Topological defect with nonzero Hopf invariant in Yang–Mills–Higgs model by Yan He, Hao Guo

“…We propose a topological defect or instanton solution with nonzero Hopf invariant to the 3+1D non-Abelian gauge theory coupled with scalar fields. …”
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PT-symmetric non-Hermitian Hopf metal by Seik Pak, Cheol Hun Yeom, Sonu Verma, Moon Jip Park

“…In this Letter, we describe the metallic counterpart of the Hopf insulator in non-Hermitian systems. While the Hopf invariant is not a stable topological index due to the additional non-Hermitian degree of freedom, we show that the PT symmetry stabilizes the Hopf invariant even in the presence of the non-Hermiticity. …”
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Lipschitz homotopies of mappings from 3-sphere to 2-sphere by Berdnikov, Aleksandr

“…The present paper resolves this problem for the first case where Hopf invariant plays a role: m = 3, n = 2, constructing a homotopy with Lipschitz constant O(L).…”
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Umbilic Lines in Orientational Order by Thomas Machon, Gareth P. Alexander

“…Their global structure leads to the definition of a self-linking number and helicity integral which relates the linking of umbilic lines to the Hopf invariant of the texture.…”
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Euler potentials for the MHD Kamchatnov-Hopf soliton solution by V. S. Semenov, D. B. Korovinski, H. K. Biernat

“…The MHD soliton solution (Kamchatnov, 1982) is based on the Hopf invariant of the mapping of a 3-D sphere into a 2-D sphere; it can have arbitrary helicity depending on control parameters. …”
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Fractional Hopfions in the Faddeev-Skyrme model with a symmetry breaking potential by A. Samoilenka, Ya. Shnir

“…In order to classify these configurations we define a counterpart of the usual position curve, which represents a collection of loops C 1 , C − 1 $$ {\mathcal{C}}_1,{\mathcal{C}}_{-1} $$ corresponding to the preimages of the points ϕ → = ± 1 , 0 , 0 $$ \overrightarrow{\phi}=\left(\pm 1,0,0\right) $$ , respectively. Then the Hopf invariant can be defined as Q = link C 1 C − 1 $$ Q=\mathrm{link}\left({\mathcal{C}}_1,{\mathcal{C}}_{-1}\right) $$ . …”
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Area-contracting maps between rectangles by Guth, Lawrence

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Fibrewise CoHopf spaces by Sunderland, A

“…</p> <p>The Thorn space is used to determine the cohomology ring of the total space of a fibrewise coHopf sphere bundle in terms of that of its base, and a generalised Hopf invariant is constructed which vanishes on fibrewise coHopf sphere bundles.…”

Hopf characterization of two-dimensional Floquet topological insulators by F. Nur Ünal, André Eckardt, Robert-Jan Slager

“…Previously found winding numbers are shown to correspond to Hopf invariants, which are associated with linking numbers describing the topology of knots in three dimensions. …”
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Some problems in algebraic topology by Gilbert, W

“…When B is of this form, we obtain conditions for the category and the weak categories of B to be less than or equal to one of the terms of Hopf invariants of $alpha$. We use these conditions to prove the examples. …”