Adjoint Polynomials of Torus Knots

• Adjoint polynomials are knot invariants defined by coloring torus knots with the adjoint representation, encapsulating complex topological and representation-theoretic data.
• Vogel’s universality unifies all simple Lie algebras using three parameters (α, β, γ), enabling explicit closed-form expressions and composite expansions.
• These polynomials bridge various invariants—from HOMFLY-PT and Kauffman to twisted Alexander and DAHA–Jones—providing practical insights into quantum dimensions and knot topology.

Adjoint polynomials of torus knots are knot invariants derived by coloring a torus knot with the adjoint representation of a simple Lie algebra, typically in the framework of quantum group invariants such as HOMFLY-PT or Kauffman polynomials, as well as from twisted Alexander polynomials and DAHA–Jones constructions. These polynomials admit a closed-form, explicitly universal description via the Rosso–Jones formula, and encode deep representation-theoretic and topological information. Their construction is characterized by Vogel’s universality—the expression of invariants in terms of three parameters $(\alpha,\beta,\gamma)$ that unify all simple Lie algebras—and composite/plethystic expansions involving only the adjoint and its descendants.

1. Definition and Rosso–Jones Construction

For a $(m,n)$ torus knot $T[m,n]$, the adjoint polynomial is obtained by the Rosso–Jones formula specialized to the adjoint representation (Adj), denoted $H_{\mathrm{Adj}}^{T[m,n]}(q,A)$ for HOMFLY-PT (with $A=q^N$ for $SU(N)$, or equivalent for other series). The formula takes the form: $$H_{\mathrm{Adj}}^{T[m,n]}(q,A) = \frac{A^{2mn}}{A^2 qD_{\mathrm{Adj}}} \left[(m-1) + \sum_{a,b=1}^m (-1)^{a+b} A^{-2n} q^{-2n(a+b-m-1)} qD_{([a,1^{m-a}],[b,1^{m-b}])} \right]$$ where $qD_{\mathrm{Adj}}$ and $qD_{([a,1^{m-a}],[b,1^{m-b}])}$ denote quantum dimensions of the adjoint and composite hook representations, and the sum extends over Young diagrams indexing hooks. This double-sum formula encapsulates all representation-theoretic data and is symmetric in $(m,n)$ (Mironov et al., 28 Dec 2025).

2. Vogel’s Universality and Universal Formulae

Vogel’s universality posits that adjoint polynomials can be universally written using three parameters $(\alpha,\beta,\gamma)$ associated with the Vogel plane. For any simple Lie algebra, one sets $u=q^\alpha$, $v=q^\beta$, $w=q^\gamma$, and expresses the polynomial as: $$P_{\mathrm{Adj}}^{[m,n]}(u,v,w) = T^{2mn/t} \sum_{Q\subset \mathrm{Adj}^{\otimes m}} c_Q^{(m,n)} \lambda_Q^{-n} D_Q(u,v,w)$$ with $T=q^{\alpha+\beta+\gamma}$, $t=\alpha+\beta+\gamma$, $D_Q$ universal quantum dimensions, $\lambda_Q$ R-matrix eigenvalues, and $c_Q$ plethysm/Adeams coefficients (Bishler et al., 6 Jun 2025, Mironov et al., 2015, Mironov et al., 2015). Specializations yield the HOMFLY (for $SL_N$) and Kauffman (for $SO_N$, $Sp_N$) polynomials, and exceptional cases.

3. Quantum Dimensions, Casimir Eigenvalues, and Racah Matrices

Quantum dimensions of the adjoint and its descendants are rational functions in $u,v,w$. For $\mathrm{Adj}$,

$$D_{\mathrm{Adj}}(u,v,w) = -\frac{\{\sqrt{u}vw\} \{\sqrt{v}uw\} \{\sqrt{w}uv\}}{\{\sqrt{u}\}\{\sqrt{v}\}\{\sqrt{w}\}}$$

Analogous expressions hold for $X_2, Y_2(\alpha), Y_2(\beta), Y_2(\gamma), \dots$ as needed for the specific $m$-strand case (Mironov et al., 2015, Mironov et al., 2015).

Eigenvalues of the quantum R-matrix correspond to $q^{\text{Casimir}_Q}$ for each irreducible channel $Q$ in $\mathrm{Adj}^{\otimes m}$, governing the topological evolution. Racah matrices controlling the mixing (fusion) are universally described, with explicit $6\times 6$ forms for arborescent knots such as two-strand cases; squares of entries are rational functions of the eigenvalues and $uvw$ (Mironov et al., 2015).

4. Explicit Formulas and Specializations

Tables of explicit formulas exist for low-strand cases:

Torus knot	Universal adjoint polynomial	Quantum Dimensions/Channels
$T[2,3]$	$$P_{\mathrm{Adj}}^{[2,3]}(u,v,w)=\frac{(uvw)^6}{D_{\mathrm{Adj}}}[\dots]$$	$$D_{\mathrm{Adj}}, D_{X_2}, D_{Y_2(\alpha,\beta,\gamma)}$$
$T[3,4]$	$$P_{\mathrm{Adj}}^{[3,4]}(u,v,w)=\frac{(uvw)^8}{D_{\mathrm{Adj}}}[\dots]$$	$$D_{\mathrm{Adj}}, D_{X_3}, D_{Y_3(\alpha,\beta,\gamma)}, D_{C(\alpha,\beta,\gamma)}$$

These formulas reduce under specialization:

• $A\to 1$ (Alexander polynomial): $P_{\mathrm{Adj}}=1$
• $q\to 1$ (special polynomial): fundamental squared $\sigma_{\mathrm{Adj}} = [\sigma_{[1]}]^2$
• $A=q^2$ (colored Jones): yields closed cube-free formula
• Reflection invariance: $$P_{\mathrm{Adj}}^{[m,-n]}(u,v,w)=P_{\mathrm{Adj}}^{[m,n]}(u^{-1},v^{-1},w^{-1})$$
• Topological symmetry: $P_{\mathrm{Adj}}^{[m,n]}=P_{\mathrm{Adj}}^{[n,m]}$

5. DAHA Approach and Homological Interpretations

Adoint-colored DAHA–Jones polynomials are constructed from Macdonald polynomials indexed by the adjoint weight $\theta$, via the DAHA automorphism $\gamma_{m,n}^\wedge$: $$J_{m,n}^{\mathrm{adj}}(q,t) = \{ \gamma_{m,n}^\wedge (P_\theta^\circ(X)) \}_{ev}$$ where $P_\theta^\circ$ is the normalized Macdonald polynomial, and $ev$ denotes evaluation. These polynomials exhibit:

• Polynomiality in $q,t$
• Duality: $$J_{m,n}^{\mathrm{adj}}(q,t) = J_{m,n}^{\mathrm{adj}}(t^{-1},q^{-1})$$
• $m\leftrightarrow n$ symmetry

In the limit $a\to 0$, $q\to 1$, these provide Betti numbers of Jacobian factors of plane curve singularities linked to $T(m,n)$, establishing a conjectural correspondence to Khovanov–Rozansky homology (Cherednik et al., 2014).

6. Twisted Alexander Polynomials and Nonabelian Torsion

The twisted Alexander polynomial in the adjoint representation for $T(p,q)$ is derived using Fox calculus: $$\Delta^{\mathrm{Adj} \circ \rho}_{T(p,q)}(t) = \frac{(t^{pq}-1)^3}{(t^p-1)(t^q-1)\big(t^{2q}-2\cos{\tfrac{2\pi k}{p}}t^q + 1\big)\big(t^{2p}-2\cos{\tfrac{2\pi l}{q}}t^p + 1\big)}$$ for representation parameters $(k,l)$. In the $t\to 1$ limit, this directly computes the nonabelian Reidemeister torsion: $$\tau_K^\rho = -\frac{p^2q^2}{16\sin^2(\pi k/p)\sin^2(\pi l/q)}$$ generalizing classical Alexander polynomials and establishing its topological and representation-theoretic character (Tran, 2013).

7. Structural Properties and Universality Phenomena

Adjoint polynomials of torus knots universally display key features:

• Degree linearity in $m,n$ for each summation term
• Full symmetry in $q\leftrightarrow q^{-1}$ for each building block
• Special polynomial factorization and Alexander property
• Divisibility of $P_{\mathrm{Adj}}-1$ by $(uvw-1)(uvw+1)$, often extended to higher differential expansions
• Recursion conjecture: For fixed $m$, $P_{\mathrm{Adj}}^{[m,n]}$ obeys linear recursion of order $m$ in $n$
• Topological $[m,n]\leftrightarrow[n,m]$ invariance throughout Vogel’s plane

This universality allows for the adjoint polynomial to serve as a unifying thread for knot invariants across Lie types ($SU_N$, $SO_N$, $Sp_N$, exceptional groups), arborescent knots, and their representation-theoretic and homological avatars (Mironov et al., 2015, Mironov et al., 2015, Bishler et al., 6 Jun 2025, Mironov et al., 28 Dec 2025).

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The explicit, universal structure of adjoint polynomials for torus knots, grounded in Vogel’s plane and Rosso–Jones plethysm, provides a systematic foundation for studying knot invariants in higher representations, extending their reach to homological algebra and topological quantum field theory. The construction accommodates all series and exceptional cases uniformly, and its symmetry and recursion properties continue to be active fields of investigation.