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=∫vs curl vs dV/4k2 of mappings S3 -> S2.