On the topological characterization of near force-free magnetic fields, and the work of late-onset visually-impaired topologists

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February 2016, 9(1): 215-234. doi: 10.3934/dcdss.2016.9.215

On the topological characterization of near force-free magnetic fields, and the work of late-onset visually-impaired topologists

P. Robert Kotiuga ^1,

1.	Boston University, ECE Dept, 8 Saint Mary's Street, Boston, MA 02215, United States

Received September 2014 Revised February 2015 Published December 2015

Abstract
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The Giroux correspondence and the notion of a near force-free magnetic field are used to topologically characterize near force-free magnetic fields which describe a variety of physical processes, including plasma equilibrium. As a byproduct, the topological characterization of force-free magnetic fields associated with current-carrying links, as conjectured by Crager and Kotiuga, is shown to be necessary and conditions for sufficiency are given. Along the way a paradox is exposed: The seemingly unintuitive mathematical tools, often associated to higher dimensional topology, have their origins in three dimensional contexts but in the hands of late-onset visually impaired topologists. This paradox was previously exposed in the context of algorithms for the visualization of three-dimensional magnetic fields. For this reason, the paper concludes by developing connections between mathematics and cognitive science in this specific context.

Keywords: topology, magnetic fields, Electromagnetism, neuroplasticity..

Mathematics Subject Classification: 37D55, 53D35, 58Z99, 78A2.

Citation: P. Robert Kotiuga. On the topological characterization of near force-free magnetic fields, and the work of late-onset visually-impaired topologists. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 215-234. doi: 10.3934/dcdss.2016.9.215

References:

[1]	F. J. Almgren and W. P. Thurston, Examples of unknotted curves which bound only surfaces of high genus within their convex hulls, Annals of Math. $2^{nd}$ Ser., 105 (1977), 527-538. doi: 10.2307/1970922.
[2]	M. F. Atiyah, Mathematics in the $20^{th}$ century, Bull. LMS, 34 (2002), 1-15. doi: 10.1112/S0024609301008566.
[3]	G. Burde and H. Zeischang, Knots, $2^{nd}$ ed., De Gruyter Studies in Math No.5, Walter de Gruyter, 2003.
[4]	J. C. Crager and P. R. Kotiuga, Cuts for the magnetic scalar potential in knotted geometries and force-free magnetic fields, IEEE Trans. Mag., 38 (2002), 1309-1312.
[5]	D. A. Ellwood, P. S. Ozsvath, A. I. Stipicz and Z. Szabo. Eds., Floer Homology, Gauge Theory, and Low-Dimensional Topology, Clay Math. Proc. Vol.5, AMS, AMS Providence, RI, 2006.
[6]	J. B. Etnyre, Lectures on open book decompositions and contact structures, in Floer Homology, Gauge Theory, and Low-Dimensional Topology, Clay Math. Proc., 5, Amer. Math. Soc., Providence, RI, 2006, 103-141.
[7]	G. K. Francis and B. Morin, Arnold Shapiro's eversion of the sphere, Math. Intelligencer, 2 (1980), 200-203. doi: 10.1007/BF03028603.
[8]	P. Frankl and L. Pontryagin, Ein Knotensatz mit Anwendung auf die Dimensionstheorie, Math. Annalen, 102 (1930), 785-789. doi: 10.1007/BF01782377.
[9]	E. Giroux, Géométrie de contact: De la dimension trois vers les dimensions supérieurs, Proc. Int'l Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, 405-414.
[10]	H. Goda, Circle valued Morse theory for knots and links, in Floer Homology, Gauge Theory, and Low-Dimensional Topology, Clay Math. Proc., 5, Amer. Math. Soc., Providence, RI, 2006, 71-99.
[11]	P. W. Gross and P. R. Kotiuga, Electromagnetic Theory and Computation: A Topological Approach, MSRI Monograph No. 48, Camb. U. Press, 2004. doi: 10.1017/CBO9780511756337.
[12]	R. Hiptmair, P. R. Kotiuga and S. Tordeux, Self-adjoint curl operators, Annali di Matematica Pura ed Applicata, 191 (2012), 431-457. doi: 10.1007/s10231-011-0189-y.
[13]	P. R. Kotiuga, On making cuts for magnetic scalar potentials in multiply connected regions, Jour. Appl. Phys., 61 (1987), 3916-3918. doi: 10.1063/1.338583.
[14]	P. R. Kotiuga, An algorithm to make cuts for magnetic scalar potentials in tetrahedral meshes based on the finite element method, IEEE Trans. Mag., 25 (1989), 4129-4131. doi: 10.1109/INTMAG.1989.690333.
[15]	P. R. Kotiuga, Helicity functionals and metric invariance in three dimensions, IEEE Trans. Mag., 25 (1989), 2813-2815. doi: 10.1109/20.34293.
[16]	P. R. Kotiuga, Topology-based inequalities and inverse problems for near force-free magnetic fields, IEEE Trans. Mag., 40 (2004), 1108-1111. doi: 10.1109/TMAG.2004.824590.
[17]	J. C. Lagarias, J. Hass and W. P. Thurston, Area inequalities for embedded disks spanning unknotted curves, Journ. Diff. Geom., 68 (2004), 1-29.
[18]	S. Smale, A classification of immersions of the two-sphere, Trans. Amer. Math. Soc., 90 (1958), 281-290.
[19]	W. P. Thurston and H. E. Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc., 52 (1975), 345-347. doi: 10.1090/S0002-9939-1975-0375366-7.
[20]	H. Whitney, Moscow 1935: Topology moving toward America, reprinted in Hassler Whitney Collected Papers, Vol I (eds. J. Eells and D. Toledo), Birkhauser, Boston, 1992, 1-21.

show all references

References:

[1]	F. J. Almgren and W. P. Thurston, Examples of unknotted curves which bound only surfaces of high genus within their convex hulls, Annals of Math. $2^{nd}$ Ser., 105 (1977), 527-538. doi: 10.2307/1970922.
[2]	M. F. Atiyah, Mathematics in the $20^{th}$ century, Bull. LMS, 34 (2002), 1-15. doi: 10.1112/S0024609301008566.
[3]	G. Burde and H. Zeischang, Knots, $2^{nd}$ ed., De Gruyter Studies in Math No.5, Walter de Gruyter, 2003.
[4]	J. C. Crager and P. R. Kotiuga, Cuts for the magnetic scalar potential in knotted geometries and force-free magnetic fields, IEEE Trans. Mag., 38 (2002), 1309-1312.
[5]	D. A. Ellwood, P. S. Ozsvath, A. I. Stipicz and Z. Szabo. Eds., Floer Homology, Gauge Theory, and Low-Dimensional Topology, Clay Math. Proc. Vol.5, AMS, AMS Providence, RI, 2006.
[6]	J. B. Etnyre, Lectures on open book decompositions and contact structures, in Floer Homology, Gauge Theory, and Low-Dimensional Topology, Clay Math. Proc., 5, Amer. Math. Soc., Providence, RI, 2006, 103-141.
[7]	G. K. Francis and B. Morin, Arnold Shapiro's eversion of the sphere, Math. Intelligencer, 2 (1980), 200-203. doi: 10.1007/BF03028603.
[8]	P. Frankl and L. Pontryagin, Ein Knotensatz mit Anwendung auf die Dimensionstheorie, Math. Annalen, 102 (1930), 785-789. doi: 10.1007/BF01782377.
[9]	E. Giroux, Géométrie de contact: De la dimension trois vers les dimensions supérieurs, Proc. Int'l Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, 405-414.
[10]	H. Goda, Circle valued Morse theory for knots and links, in Floer Homology, Gauge Theory, and Low-Dimensional Topology, Clay Math. Proc., 5, Amer. Math. Soc., Providence, RI, 2006, 71-99.
[11]	P. W. Gross and P. R. Kotiuga, Electromagnetic Theory and Computation: A Topological Approach, MSRI Monograph No. 48, Camb. U. Press, 2004. doi: 10.1017/CBO9780511756337.
[12]	R. Hiptmair, P. R. Kotiuga and S. Tordeux, Self-adjoint curl operators, Annali di Matematica Pura ed Applicata, 191 (2012), 431-457. doi: 10.1007/s10231-011-0189-y.
[13]	P. R. Kotiuga, On making cuts for magnetic scalar potentials in multiply connected regions, Jour. Appl. Phys., 61 (1987), 3916-3918. doi: 10.1063/1.338583.
[14]	P. R. Kotiuga, An algorithm to make cuts for magnetic scalar potentials in tetrahedral meshes based on the finite element method, IEEE Trans. Mag., 25 (1989), 4129-4131. doi: 10.1109/INTMAG.1989.690333.
[15]	P. R. Kotiuga, Helicity functionals and metric invariance in three dimensions, IEEE Trans. Mag., 25 (1989), 2813-2815. doi: 10.1109/20.34293.
[16]	P. R. Kotiuga, Topology-based inequalities and inverse problems for near force-free magnetic fields, IEEE Trans. Mag., 40 (2004), 1108-1111. doi: 10.1109/TMAG.2004.824590.
[17]	J. C. Lagarias, J. Hass and W. P. Thurston, Area inequalities for embedded disks spanning unknotted curves, Journ. Diff. Geom., 68 (2004), 1-29.
[18]	S. Smale, A classification of immersions of the two-sphere, Trans. Amer. Math. Soc., 90 (1958), 281-290.
[19]	W. P. Thurston and H. E. Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc., 52 (1975), 345-347. doi: 10.1090/S0002-9939-1975-0375366-7.
[20]	H. Whitney, Moscow 1935: Topology moving toward America, reprinted in Hassler Whitney Collected Papers, Vol I (eds. J. Eells and D. Toledo), Birkhauser, Boston, 1992, 1-21.

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