Kauffman invariant

L. Kauffman, Invariants of graphs in three-space. Trans. Am. Math. Soc. 1989, 311, 697-710. 

The attempt to apply the Kauffman invariants of regular isotopy to investigating statistical properties of random walks with topological constraints in a thin slit was made in a recent work [47],... 

For the identification of the topological state of the knot we use the Kauffman algebraic invariant K(A), which is the Laurent polynomial in A variable. We have shown the Kauffman invariant to be equal to the partition function of some special disordered Potts model [3, 4]. The number of equivalent states, and the nearest neighbor interaction constant, J /, are defined as follows ... 

We have constructed the upper estimation for the probability Pq to find the trivial knot. Since Kauffman invariant is not complete some nontrivial knots with K A) = 1 may exist (K A) = 1 for the trivial knot). Then Pq is less than the probability to find a knot with K A) = 1 for all A-values which, in turn, is less than the probability to find a knot with invariant K A) being equal to 1 for some fixed value A = A ... 

The Kauffman invariant in terms of Potts partition function reads... 

A.Grosberg, S.Nechaev Averaged Kauffman invariant and quasi-knot concept for linear polymers , Europhys. Lett. 20 613 (1992). 

We would like to describe in this section the very beautiful idea proposed by L.H. Kauffman for the analytical construction of powerful polynomial invariants of knots and links. 

Now, when we have formulated the Reidemeister theorem, we can describe the construction of a powerful polynomial bracket invariant proposed by L.H. Kauffman [40,41]. This invariant can be introduced as a certain partition function, which is the sum over the set of some formal degrees of freedom. 

Grishanov S, Meshkov V and Omelchenko A (2007), Kauffman-type polynomial invariants of doubly-periodic structures , J Knot Theor Ramif, 16(6), 779-788. 

Very recently the topological quantum computation has been oriented to the computation of invariant polynomials for knots and links, such as Jones, HOMELY, Kauffman polynomials [4]. The physical referent that is keeping on mind in this application of TQC is certain kinds of anyon structures which are realized as... 

From the oilier side, it is known that link invariants also appear in quantum gravity as the amplitudes for certain gravitational process. Specifically the Jones, HOMFLY and Kauffman polynomials can be obtained as the vacuum states of the quantum gravitational field [6]. In other words the quantum gravity is able to compute Jones polynomials. 

In [8] Witten was able to obtain the Jones polynomial within the context of the topological quantum field theory of the Chern-Simons kind. Nearly immediately other polynomials were obtained such as the HOMFLY and the Kauffman polynomials [9]. The main idea consists in that the observables (vacuum expectation values) of the TQFT s are by itself topological invariants for knots, links, tangles and three-manifolds. 

As is well known, the bracket polynomial (1) is invariant under Reidemeister moves II and III but it is not invariant under Reidemeister move I. But the normalized Kauffman bracket (2) is invariant under all the Reidemeister moves and when certain change of variable is made, the Jones polynomial is obtained, namely... 

Kauffman, L.H. Temperley-Lieb Recoupling Theory and Invariants ofThree-Manifolds. In Annals Studies, vol. 114, Princeton University Press, Princeton, NJ (1994)... 

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