Moduli-space structure of knots with intersections

Abstract

It is well known that knots are countable in ordinary knot theory. Recently, knots {\it with intersections} have raised a certain interest, and have been found to have physical applications. We point out that such knots --equivalence classes of loops in $R^3$ under diffeomorphisms-- are not countable; rather, they exhibit a moduli-space structure. We characterize these spaces of moduli and study their dimension. We derive a lower bound (which we conjecture being actually attained) on the dimension of the (non-degenerate components) of the moduli spaces, as a function of the valence of the intersection.

15 pages, latex-revtex, no figures

Cited by in corpus (28)

• Loop Quantum Gravity
• ``Sum over Surfaces'' form of Loop Quantum Gravity
• The Phoenix Project: Master Constraint Programme for Loop Quantum Gravity
• Loop Quantum Gravity: An Inside View
• Quantum gravity with a positive cosmological constant
• Quantum tetrahedra and simplicial spin networks
• Quantum Spin Dynamics VIII. The Master Constraint
• Spin Foams and Canonical Quantization
• Consistency Check on Volume and Triad Operator Quantisation in Loop Quantum Gravity I
• Consistency Check on Volume and Triad Operator Quantisation in Loop Quantum Gravity II
• Towards an Anomaly-Free Quantum Dynamics for a Weak Coupling Limit of Euclidean Gravity
• Separable Hilbert space in Loop Quantum Gravity
• Properties of the Volume Operator in Loop Quantum Gravity I: Results
• Critical Overview of Loops and Foams
• Matrix Elements of Thiemann's Hamiltonian Constraint in Loop Quantum Gravity
• Bohr-Sommerfeld Quantization of Space
• The century of the incomplete revolution: searching for general relativistic quantum field theory
• Loop Quantum Gravity
• Loop Quantum Gravity a la Aharonov-Bohm
• Combinatorial space from loop quantum gravity
• Loop Quantum Gravity Phenomenology: Linking Loops to Observational Physics
• Gravitational Statistical Mechanics: A model
• The constraint algebra in Smolins' $G\to 0$ limit of 4d Euclidean Gravity
• Modelling Space with an Atom of Quantum Geometry
• Angle and Volume Studies in Quantized Space
• New Operators for Spin Net Gravity: Definitions and Consequences
• Knotted 4-regular graphs: polynomial invariants and the Pachner moves
• Moduli Structures, Separability of the Kinematic Hilbert Space and Frames in Loop Quantum Gravity