3D magnetic textures with mixed topology: Unlocking the tunable Hopf index

Knots and links play a crucial role in understanding topology and discreteness in nature. In magnetic systems, twisted, knotted and braided vortex tubes manifest as Skyrmions, Hopfions, or screw dislocations. These complex textures are characterized by topologically non-trivial quantities, such as a Skyrmion number, a generalized Hopf index H <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>H</mml:mi> </mml:math> , a Burgers vector (quantified by an integer \nu <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>ν</mml:mi> </mml:math> ), and linking numbers. In this work, we introduce a discrete geometric formulation of H <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>H</mml:mi> </mml:math> for periodic magnetic textures, expressed in terms of the self-linking and inter-linking of flux tubes of the emergent magnetic field. For configurations with cuniform boundary conditions, this formulation reproduces the standard integer-valued Hopf invariant classified by \pi_3(S^2) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> . For textures embedded in non-uniform backgrounds, where compactification to S^3 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:math> is not possible, the same flux-linking invariant provides a boundary-condition-dependent extension that can assume non-integer values while remaining constant under boundary-preserving deformations. We interpret such configurations as states of “mixed topology”, continuously connected to integer Hopf sectors when the background becomes uniform. Our results establish a unified geometric framework for characterizing three-dimensional magnetic textures beyond the compactified homotopy setting.