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

1. 

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

Received  September 2014 Revised  February 2015 Published  December 2015

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.
Citation: P. Robert Kotiuga. On the topological characterization of near force-free magnetic fields, and the work of late-onset visually-impaired topologists. Discrete & 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.  doi: 10.2307/1970922.  Google Scholar

[2]

M. F. Atiyah, Mathematics in the $20^{th}$ century,, Bull. LMS, 34 (2002), 1.  doi: 10.1112/S0024609301008566.  Google Scholar

[3]

G. Burde and H. Zeischang, Knots, $2^{nd}$ ed.,, De Gruyter Studies in Math No.5, (2003).   Google Scholar

[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.   Google Scholar

[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, (2006).   Google Scholar

[6]

J. B. Etnyre, Lectures on open book decompositions and contact structures,, in Floer Homology, (2006), 103.   Google Scholar

[7]

G. K. Francis and B. Morin, Arnold Shapiro's eversion of the sphere,, Math. Intelligencer, 2 (1980), 200.  doi: 10.1007/BF03028603.  Google Scholar

[8]

P. Frankl and L. Pontryagin, Ein Knotensatz mit Anwendung auf die Dimensionstheorie,, Math. Annalen, 102 (1930), 785.  doi: 10.1007/BF01782377.  Google Scholar

[9]

E. Giroux, Géométrie de contact: De la dimension trois vers les dimensions supérieurs,, Proc. Int'l Congress of Mathematicians, (2002), 405.   Google Scholar

[10]

H. Goda, Circle valued Morse theory for knots and links,, in Floer Homology, (2006), 71.   Google Scholar

[11]

P. W. Gross and P. R. Kotiuga, Electromagnetic Theory and Computation: A Topological Approach,, MSRI Monograph No. 48, (2004).  doi: 10.1017/CBO9780511756337.  Google Scholar

[12]

R. Hiptmair, P. R. Kotiuga and S. Tordeux, Self-adjoint curl operators,, Annali di Matematica Pura ed Applicata, 191 (2012), 431.  doi: 10.1007/s10231-011-0189-y.  Google Scholar

[13]

P. R. Kotiuga, On making cuts for magnetic scalar potentials in multiply connected regions,, Jour. Appl. Phys., 61 (1987), 3916.  doi: 10.1063/1.338583.  Google Scholar

[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.  doi: 10.1109/INTMAG.1989.690333.  Google Scholar

[15]

P. R. Kotiuga, Helicity functionals and metric invariance in three dimensions,, IEEE Trans. Mag., 25 (1989), 2813.  doi: 10.1109/20.34293.  Google Scholar

[16]

P. R. Kotiuga, Topology-based inequalities and inverse problems for near force-free magnetic fields,, IEEE Trans. Mag., 40 (2004), 1108.  doi: 10.1109/TMAG.2004.824590.  Google Scholar

[17]

J. C. Lagarias, J. Hass and W. P. Thurston, Area inequalities for embedded disks spanning unknotted curves,, Journ. Diff. Geom., 68 (2004), 1.   Google Scholar

[18]

S. Smale, A classification of immersions of the two-sphere,, Trans. Amer. Math. Soc., 90 (1958), 281.   Google Scholar

[19]

W. P. Thurston and H. E. Winkelnkemper, On the existence of contact forms,, Proc. Amer. Math. Soc., 52 (1975), 345.  doi: 10.1090/S0002-9939-1975-0375366-7.  Google Scholar

[20]

H. Whitney, Moscow 1935: Topology moving toward America,, reprinted in Hassler Whitney Collected Papers, (1992), 1.   Google Scholar

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.  doi: 10.2307/1970922.  Google Scholar

[2]

M. F. Atiyah, Mathematics in the $20^{th}$ century,, Bull. LMS, 34 (2002), 1.  doi: 10.1112/S0024609301008566.  Google Scholar

[3]

G. Burde and H. Zeischang, Knots, $2^{nd}$ ed.,, De Gruyter Studies in Math No.5, (2003).   Google Scholar

[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.   Google Scholar

[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, (2006).   Google Scholar

[6]

J. B. Etnyre, Lectures on open book decompositions and contact structures,, in Floer Homology, (2006), 103.   Google Scholar

[7]

G. K. Francis and B. Morin, Arnold Shapiro's eversion of the sphere,, Math. Intelligencer, 2 (1980), 200.  doi: 10.1007/BF03028603.  Google Scholar

[8]

P. Frankl and L. Pontryagin, Ein Knotensatz mit Anwendung auf die Dimensionstheorie,, Math. Annalen, 102 (1930), 785.  doi: 10.1007/BF01782377.  Google Scholar

[9]

E. Giroux, Géométrie de contact: De la dimension trois vers les dimensions supérieurs,, Proc. Int'l Congress of Mathematicians, (2002), 405.   Google Scholar

[10]

H. Goda, Circle valued Morse theory for knots and links,, in Floer Homology, (2006), 71.   Google Scholar

[11]

P. W. Gross and P. R. Kotiuga, Electromagnetic Theory and Computation: A Topological Approach,, MSRI Monograph No. 48, (2004).  doi: 10.1017/CBO9780511756337.  Google Scholar

[12]

R. Hiptmair, P. R. Kotiuga and S. Tordeux, Self-adjoint curl operators,, Annali di Matematica Pura ed Applicata, 191 (2012), 431.  doi: 10.1007/s10231-011-0189-y.  Google Scholar

[13]

P. R. Kotiuga, On making cuts for magnetic scalar potentials in multiply connected regions,, Jour. Appl. Phys., 61 (1987), 3916.  doi: 10.1063/1.338583.  Google Scholar

[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.  doi: 10.1109/INTMAG.1989.690333.  Google Scholar

[15]

P. R. Kotiuga, Helicity functionals and metric invariance in three dimensions,, IEEE Trans. Mag., 25 (1989), 2813.  doi: 10.1109/20.34293.  Google Scholar

[16]

P. R. Kotiuga, Topology-based inequalities and inverse problems for near force-free magnetic fields,, IEEE Trans. Mag., 40 (2004), 1108.  doi: 10.1109/TMAG.2004.824590.  Google Scholar

[17]

J. C. Lagarias, J. Hass and W. P. Thurston, Area inequalities for embedded disks spanning unknotted curves,, Journ. Diff. Geom., 68 (2004), 1.   Google Scholar

[18]

S. Smale, A classification of immersions of the two-sphere,, Trans. Amer. Math. Soc., 90 (1958), 281.   Google Scholar

[19]

W. P. Thurston and H. E. Winkelnkemper, On the existence of contact forms,, Proc. Amer. Math. Soc., 52 (1975), 345.  doi: 10.1090/S0002-9939-1975-0375366-7.  Google Scholar

[20]

H. Whitney, Moscow 1935: Topology moving toward America,, reprinted in Hassler Whitney Collected Papers, (1992), 1.   Google Scholar

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