New Invariant Quantity To Measure The Entanglement In The Braids


  • Faik Mayah Department of Software, College of Computer Science and IT, Wasit University, Wasit, Iraq
  • Nisreen Alokbi Department of Physics, College of Science, Wasit University, Wasit, Iraq
  • Ali Sabeeh Rasheed Ministry of Higher Education & Scientific Research, Baghdad, Iraq


Linkingnumber, Sato-Levineinvariant, Evans-Bergerformula, Magnetichelicity


In this work, we demonstrate  that the integral formula for a generalised Sato-Levine invariant is consistent in certain situations with Evans and Berger's formula for the fourth-order winding number. Also, we found that, in principle, one can derive analogous  high-order winding numbers by which one can calculate the entanglement of braids. The winding number for the Brunnian 4-braid is calculated algebraically using the cup product  on the cohomology of a finite regular CW-space which is the complement $\mathbb{R}^3\backslash \mathcal{B}_4$.



H. Baty, “On the Sweet-Parker model for incompressible visco-resistive magnetic reconnection in two dimensions associated to ideal magnetohydrodynamic instabilities”,

M. Berger, “Introduction to magnetic helicity”, IOP Publishing: Plasma Physics and Controlled Fusion 41(1999) 167. . DOI:

H. K. Moffatt, “The degree of knottedness of tangled vortex lines”, Cambridge University Press: Journal of Fluid Mechanics, 35 (1969) 117. DOI:

M. A. Berger, “Third-order braid invariants”, IOP Publishing, Journal of Physics A: Mathematical and General 24 (1991) 4027. DOI:

M. I. Monastyrsky & V. Retakh, “Topology of linked defects in condensed matter”, Commun. Math. Phys. 3 (1986) 445. DOI:

Ch. Mayer, Topological Link Invariants of Magnetic Fields, PhD Thesis, Bochum University, 2003.

M. I. Monastyrsky & P. V. Sasorov, “Topological invariants in magnetohydrodynamics”, Soviet Physics JETP 4 (1987) 66. f.

P. Akhmetiev, “On a new integral formula for an invariant of 3-component oriented links”, J. Geom. Phys. 53 (2005) 180. DOI:

N. Evans & M. A. Berger, Hierarchy of linking integrals, Kluwer Acad. Publ.1992. DOI:

P. Akhmetiev & O. Kunakovskaya, “Integral formula for a generalized Sato-Levine invariant in magnetic hydrodynamics”, Mathematical Notes 85 (2009) 503. DOI:

A. B. Familus & E. O. Omoleb & L A. Ukpebor, ‘A Higherorder Block Method for Numerical Approximation of Third-order Boundary Value Problems in ODEs”, J. Nig. Soc. Phys. Sci. 4,, (2022) 706. DOI:

R. Bott & W. Loring, Differential Forms in Algebraic Topology, SpringerVerlag New York Incp,, (1982). DOI:

W. S. Massey, Some higher order cohomology operations, Symposium internacional de topologia algebraica, UNESCO, Mexico., (1958).

R. A. Fenn, Techniques of Geometric Topology, Cambridge University 8

Press: London Maths. Soc. Lecture Note Series, 57, (1983).

H. K. Moffatt, ‘Magnetic Field Generation in Electrically Conducting Fluids, Cambridge University Press, ISBN 0521216400, (1983).

N. Sato, “Cobordisms of semiboundary links”, Topology Appl, 18 (1984) 225. DOI:

D. Ruberman, “Concordance of links in S4”, Contemp. Math. 35 (1984) 481. DOI:

T. Cochran, “On an invariant of link cobordism in dimension four”, Topology and its Applications, 18 (1984) 97., 18. DOI:

T. Cochran, “Geometric invariants of link cobordism”, Comment. Math. Helv. pp. 291-311, 60, DOI:

P. Akhmetiev & A. Ruzmaikin, “A fourth-order topological invariant of magnetic or vortex lines”, J Geom Phys 15 (1995) 5. DOI:

G. Ellis & F. Hegarty, “Computational homotopy of finite regular CWspaces”, Journal of Homotopy and Related Structures 9 (2014) 25. DOI:

P. Brendel, P. D?otko, G. Ellis, M. Juda & M. Mrozek, “Computing fundamental groups from point clouds”, Applicable Algebra in Engineering, Communication and Computing 26 (2015) 27. DOI:

Nisreen Alokbi & G. Ellis, “Distributed computation of low-dimensional cup products”, International Press of Boston: Homology, Homotopy and Applications 20 (2018) 41, DOI:

GAP Version 4.5.6, Manual for GAP - Groups, Algorithms, and Programming, The GAP Group 2013.

Nisreen Alokbi, FpGd – Finitely Presented Groupoid (GAP package), 2019.

M. Morishita, Knots and primes, Springer, London, 2012

G. Ellis, HAP - Homological Algebra Programming, Version 1.10.13. (2013).

Jerrold Franklin, Understanding Vector Calculus: Practical Development and Solved Problems, Dover Publications, ISBN. 9780486835907, 2020.



How to Cite

Mayah, F., Alokbi, N., & Rasheed, A. S. (2022). New Invariant Quantity To Measure The Entanglement In The Braids. Journal of the Nigerian Society of Physical Sciences, 4(4), 1051.



Original Research