##
**Spin model on knot projections.**
*(English)*
Zbl 0853.46079

Herman, Richard (ed.) et al., Operator algebras, mathematical physics, and low dimensional topology. Wellesley, MA: A. K. Peters. Res. Notes Math., Boston, Mass. 5, 195-200 (1993).

The paper concerns the discussion of two models from statistical mechanics, namely vertex model and spin one which, when applied to planar diagrams, lead to link invariants.

In both models one introduces the notion of a state \(\sigma\) of the diagram with its energy \(E(\sigma)\) and the partition function \[ Z= \sum_\sigma \exp\Biggl(- {E(\sigma)\over KT}\Biggr), \] with \(K\) and \(T\) as the Boltzmann constant and the temperature, respectively. The main problem is to determine when the invariants of diagrams are actually invariants of the underlying links. In the case of vortex models, a state of an oriented diagram is an assignment of a spin to each edge of the diagram (where the spins lie in a finite set \(\{1, 2,\dots, n\}\)). The energy of the state is the sum of local contributions coming from each vertex. These local energies depend on the sign of the crossing and the spins of the 4 adjoint edges.

On the other hand, the diagrams used in spin models are unoriented and a state is an assignment of spins to vertices of a planar graph associated to the diagram. The total energy of the state again is the sum of local energies. The local energies come from the edges and depend on the sign of the edge and the spins assigned to two vertices of the edge.

From the knot theoretical point of view, the spin and vertex models are not equivalent (although physicists have tended to treat both models as equivalent).

In the paper, the author presents a new spin model due to Jaeger and its particular example – the Potts model. Then it is noted that in the last case the energy is invariant under the permutation group of the spins. In this way, one gets a knot invariant which turns out to be determined by the homology of the 2-fold bracket cover of the knot, up to a phase factor. In drawing these conclusions the author has noticed an interesting equivalence between the Potts model for all values of \(n\) and the vertex model.

For the entire collection see [Zbl 0788.00059].

In both models one introduces the notion of a state \(\sigma\) of the diagram with its energy \(E(\sigma)\) and the partition function \[ Z= \sum_\sigma \exp\Biggl(- {E(\sigma)\over KT}\Biggr), \] with \(K\) and \(T\) as the Boltzmann constant and the temperature, respectively. The main problem is to determine when the invariants of diagrams are actually invariants of the underlying links. In the case of vortex models, a state of an oriented diagram is an assignment of a spin to each edge of the diagram (where the spins lie in a finite set \(\{1, 2,\dots, n\}\)). The energy of the state is the sum of local contributions coming from each vertex. These local energies depend on the sign of the crossing and the spins of the 4 adjoint edges.

On the other hand, the diagrams used in spin models are unoriented and a state is an assignment of spins to vertices of a planar graph associated to the diagram. The total energy of the state again is the sum of local energies. The local energies come from the edges and depend on the sign of the edge and the spins assigned to two vertices of the edge.

From the knot theoretical point of view, the spin and vertex models are not equivalent (although physicists have tended to treat both models as equivalent).

In the paper, the author presents a new spin model due to Jaeger and its particular example – the Potts model. Then it is noted that in the last case the energy is invariant under the permutation group of the spins. In this way, one gets a knot invariant which turns out to be determined by the homology of the 2-fold bracket cover of the knot, up to a phase factor. In drawing these conclusions the author has noticed an interesting equivalence between the Potts model for all values of \(n\) and the vertex model.

For the entire collection see [Zbl 0788.00059].

Reviewer: W.Kosiński (Warszawa)

### MSC:

46N55 | Applications of functional analysis in statistical physics |

05C99 | Graph theory |

57N20 | Topology of infinite-dimensional manifolds |

46L60 | Applications of selfadjoint operator algebras to physics |