The Heegaard splitting of
is a particularly useful representation for quantum phases of spin j
-representation arising in the mapping
, which can be related to
torus knots in Hilbert space. We show that transitions to homotopically equivalent knots can be associated with gauge invariance, and that the same mechanism is at the heart of quantum entanglement. In other words, (minimal) interaction causes entanglement. Particle creation is related to cuts in the knot structure. We show that inner twists can be associated with operations with the quaternions
, which are crucial to understand the Hopf mapping
. We discuss the relationship between observables on the Bloch sphere
, and knots with inner twists in Hilbert space. As applications, we discuss selection rules in atomic physics, and the status of virtual particles arising in Feynman diagrams. Using a simple paper strip model revealing the knot structure of quantum phases in Hilbert space including inner twists, a
model of entanglement and gauge symmetries is proposed, which may also be valid for physics education.
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