtained

$t^i$7

and for a system of boundary curves $t^i$8,

$t^i$9

with $K\subset S^3$0 the Gram matrices of the truncated hyperideal tetrahedra and $K\subset S^3$1 logarithmic holonomies. Their Dehn-filling formulas add the expected surgery correction factors

$K\subset S^3$2

These expressions show that adjoint torsion is algebraic in logarithmic holonomies and Gram determinants, and therefore behaves as a natural higher-rank torsion function in hyperbolic coordinates (Wong et al., 2021).

This higher-rank picture suggests that the adjoint hyperbolic torsion polynomial is only the rank-$K\subset S^3$3 instance of a wider family of adjoint torsion functions associated with principal embeddings, exponents of semisimple Lie algebras, and explicit holonomy-coordinate formulas.

6. Detection results, structural identities, and related invariants

For hyperbolic double twist knots $K\subset S^3$4, Tran proved that the adjoint hyperbolic torsion polynomial detects both genus and fibering: $K\subset S^3$5 and

$K\subset S^3$6

The proof combines explicit leading and trailing terms of $K\subset S^3$7 with algebraic-integer arguments for the holonomy character parameters. The same source records that Dunfield–Friedl–Jackson observed that $K\subset S^3$8 does not always realize the maximal degree for general hyperbolic knots, but they numerically checked on all hyperbolic knots up to $K\subset S^3$9 crossings that

$\mathrm{Ad}\circ \rho_0$00

(Tran, 20 Aug 2025).

For hyperbolic twist knot exteriors, the adjoint torsion function also satisfies a nontrivial residue-type identity. Writing the irreducible character variety as

$\mathrm{Ad}\circ \rho_0$01

in variables given by the meridian eigenvalue $\mathrm{Ad}\circ \rho_0$02 and a trace parameter $\mathrm{Ad}\circ \rho_0$03, and letting $\mathrm{Ad}\circ \rho_0$04 be a boundary slope, the torsion is

$\mathrm{Ad}\circ \rho_0$05

while in the meridional case,

$\mathrm{Ad}\circ \rho_0$06

For generic $\mathrm{Ad}\circ \rho_0$07 one has the vanishing identity

$\mathrm{Ad}\circ \rho_0$08

proved for all hyperbolic twist knot exteriors by reduction to Jacobi’s residue theorem (Yoon, 2020).

A related but non-adjoint invariant is the hyperbolic torsion polynomial attached directly to a lift of the holonomy representation in $\mathrm{Ad}\circ \rho_0$09. For hyperbolic knots in an integer homology 3-sphere, this polynomial is an unambiguous symmetric Laurent polynomial with coefficients in the trace field, and numerical calculations for all $\mathrm{Ad}\circ \rho_0$10 hyperbolic knots in $\mathrm{Ad}\circ \rho_0$11 with at most $\mathrm{Ad}\circ \rho_0$12 crossings found that it gave a sharp bound on genus and determined both fibering and chirality, while sometimes detecting mutation (Dunfield et al., 2011). This comparison shows that adjoint torsion polynomials belong to a broader family of holonomy-based torsion invariants, but with distinct degree behavior, symmetry, and higher-rank deformation-theoretic content.

The resulting picture is structurally rigid. The adjoint hyperbolic torsion polynomial is a normalized adjoint twisted Alexander polynomial, a nonabelian Reidemeister torsion polynomial, a determinant of infinitesimal gluing or Ptolemy Jacobians in triangulated settings, and, in higher rank, a multiplicative torsion function controlled by principal embeddings and Lie-theoretic exponents. This suggests that its most natural habitat is the interface of hyperbolic geometry, character varieties, and quantum 1-loop structures rather than knot theory alone.