Torus Link Complements

• Torus link complements are topological spaces formed by removing torus links from manifolds, characterized by recursive group presentations and rich geometric structures.
• Their study integrates explicit presentations of fundamental groups and character varieties with detailed analyses of skein modules and quantum invariants.
• Advanced methods like Floer homology and canonical decompositions reveal deeper interconnections in 3- and higher-dimensional topology.

A torus link complement is the topological space obtained by removing a (finite) torus link from a manifold, typically $S^3$, and studying the resulting structure. Torus links, denoted $T(p,q)$ or $T(p,q,n)$ for multiple components, are embedded as unions of simple closed curves on an auxiliary torus in $S^3$, and their complements are central to the study of 3-manifold topology, character varieties, skein modules, Floer homology, and link invariants. The algebraic, geometric, and combinatorial properties of these link complements exhibit recursive structure and deep interconnections with quantum invariants, representation theory, and geometric structures in higher dimensions.

1. Fundamental Groups and Presentations

The fundamental group of a torus $n$-link complement in $S^3$ admits explicit, economical presentations via the groupoid Seifert–van Kampen theorem. For the torus $n$-link $L_n$ with $n$ parallel copies of $T_{p,q}$, the fundamental group is: $$\pi_1(S^3 \setminus L_n) = \langle \alpha, \beta, f_1,\ldots,f_{n-1} \mid \alpha^p f_k \beta^{-q} f_k^{-1} = 1,\; k=1,\dots,n-1 \rangle,$$ where $\alpha,\beta$ are meridian and longitude generators of the auxiliary torus, and $f_k$ are extra generators arising from components of the intersection with the torus. This approach generalizes to nested torus links through inductive applications, with additional meridian-longitude pairs and commutator relations encoding the recursive decomposition topology (Argyres et al., 2019).

For torus links in $T^3$, the link group presentation involves Wirtinger generators for each overpass in a canonical diagram on the flat torus, together with “boundary identification” and “torus” relations capturing the $T^3$ topology (Vuong, 2023).

2. Character Varieties and Representation Spaces

The $G$-character variety of a torus link complement, $\mathcal{X}_G(T(p,q,n))$, parametrizes conjugacy classes of representations of the link group into a reductive group $G$, such as $SL_2(\mathbb{C})$ or $SL_3(\mathbb{C})$ (González-Prieto et al., 2024). For $G=SL_2(\mathbb{C})$, the character variety admits a stratification into irreducible, totally-reducible, and partially-reducible loci, with explicit birational models and formulas for mixed Hodge polynomials ($E$-polynomials). The key input is the fundamental group presentation,

$$\Gamma^n(p,q) = \langle a, b, f_1,\dots,f_{n-1} \mid a^p = b^q,\; a^p f_i = f_i b^q \rangle,$$

whose representation variety is

$$\mathrm{Hom}(\Gamma^n(p,q), G) = \{ (A,B,F_1,\ldots, F_{n-1}) \in G^{n+1} : A^p=B^q,\; A^p F_i=F_i B^q \}.$$

Closed-form expressions for $E$-polynomials reveal that the number of link components appears only as exponents; in the $SL_2$-case, the expression depends only polynomially on $n-1$, reflecting a fibrational structure and the predictable growth of the representation moduli (González-Prieto et al., 2024).

3. Quantum and Classical Invariants: Skein Modules

The Kauffman skein module of a torus knot or link complement serves as a quantization of the character variety, with a grading determined by an incompressible annulus splitting the complement. The module $S(X)$ of the $T_{p,q}$-knot complement admits a filtration by intersection number with the annulus, whose associated graded pieces correspond to trace-functions on the $SL(2,\mathbb{C})$ character variety (Marche, 2010). For torus links ($gcd(p,q)>1$), similar techniques apply, with an expected isomorphism

$$S(S^3 \setminus T_{p,q}) \cong \mathbb{C}[\chi(\pi_1(S^3 \setminus T_{p,q}))] \otimes \mathbb{C}[A^{\pm1}],$$

and graded ranks matching dimensions of irreducible representation classes.

4. Floer Homology and Bordered Structures

The complement of the $(2,2n)$-torus link in $S^3$ admits a complete computation of its bordered Floer bimodule $\mathrm{CFDD}(S^3 \setminus \nu(L))$ (Lee, 2013). Decomposing the complement as a doubly-bordered manifold, the bimodule is constructed over the tensor product of two copies of the torus algebra, with explicit generators and a $\delta^1$ structure map given by counting Maslov-index-one holomorphic curves in the Heegaard diagram. The combinatorics of the module encode the topological and smooth structure of the link complement.

Key points include:

• The bimodule structure reflects the gluing of bordered pieces, enabling recovery of Heegaard–Floer invariants for manifolds obtained by Dehn surgery on the torus link.
• Explicit homotopy reduction yields minimal models with only essential differentials.
• The construction generalizes the surgery exact triangle and is compatible with classical lens space invariants, providing a deep algebraic encoding of the topological data (Lee, 2013).

5. Homotopy and Loop Classification

For $(2,p)$-torus link complements viewed as 2-bridge link exteriors, the classification of essential homotopy classes of simple loops on the standard 2-bridge sphere $\Sigma$ is governed by a precise arithmetical relationship (Lee et al., 2010). Two simple loops $\alpha_s$ and $\alpha_{s'}$ (with slope $s$ and $s'$ in the projective Farey graph) are homotopic in the complement if and only if

$\frac{1}{s} + \frac{1}{s'} = p,$

or equivalently, if their $S$-sequences satisfy the relevant orbits under the extended symmetry group $\tilde\Gamma_{1/p}$. This provides a complete, combinatorial invariant for the homotopy classes of such loops, with implications for the geometry of the link complement and connections to sum identities for lengths of geodesics in the hyperbolic case.

6. Higher-Dimensional and Nontrivial Ambients

Torus link complements in nonstandard settings, such as in the $3$-torus $T^3$ or the $4$-sphere $S^4$, display novel topological and geometric features:

• In $T^3$, diagrammatics with generalized Reidemeister moves allow for explicit presentations of the link group, computation of homology, and multivariable (and twisted) Alexander polynomials via Fox calculus. The homological structure is dictated by the arithmetical data of how each component represents a class in $H_1(T^3;\mathbb{Z})$ (Vuong, 2023).
• In $S^4$, the complement of a link of five 2-tori is an integral arithmetic hyperbolic 4-manifold. Its fundamental group admits a 5-generator, 10-relator “Borromean-style” commutator presentation; the geometry is distinguished by a double cover with $\mathbb{H}^2 \times \mathbb{H}^2$ structure, perfect Morse functions for every cohomology class, and explicit Alexander ideal computation (with trivial Alexander polynomial) (Martelli, 2024).

7. Canonical Decompositions and Geometric Structures

Hyperbolic torus link complements, especially those arising as fully augmented links in thickened tori $T^2 \times I$, admit canonical decompositions into right-angled ideal torihedra via the circle-pattern theorem of Bobenko–Springborn (Kwon, 2020). For diagrams satisfying weak primeness and twist-reduction, the complement splits into two isometric ideal torihedra, and the decomposition is canonical, with each face mapping unambiguously to a geometric piece. This structure underlies volume density results and the asymptotic behavior of geometric invariants for sequences of links converging to biperiodic (infinite) ones. The circle-pattern methodology generalizes to thickened surfaces $S \times I$, offering a unifying geometric framework for canonical decompositions of link complements in a wide variety of ambient manifolds.

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These results illustrate the algebraic, topological, and geometric richness of torus link complements and demonstrate the deep connections between combinatorial presentations, representation spaces, quantum invariants, Floer theory, and geometric decompositions in both three and higher dimensions (Lee, 2013, Martelli, 2024, Kwon, 2020, Marche, 2010, González-Prieto et al., 2024, Vuong, 2023, Lee et al., 2010, Argyres et al., 2019).