Two-Bridge Knots: Classification & Invariants

• Two-bridge knots are knots in S³ with bridge index two, defined via rational tangle closures and classified using Schubert and Conway’s notation.
• They play a pivotal role in exploring invariants such as Seifert genus, four-genus, and state invariants, facilitating algorithmic and combinatorial analyses.
• Their study extends to ribbon and sliceness phenomena, unknotting via local moves, and concordance structures, providing insights into topological orderability and manifold invariants.

A two-bridge knot is a knot in $S^3$ admitting a diagram with bridge index two, equivalently arising as the closure of a rational tangle with slope $p/q$, where $p > 0$, $p$ odd, and $\gcd(p,q) = 1$. Two-bridge knots possess a canonical classification via Schubert notation $K(p,q)$ or Conway’s normal form $C(a_1, a_2, ..., a_n)$, where the continued-fraction expansion of $p/q$ encodes the twist sequence in the 4-plat diagram. The family is foundational in knot theory, serving as a testing ground for knot invariants, topological properties, cobordism, orderability, and covering space constructions. The research literature details their combinatorial, geometric, and algebraic invariants, ribbon and slice structures, random diagram statistics, polynomial parameterizations, and placement in concordance and partial order frameworks.

1. Classification and Parametrization

Two-bridge knots are described in three equivalent frameworks:

• Schubert notation: $K(p,q)$ where $p > 0$ is odd, $q \in (0, p)$, and $\gcd(p,q) = 1$. The knot type is determined up to orientation and mirroring by $(p,q) \sim (p, p-q)$ (Horigome et al., 2024).
• Continued-fraction expansion: $p/q = [a_1, a_2, \ldots, a_n]$ gives the twist sequence. Conway’s normal form $C(a_1, ..., a_n)$ prescribes alternating twist boxes according to these coefficients, up to symmetries (Baader et al., 2019, Cohen et al., 2016).
• 4-plat closure: For $K = K(2a_1, ..., 2a_{2m})$, the number of even “twist regions” $2m$ determines the plat diagram (Baader et al., 2019).

This classification allows explicit generation of knot diagrams, computation of invariants, realizations of invariants such as knot group presentations, and direct study of families (e.g. genus one knots where $[2m, 2n]$) (Tran, 2015).

2. Knot Invariants and State Surfaces

Two-bridge knots are central in the study of classical and generalized invariants:

• Seifert genus: For a 4-plat $K(2a_1, ..., 2a_{2m})$, $g(K) = m$. Seifert genus coincides with minimal spanning surface genus in $S^3$ (Baader et al., 2019).
• Four-genus: The smooth four-genus $g_4(K)$ satisfies $g_4(K) \leq g(K)$, and for large crossing number, the average ratio $\langle g_4/g \rangle_n$ tends to zero, i.e., most large two-bridge knots have sublinear four-genus relative to Seifert genus (Baader et al., 2019).
• State invariants (Gordon–Litherland theory): Essential state surfaces, classified by $|n_i| \geq 2$ expansions, give rise to a collection of state polynomials $\Delta_S(t)$ and state signatures $\sigma_S$. These extend Alexander polynomial and signature to all incompressible surfaces and realize boundary slopes as signature differences: $\text{slope}(S) = 2(\sigma_S - \sigma(K))$ (Curtis et al., 2017).

Classical invariants can be algorithmically computed and related to the combinatorics of the continued fraction, with explicit skein-type recurrences and symmetry relations (Curtis et al., 2017).

3. Ribbon and Sliceness Phenomena

A two-bridge knot is ribbon if it bounds an immersed disk in $B^4$ with only ribbon singularities. Lisca classified all slice/ribbon two-bridge knots as those of form $K(m^2, mk \pm 1)$, $K(m^2, d(m \pm 1))$ (for divisors $d$ of $2m\mp 1$ or $m \pm 1$) with odd $m$, $k$, and $(m, k) = 1$ (Horigome et al., 2024).

Symmetric union presentations are explicitly constructed for all ribbon two-bridge knots, where the diagram is a union of a knot and its mirror with twist bands; these unions encode the ribbon structure. The partial knot in the union is a simpler two-bridge knot $K(m,k)$ or $K(m,d)$. For every slice two-bridge knot, a symmetric union is constructed, contributing to the validation of the slice–ribbon conjecture in this context (Horigome et al., 2024).

4. Unknotting via Local Moves

The $\Delta$-unknotting number $u^\Delta(K)$ for two-bridge knots is the minimal number of $\Delta$-moves needed to reduce $K$ to the unknot. For knots with Conway parameters all even or all-but-last-even, $u^\Delta(K) = |a_2(K)|$, where $a_2(K)$ is the second Conway coefficient (Nakamura, 28 Dec 2025). This result parallels the computation of classical unknotting number in the torus and pretzel knot families.

Explicit formulas for $u^\Delta$ are given for various parameter types and exact criteria for two-bridge knots with $\Delta$-unknotting number one are derived, specifying palindromic Conway forms (Nakamura, 28 Dec 2025).

5. Algebraic and Concordance Structures

• Equivariant concordance: Two-bridge knots generate infinite order in the strongly equivariant concordance group and are never equivariantly slice; this is proven via link invariants (butterfly and moth polynomials), axis-linking number, and covering space arguments. Invariants such as Kojima–Yamasaki’s $\eta$-function admit closed formulas reliant on continued fraction parameters (Prisa et al., 2023).
• Twisted invariants: The twisted Alexander polynomial for genus one two-bridge knots, including twist knots, is explicitly computed for $SL_2(\mathbb{C})$ representations. These formulas are sophisticated functions of the trace variables and Chebyshev polynomials and, when specialized, recover classical Alexander polynomials and Reidemeister torsion of surgery manifolds (Tran, 2015).
• Partial order and covers: The Ohtsuki–Riley–Sakuma (ORS) partial order enables epimorphisms between knot groups of two-bridge knots. Precise combinatorial criteria for upper bounds in the ORS order are established via expanded even continued fraction vectors. Some pairs (e.g., trefoil and figure-eight) lack a common ORS upper bound due to incompatible parity patterns (Garrabrant et al., 2010).

6. Statistical and Computational Aspects

• Random models: Probability distributions for crossing number in random two-bridge knots, constructed via billiard table models, are derived analytically. For large $n$, the probability of encountering any fixed knot type decays exponentially, while the normalized minimal crossing number converges to $(\sqrt{5}-1)/4 \approx 0.309$ (Cohen et al., 2016).
• Symmetric diagrams and crossing numbers: The equivariant crossing number $c_t(K)$ is bounded above by $c_2(K)$, computed via restricted continued fractions of type (A) (even parity) or (B) (palindromic and odd-central entry). Computations up to 14 crossings demonstrate that $c_2(K)-c(K) \le 3$, with exact symmetric diagrams obtainable for at least 20 knots with up to 10 crossings (Nanasawa, 2023).
• Lexicographic degree: The minimal degree triple of polynomial parametrizations (“lexdeg”) is classified for all two-bridge knots up to 11 crossings: torus knots achieve $(2, n, 2n-1)$ and non-torus examples attain $(3, d_y, 2d_y-1)$, where $d_y$ is informed by the continued fraction (Brugallé et al., 2015).

Knot Family	Invariant/Property	Quantitative Formula
Genus one	Twisted Alexander poly.	$\Delta_{J(2m,2n),\rho}(t)$ explicit in $s,y$
Ribbon	Slice classification	$K(m^2, mk \pm 1)$, $d(m \pm 1)$ forms
Delta-unknotting	$\Delta$-unknotting number	$u^\Delta(K) = |a_2(K)|$ for specific families

These computations facilitate further analyses of fiberedness, hyperbolic geometry, surgery orderability, and polynomial knot recognition.

7. Geometric, Topological, and Orderability Implications

Two-bridge knots furnish explicit constructions for branched cyclic covers using bi-twist face-pairing methods. For $K(p,q)$, the $n$-fold cover is obtained by cyclically banding copies of faceted balls with prescribed twist sequences; group presentations and homology (e.g., Sieradski and Fibonacci manifolds) are computed directly (Cannon et al., 2013).

Orderability under Dehn surgery is established for large families such as $[1,1,2,2,2j]$, relying on Alexander polynomial roots and cohomological rigidity. These results link to the L-space conjecture and exhibit intervals of left-orderable fillings for two-bridge knot complements, even in cases without fibering or real trace field places (Le, 2021).

Ribbon and slice structures manifest in concrete diagrammatic realizations (symmetric unions) and classification schemes. Invariants derived from essential state surfaces deepen the correspondence between combinatorial, algebraic, and geometric data. Delta-unknotting and crossing number refinements interplay with minimal diagrammatic symmetry and the effective recognition of knot type.

In summary, the theory of two-bridge knots is an amalgam of combinatorial classification, geometric construction, invariant computation, and topological application. Its rich interplay between diagrammatic, algebraic, and geometric structures drives ongoing advances in knot theory, low-dimensional topology, and manifold invariants.