Links–Gould Polynomial Overview

• Links–Gould polynomial is a two-variable Laurent invariant for oriented links derived from quantum superalgebras, generalizing the Alexander–Conway polynomial.
• It uses cubic quotients of braid group algebras and skein relations to provide refined genus bounds and detect fiberedness in knots and links.
• Recent colored generalizations like LG^(n) and the V_n-series extend its application in quantum invariants, categorification, and combinatorial topology.

The Links–Gould polynomial is a two-variable Laurent polynomial invariant of oriented links, constructed using the representation theory of quantum superalgebras (notably U_q(sl(2|1)) and U_q(gl(2|1))). It generalizes the classical Alexander–Conway polynomial and exhibits quantum-topological features such as refined genus bounds, skein relations, and an explicit algebraic characterization via cubic quotients of braid group algebras. In addition, its recent colored and higher-dimensional generalizations (such as the LG^{(n)} polynomial and the V_n-series from Nichols algebras) connect it to a broader family of quantum invariants, categorifications, and combinatorial structures.

1. Algebraic and Skein-Theoretic Foundations

The Links–Gould polynomial LG(L; t₀, t₁) is defined by assigning to each oriented link L a Laurent polynomial in two variables, often via the Reshetikhin–Turaev functor applied to typical 4-dimensional representations of the quantum superalgebra U_q(sl(2|1)) or U_q(gl(2|1)) (Marin et al., 2012). The algebraic construction proceeds by encoding the invariant as a Markov trace on a sequence of finite-dimensional cubic quotients Aₙ of the braid group algebra. For n-strand braids, one imposes a generic cubic relation

$(s_i - a)(s_i - b)(s_i - c) = 0,$

together with further three-strand and four-strand relations (r₂ and r₃). The resulting defining algebra supports a unique Markov trace vanishing on split links, ensuring that LG is well-defined and link invariant under Markov moves.

Key algebraic properties include:

• Surjectivity of the braid group algebra map into the centralizer algebra LGₙ = End_{U_q(sl(2|1))}(V(0,α)^{\otimes n}),
• An explicit conjectural dimension formula connected to Catalan numbers,
• A bimodule structure, admitting recursive decomposition analogous to analogous structures in the BMW/Kauffman setting.

The skein-theoretic structure is characterized by cubic relations amongst braid generators and additional local relations, forming a reduction system that can be used to calculate LG for any oriented link (Garoufalidis et al., 25 May 2025).

2. Connection to Classical Invariants and Generalizations

The Links–Gould polynomial fundamentally generalizes the Alexander–Conway polynomial. Under suitable specializations, one finds:

• LG(L; t₀, t₁) evaluated at t₁ = –t₀ yields the Alexander polynomial, up to variable change, i.e., LG(L; t₀, –t₁) = A_L(t₁²),
• LG(L; t₀, t₀) = (A_L(t₀))² (Kohli, 13 May 2025, Kohli, 2015, Kohli et al., 2016),
• Higher variants LG_{m,n}(L; T, e^{2πi/n}) conjecturally specialize to powers of the Alexander polynomial: LG_{m,n}(L; T, e^{2πi/n}) = A_L(T)^m.

This specialization extends not just to the classical Alexander polynomial, but also to quantum invariants such as the single-colored ADO_3-invariant (at roots of unity), with

$$ADO_3(L; t) = LG(L; t^2, \omega^2 t^{-2}), \qquad \omega = e^{\pi i/3}$$

for all closures of 5-braids and, conjecturally, all links (Takenov, 2022). Furthermore, the invariant vanishes on split links, exhibiting symmetry properties consistent with the vanishing of the Alexander polynomial for split links.

The polynomial also connects to broader quantum algebraic frameworks, including representations from Nichols algebras and colored quantum invariants. Recent work establishes a cabling formula expressing LG^{(n)} for a knot colored by a 4n-dimensional irreducible representation in terms of LG-polynomials colored by lower-dimensional representations, showing equivalence between LG^{(2)} and the V_2-polynomial (Garoufalidis et al., 13 Sep 2025, Garoufalidis et al., 5 Sep 2024). These colored forms satisfy genus bounds and specialize to the Alexander polynomial independently of n.

3. Genus Bounds and Topological Criteria

A central topological property of the Links–Gould polynomial is its effectiveness in bounding the Seifert genus of knots and links. For a knot K,

$$\operatorname{span}\left(LG(K; t_0, t_1)\right) \leq 4g(K)$$

where span denotes the difference between maximal and minimal degrees in the Laurent polynomial (Kohli et al., 2023). Computational evidence establishes sharpness for alternating knots and, for the colored LG^{(2)} (V_2) polynomial, for all knots with up to 16 crossings (Garoufalidis et al., 13 Sep 2025, Garoufalidis et al., 5 Sep 2024).

Unlike the Alexander polynomial, which may fail in cases of mutants (e.g., the Kinoshita–Terasaka and Conway knots where Δ_K(t) = 1 gives no genus bound), the Links–Gould polynomial provides a strictly better lower bound in such cases, detecting genus information where classical invariants do not.

In addition, there is a conjectured fiberedness criterion: for a knot K, LG(K; t₀, t₁) is monic (i.e., the top coefficient in t₀ is ±1) if and only if K is fibered, at least for alternating knots. This property generalizes the classical criterion for the Alexander polynomial and is supported by extensive computational data (Kohli, 13 May 2025).

4. Colored Invariants, Cabling Formula, and Higher Vₙ-Polynomials

Recent developments introduce colored versions of the Links–Gould polynomial, assigning invariants to knots colored by higher-dimensional irreducible modules over U^H_q(sl(2|1)). These LG^{(n)} invariants are related via a cabling formula to cables of the knot with lower colors, with explicit coefficients given by ratios of modified dimensions in the module category (Garoufalidis et al., 13 Sep 2025):

$$LG^{(1)}_{K^{(n,0)}}(\hat{q}, q) = \sum_{k+\ell \leq n-1} m^{(n)}_{k,\ell} \cdot A^{(n)}_{k,\ell}(\hat{q}, q) \cdot LG^{(k+1)}_K(q^{n+\ell}, q)$$

For n = 2, the invariants LG^{(2)} coincide (up to change of variables) with the V_2-polynomial constructed from the R-matrix of a rank 2 Nichols algebra. These "V_n" invariants exhibit symmetry, satisfy sharp genus bounds (deg_t V_n ≤ 4g(K)), and specialize to the square of the Alexander polynomial at q = 1 (Garoufalidis et al., 5 Sep 2024, Garoufalidis et al., 13 Sep 2025).

Patterns observed include mutation invariance at n = 1 (LG, V_1); for n = 2, while genus bounds are sharp, the polynomial still sometimes fails to distinguish Conway mutants. Even categorified invariants such as Heegaard Floer homology and Khovanov homology coincide on such pairs, indicating deep symmetry.

5. Plumbing Multiplicativity and Topological Obstructions

The multiplicativity of the "top coefficient" of LG(L; p, q) with respect to plumbing operations on Seifert surfaces is a notable property (Lopez-Neumann et al., 18 Feb 2025). Specifically, for Seifert surfaces Σ₁, Σ₂ plumbed to form Σ, one has: $$\operatorname{top}(\partial \Sigma, q) = \operatorname{top}(\partial \Sigma_1, q) \cdot \operatorname{top}(\partial \Sigma_2, q)$$ where "top" denotes the coefficient of maximal p-degree (with p determined by Euler characteristic).

This property allows algebraic detection of fiberedness, provides obstructions for links bounding annular or plumbed surfaces built from twisted annuli, and determines (up to normalization) the Hopf invariant of the tangent plane field of a fibered link's surface.

In particular, the top coefficient factors uniquely as products of polynomials a_n(q) corresponding to annuli with n full twists, showing the decomposition into annular building blocks is unique and that links whose LG top coefficient does not factor appropriately cannot bound such plumbed surfaces.

6. Broader Combinatorial and Quantum-Topological Context

Recent work bridges the Links–Gould polynomial with other combinatorial and algebraic frameworks:

• The polynomial ring construction connecting central binomial coefficients and Gould's sequence via recursive quotient rings demonstrates new algebraic links between integer sequences and invariants, with potential applications for binomial transforms and combinatorics (Shunia, 2023).
• Graph-theoretic perspectives connect link polynomials to the Bollobás–Riordan polynomial, cyclic signed graphs, and virtual links, situating LG in a wider family of invariants (such as the Jones and HOMFLY polynomials) sharing recursive and spanning subgraph expansions (Deng et al., 2017, Jiang et al., 2016).
• Differential hierarchy (Z-expansion) and evolution methodologies applied to quantum knot invariants like HOMFLY suggest potential for similar recursion and mirror symmetry behavior in the paper of the Links–Gould polynomial (Arthamonov et al., 2013).

7. Current Trends, Comparisons, and Open Problems

The skein-theoretic characterization of LG and its identification with the V_1 polynomial unify quantum-group and combinatorial approaches, establishing a robust computational foundation (Garoufalidis et al., 25 May 2025). The sharp genus detection established for the colored LG^{(2)} and V_2 polynomials up to high crossing number knots sets a benchmark for the topological sensitivity of quantum invariants (Garoufalidis et al., 13 Sep 2025, Garoufalidis et al., 5 Sep 2024).

Mutation invariance at n = 1 and (often) n = 2, symmetry properties, and explicit cabling formulas position the LG family as a central object in quantum knot theory, but also highlight limitations: certain subtle knot equivalences (mutants) are undetectable even by these refined invariants and their categorified analogs.

Open questions remain regarding the full scope of mutation detection for higher n, the extension of genus sharpness, the detailed algebraic structure of the cubic defining algebras (their completeness and dimension formulas), and further generalizations connecting LG to integer sequence algebraic frameworks.

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Table: Key Specializations and Properties

Invariant Type	Specialization	Genus Bound
Links–Gould LG(L; t₀, t₁)	t₁ = –t₀ ⇒ Alexander	span ≤ 4g(K)
LG^{(2)} = V₂-polynomial	Cabling formula	Equality
LG(L; t₀, t₀)	(Alexander)^2	Sharp when alternating
ADO₃(L; t)	LG(L; t², ω² t⁻²)	Conjecturally general
Top coefficient of LG	Plumbing multiplicative	Detects fibredness

This structure summarizes key relationships among Links–Gould polynomial variants, their genus bounds, and specialization properties established in the literature.