Laurent polynomial

One approach to the problem of establishing a knot s chirality or achirality is through the use of knot invariants. The first invariant capable of distinguishing between enantiomorphs, a one-variable Laurent polynomial (a polynomial that has both positive and negative powers), was discovered only as recently as 1985, by Vaughan Jones. 00 More powerful two-variable polynomials have subsequently been developed by others.101 102... 

Laurent Polynomial where the exponent in a power of z can be negative as well as positive or zero. 

The Laurent polynomial corresponding to a given sequence is called its Generating Function, its z-transform, or its symbol. 

In fact we can use generating functions which are not polynomial or Laurent polynomial, provided that they have a formal Taylor expansion about z = 0. We can then think of them as shorter notations for that expansion, and this can be extremely useful for denoting concisely sequences with an unbounded number of non-zero entries. 

This viewpoint does help to make the Laurent Polynomial idea much less outlandish. You can do long multiplication with decimal fractions just as well as with integers, and the actual manipulation of the coefficients is more or less independent of where you put the decimal point. 

Generating Function The fourth representation is made by treating the values in the mask as the coefficients of a Laurent Polynomial. This is the -transform, which maps a sequence of values into a function. The Laurent polynomial is also called the symbol of the scheme. 

Note that because the spacing of the old vertices is twice as sparse as that of the new ones, the old Laurent polynomial is in z2. 

If the scheme is not primal, the 2-transform then becomes a generalisation of a Laurent polynomial a Laurent polynomial with a shift of a fractional power of 2. This shift is not important for the algebra the entire equation is just shifted a little to one side, but the shift in the algebra is actually achieving the maintenance of symmetry. 

D. Levin Using Laurent polynomial representation for the analysis of non-uniform binary subdivision schemes. Adv. Comput. Math 11, pp41-54, 1999... 

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 ... 

Alternative higher-order methods are available in the literature, and most of them share the salient characteristics of the spatial discretization schemes used in finite-volume codes (such as, for example, QUICK and MUSCL). Some of them are based on polynomial reconstructions with some minor modifications in order to ensure positivity of the NDF, as explained in Laurent (2006), whereas others are based on the maximization of Shannon entropy (see, for example, Massot et at. (2010)). 

The Laurent expansion is often not obvious in many practical applications, so additional procedures are needed. Often, the complex function appears as a ratio of polynomials... 

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