core.klein_curve

klein_curve

Klein quartic curve: structural parameters, PSL(2,7) generators, and matrix representations.

The Klein quartic is the unique genus-3 Riemann surface with maximal automorphism group PSL(2,7) of order 168. It is the quotient H/Gamma(2,3,7) where Gamma(2,3,7) is the (2,3,7) triangle group. The curve satisfies the Gauss-Bonnet relation Area = 4pi(g-1).

KleinCurve

KleinCurve(genus: int = 3, K: float = -1.0, psl_order: int = 168, lambda_1: float = 3.838)

Klein quartic curve with all structural invariants.

Attributes:

Name	Type	Description
genus		Curve genus g = 3.
K		Gaussian curvature K = -1 (hyperbolic).
R		Scalar curvature R = 2K = -2.
area		Total area = 4pi(g-1) = 8*pi (Gauss-Bonnet).
psl_order		|PSL(2,7)| = 168.
sl_order		|SL(2,7)| = 336.
lambda_1		First eigenvalue of scalar Laplacian (Bourque-Strohmaier 2024).

Source code in src/core/klein_curve.py

def __init__(self, genus: int = 3, K: float = -1.0,
             psl_order: int = 168, lambda_1: float = 3.838):
    self.genus = genus
    self.K = K
    self.R = 2 * K
    self.area = 4 * np.pi * (genus - 1)
    self.psl_order = psl_order
    self.sl_order = 2 * psl_order
    self.lambda_1 = lambda_1
    logger.info(
        f"Klein curve initialized: g={genus}, K={K}, R={self.R}, "
        f"Area={self.area:.4f}, |PSL(2,7)|={psl_order}, λ₁={lambda_1}"
    )

generators

generators() -> tuple[ndarray, ndarray, ndarray]

Compute the Gamma(2,3,7) generators A, B, C in SL(2,R).

A has order 4 in SL(2,R) (order 2 in PSL), B has order 6 in SL (order 3 in PSL), and C = AB has order 7 in SL (order 7 in PSL). The relations A^2 = B^3 = (AB)^7 = -I hold in SL(2,R).

Returns:

Type	Description
tuple[ndarray, ndarray, ndarray]	Tuple (A, B, C) of 2x2 numpy arrays.

Source code in src/core/klein_curve.py

def generators(self) -> tuple[np.ndarray, np.ndarray, np.ndarray]:
    """Compute the Gamma(2,3,7) generators A, B, C in SL(2,R).

    A has order 4 in SL(2,R) (order 2 in PSL), B has order 6 in SL
    (order 3 in PSL), and C = AB has order 7 in SL (order 7 in PSL).
    The relations A^2 = B^3 = (AB)^7 = -I hold in SL(2,R).

    Returns:
        Tuple (A, B, C) of 2x2 numpy arrays.
    """
    A = np.array([[0, 1], [-1, 0]], dtype=float)

    c_const = 4 * np.cos(np.pi / 7) / np.sqrt(3)
    lam = (c_const + np.sqrt(c_const**2 - 4)) / 2
    B = np.array([
        [np.cos(np.pi / 3), lam * np.sin(np.pi / 3)],
        [-np.sin(np.pi / 3) / lam, np.cos(np.pi / 3)]
    ])

    C = A @ B
    logger.info("PSL(2,7) generators A, B, C computed in SL(2,R)")
    return A, B, C

verify_relations

verify_relations(A: ndarray, B: ndarray, C: ndarray) -> dict

Verify the defining relations A^2 = B^3 = (AB)^7 = -I.

Parameters:

Name	Type	Description	Default
A, B, C		Generator matrices from self.generators().	required

Returns:

Type	Description
dict	Dict with boolean verification results and norms.

Source code in src/core/klein_curve.py

def verify_relations(self, A: np.ndarray, B: np.ndarray,
                     C: np.ndarray) -> dict:
    """Verify the defining relations A^2 = B^3 = (AB)^7 = -I.

    Args:
        A, B, C: Generator matrices from self.generators().

    Returns:
        Dict with boolean verification results and norms.
    """
    I2 = np.eye(2)
    nI = -I2
    results = {
        "A_sq_eq_negI": bool(np.allclose(A @ A, nI)),
        "A_sq_norm": float(np.linalg.norm(A @ A - nI)),
        "B_cub_eq_negI": bool(np.allclose(np.linalg.matrix_power(B, 3), nI)),
        "B_cub_norm": float(np.linalg.norm(np.linalg.matrix_power(B, 3) - nI)),
        "C_sev_eq_I": bool(np.allclose(np.linalg.matrix_power(C, 7), I2)),
        "C_sev_norm": float(np.linalg.norm(np.linalg.matrix_power(C, 7) - I2)),
    }
    logger.info(f"Relation verification: {results}")
    return results

as_dict

as_dict() -> dict

Serialize curve parameters.

Source code in src/core/klein_curve.py

def as_dict(self) -> dict:
    """Serialize curve parameters."""
    return {
        "genus": self.genus,
        "K": self.K,
        "R": self.R,
        "area": self.area,
        "psl_order": self.psl_order,
        "sl_order": self.sl_order,
        "lambda_1": self.lambda_1,
    }