Yu. Makhlin ¹ and T. Misirpashaev ^1,2
1Landau Institute for Theoretical Physics, Kosygin St. 2, 117334 Moscow, Russia
²Isaac Newton Institute for Mathematical Sciences, University of Cambridge, Cambridge, CB3 OEH, U.K.

Topology of vortex-soliton intersection: Invariants and Torus homotopy

JETP Lett. 61, 49 (1995) [Pisma v ZhETF 61, 48 (1995)]

The topology relevant to the intersection of nonsingular 4-pi vortex lines with a planar transverse soliton in superfluid ³He-A, which was recently observed in Helsinki, is discussed. The essential part of the problem consists in finding a homotopy classification of mappings S² x S¹ -> S² and S¹ x S¹ x S¹ -> S². This classification is achieved, and an analytical expression for the topological invariant is found. This expression is analogous to that for the Hopf invariant H=∫v_s curl v_s dV/4k² of mappings S³ -> S².