A Formula for the Linking Number in Terms of Isometry Invariants of Straight Line Segments View Full Text


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Article Info

DATE

2022-08

AUTHORS

M. Bright, O. Anosova, V. Kurlin

ABSTRACT

The linking number is usually defined as an isotopy invariant of two non-intersecting closed curves in 3-dimensional space. However, the original definition in 1833 by Gauss in the form of a double integral makes sense for any open disjoint curves considered up to rigid motion. Hence the linking number can be studied as an isometry invariant of rigid structures consisting of straight line segments. For the first time this paper gives a complete proof for an explicit analytic formula for the linking number of two line segments in terms of six isometry invariants, namely the distance and angle between the segments and four coordinates of their endpoints in a natural coordinate system associated with the segments. Motivated by interpenetration of crystalline networks, we discuss potential extensions to infinite periodic structures and review recent advances in isometry classifications of periodic point sets. More... »

PAGES

1217-1233

References to SciGraph publications

  • 2022-07-26. Fast Predictions of Lattice Energies by Continuous Isometry Invariants of Crystal Structures in DATA ANALYTICS AND MANAGEMENT IN DATA INTENSIVE DOMAINS
  • 1979-07. Statistical mechanics of supercoils and the torsional stiffness of the DNA double helix in NATURE
  • 2021-05-16. An Isometry Classification of Periodic Point Sets in DISCRETE GEOMETRY AND MATHEMATICAL MORPHOLOGY
  • 2021-05-07. A Proof of the Invariant-Based Formula for the Linking Number and Its Asymptotic Behaviour in NUMERICAL GEOMETRY, GRID GENERATION AND SCIENTIFIC COMPUTING
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