# American Institute of Mathematical Sciences

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
 [1] Carlos J. García-Cervera, Sookyung Joo. Reorientation of smectic a liquid crystals by magnetic fields. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1983-2000. doi: 10.3934/dcdsb.2015.20.1983 [2] Serge Nicaise, Simon Stingelin, Fredi Tröltzsch. Optimal control of magnetic fields in flow measurement. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 579-605. doi: 10.3934/dcdss.2015.8.579 [3] Diego Rapoport. Random representations of viscous fluids and the passive magnetic fields transported on them. Conference Publications, 2001, 2001 (Special) : 327-336. doi: 10.3934/proc.2001.2001.327 [4] Naoufel Ben Abdallah, Raymond El Hajj. Diffusion and guiding center approximation for particle transport in strong magnetic fields. Kinetic & Related Models, 2008, 1 (3) : 331-354. doi: 10.3934/krm.2008.1.331 [5] Mihai Bostan. On the Boltzmann equation for charged particle beams under the effect of strong magnetic fields. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 339-371. doi: 10.3934/dcdsb.2015.20.339 [6] Xueke Pu. Quasineutral limit of the Euler-Poisson system under strong magnetic fields. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2095-2111. doi: 10.3934/dcdss.2016086 [7] Gareth Ainsworth, Yernat M. Assylbekov. On the range of the attenuated magnetic ray transform for connections and Higgs fields. Inverse Problems & Imaging, 2015, 9 (2) : 317-335. doi: 10.3934/ipi.2015.9.317 [8] Chenxi Guo, Guillaume Bal. Reconstruction of complex-valued tensors in the Maxwell system from knowledge of internal magnetic fields. Inverse Problems & Imaging, 2014, 8 (4) : 1033-1051. doi: 10.3934/ipi.2014.8.1033 [9] Pedro Caro. On an inverse problem in electromagnetism with local data: stability and uniqueness. Inverse Problems & Imaging, 2011, 5 (2) : 297-322. doi: 10.3934/ipi.2011.5.297 [10] Frank Jochmann. A singular limit in a nonlinear problem arising in electromagnetism. Communications on Pure & Applied Analysis, 2011, 10 (2) : 541-559. doi: 10.3934/cpaa.2011.10.541 [11] Denis Serre. Non-linear electromagnetism and special relativity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 435-454. doi: 10.3934/dcds.2009.23.435 [12] Fernando Miranda, José-Francisco Rodrigues, Lisa Santos. On a p-curl system arising in electromagnetism. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 605-629. doi: 10.3934/dcdss.2012.5.605 [13] Bruce Hughes. Geometric topology of stratified spaces. Electronic Research Announcements, 1996, 2: 73-81. [14] Guizhen Cui, Wenjuan Peng, Lei Tan. On the topology of wandering Julia components. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 929-952. doi: 10.3934/dcds.2011.29.929 [15] Fengbo Hang, Fanghua Lin. Topology of Sobolev mappings IV. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1097-1124. doi: 10.3934/dcds.2005.13.1097 [16] D. Alderson, H. Chang, M. Roughan, S. Uhlig, W. Willinger. The many facets of internet topology and traffic. Networks & Heterogeneous Media, 2006, 1 (4) : 569-600. doi: 10.3934/nhm.2006.1.569 [17] Flavio Abdenur, Lorenzo J. Díaz. Pseudo-orbit shadowing in the $C^1$ topology. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 223-245. doi: 10.3934/dcds.2007.17.223 [18] M. Delgado-Téllez, Alberto Ibort. On the geometry and topology of singular optimal control problems and their solutions. Conference Publications, 2003, 2003 (Special) : 223-233. doi: 10.3934/proc.2003.2003.223 [19] Robert J. McCann. A glimpse into the differential topology and geometry of optimal transport. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1605-1621. doi: 10.3934/dcds.2014.34.1605 [20] Juan Manuel Pastor, Silvia Santamaría, Marcos Méndez, Javier Galeano. Effects of topology on robustness in ecological bipartite networks. Networks & Heterogeneous Media, 2012, 7 (3) : 429-440. doi: 10.3934/nhm.2012.7.429

2018 Impact Factor: 0.545