### 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.

### Similar content being viewed by others

## REFERENCES

R. Ahmad, S. Paul, and S. Basu, “Characterization of entanglements in glassy polymeric ensembles using the Gaussian linking number,” Phys. Rev. E

**101**(2), 022503 (2020).O. Anosova and V. Kurlin, “Introduction to periodic geometry and topology” (2021). arXiv:2103.02749.

O. Anosova and V. Kurlin, “An isometry classification of periodic point sets,” in

*Proceedings of Discrete Geometry and Mathematical Morphology*(2021).Z. Arai, “A rigorous numerical algorithm for computing the linking number of links,” Nonlinear Theory Appl.

**4**(1), 104–110 (2013).T Banchoff, “Self-linking numbers of space polygons,” Indiana U. Math. J.

**25**, 1171–1188 (1976).E. Bertolazzi, R. Ghiloni, and R. Specogna, “Efficient computation of linking number with certification” (2019). arXiv:1912.13121.

M. Bright and V. Kurlin, “Encoding and topological computation on textile structures,” Comput. Graphics

**90**, 51–61 (2020).M. Bright, O. Anosova, and V. Kurlin, “A proof of the invariant-based formula for the linking number and its asymptotic behavior,” in

*Proceedings of Numerical Geometry, Grid Generation and Scientific Computing*(2020). https://arxiv.org/abs/2011.04631M. Bright, A. I. Cooper, and V. Kurlin, “A complete and continuous map of the lattice isometry space for all 3-dimensional lattices” (2021). arXiv:2109.11538.

M. Bright, A. I. Cooper, and V. Kurlin, “Easily computable continuous metrics on the space of isometry classes of 2-dimensional lattices” (2021). arXiv:2109.10885.

P. Cui, D. McMahon, P. Spackman, B. Alston, M. Little, G. Day, and A. Cooper, “Mining predicted crystal structure landscapes with high throughput crystallization: Old molecules, new insights,” Chem. Sci.

**10**, 9988–9997 (2019).D. DeTurck, H. Gluck, R. Komendarczyk, P. Melvin, C. Shonkwiler, and D. Vela-Vick, “Pontryagin invariants and integral formulas for Milnor’s triple linking number” (2011). arXiv:1101.3374.

C. F. Gauss, “Integral formula for linking number,”

*Zur Mathematischen Theorie der Electrodynamische Wirkungen*, Collected Works (1833), Vol. 5, p. 605.K. Klenin and J. Langowski, “Computation of writhe in modeling of supercoiled DNA,” Biopolym.: Orig. Res. Biomol.

**54**(5), 307–317 (2000).M. Kontsevich, “Vassiliev’s knot invariants,” Adv. Sov. Math.

**16**, 137–150 (1993).V. Kurlin, “Compressed Drinfeld associators,” J. Algebra

**292**, 184–242 (2005).V. Kurlin, “The Baker–Campbell–Hausdorff formula in the free metabelian Lie algebra,” J. Lie Theory

**17**(3), 525–538 (2007).J. C. Maxwell,

*A Treatise on Electricity and Magnetism I*(Dover, New York, 1954).M. Mosca and V. Kurlin, “Voronoi-based similarity distances between arbitrary crystal lattices,” Cryst. Res. Technol.

**55**(5), 1900197 (2020).E. Panagiotou, “The linking number in systems with periodic boundary conditions,” J. Comput. Phys.

**300**, 533–573 (2015).E. Panagiotou and L. H. Kauffman, “Knot polynomials of open and closed curves,” Proc. R. Soc. A

**476**, 20200124 (2020). arXiv:2001.01303.R. L. Ricca and B. Nipoti, “Gauss’ linking number revisited,” J. Knot Theory Its Ramifications

**20**(10), 1325–1343 (2011).J. Ropers, M. M. Mosca, O. Anosova, V. Kurlin, and A. I. Cooper, “Fast predictions of lattice energies by continuous isometry invariants of crystal structures,” in

*Proceedings of DACOMSIN*(2021). https://arxiv.org/abs/2108.07233.T. Vogel, “On the asymptotic linking number,” Proc. Am. Math. Soc.

**131**, 2289–2297 (2003).A. V. Vologodskii, V. V. Anshelevich, A. V. Lukashin, and M. D. Frank-Kamenetskii, “Statistical mechanics of supercoils and the torsional stiffness of the DNA double helix,” Nature

**280**(5720), 294–298 (1974).D. Widdowson and V. Kurlin, “Pointwise distance distributions of periodic sets” (2021). arXiv:2108.04798.

D. Widdowson, M. Mosca, A. Pulido, V. Kurlin, and A. Cooper, “Average minimum distances of periodic point sets,” MATCH Commun. Math. Comput. Chem.

**87**(3), 529–559 (2022). https://arxiv.org/abs/2009.02488.

## Funding

This work was supported by the UK Engineering and Physical Sciences Research Council under the grant £3.5M “Application-driven Topological Data Analysis” (EP/R018472/1).

## Author information

### Authors and Affiliations

### Corresponding authors

## Ethics declarations

The authors declare that they have no conflicts of interest.

## Rights and permissions

## About this article

### Cite this article

Bright, M., Anosova, O. & Kurlin, V. A Formula for the Linking Number in Terms of Isometry Invariants of Straight Line Segments.
*Comput. Math. and Math. Phys.* **62**, 1217–1233 (2022). https://doi.org/10.1134/S0965542522080024

Received:

Revised:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1134/S0965542522080024