The jones polynomial of rational links request pdf. For a link l, the jones polynomial of lis a laurent polynomial in t12. Download scientific diagram the jones polynomial of the negative hopf link from. By examining some particular examples we show how numerical.
We construct arbitrarily many 2bridge knots or links with the same q polynomial but. The jones polynomial university of california, berkeley. Jones polynomial invariants for knots and satellites. Let us consider the upper crossing point, see picture 1.
This polynomial is a remarkably good method of distinguishing knots. The jones polynomial now recall that the jones polynomial s is defined by the identities. Before proving this result, some commentary on terminology. The jones polynomial of the negative hopf link download. Jones polynomial in a kth primitive root of unity is unchanged under a kmove, if k is odd, and changes. For links of more than one component, however, the jones polynomial may change depending on the choice of orientation for each component. The hopf link on the left and the righthanded trefoil. Jonesconway polynomial encyclopedia of mathematics. Figure 4 illustrates the calculation of the bracket for the hopf link and for the. In particular, we can readily deduce the result due to yamada17 on the jones polynomial of 2parallels.
We start with the skein relation for one crossing of the trefoil, which gives. A primer of hopf algebras 3 basis, and the multiplication in gis extended to kgby linearity. Then, for the sake of illustration, explicit calculations of the jones polynomial for the lefthanded and righthanded trefoil knot and for the whitehead link via the. Here is the kauffman bracket calculation for the hopf link. The jones polynomial and its limitations masters thesis arxiv. Because the connected sum of links may depend on the choice of connected components, one can use the connected sum formula to find different links with the same jonesconway polynomial, for example the connected sum of three hopf links can give two different results, both with the polynomial equal to. Then our initial hopf link gives l so we will write l for that link. To see this, add a crossing connecting the two components.
On the khovanov homology of 2 and 3strand braid links. Having done this, we shall present the results on satellite knots in a form which is strikingly reminiscent of the classical fox formula for alexander polynomials, and which gives a key role to the hopf link and its invariants. Jones lq computes the jones polynomial of a knot or link l as a function of the variable q. The jones polynomial can also be derived from thethe jones polynomial can also be derived from the bracket polynomial of kauffman 1987 the bracket polynomial is identical to a qstate nonintersecting string model of perk and wu 1986. We extend this result to links, nding that the jones polynomial doesnt detect hopf link factors. By using satellites of the hopf link it is possible to produce an infinite family of links with the same jones polynomial of the trivial link. Cs e6204 lectures 9b and 10 alexanderconway and jones. The rst step is to assign a bracket to every knot or link diagram k. Introduction this paper is an introduction to the subject of virtual knot theory, a generalization of classical knot theory that i discovered in 1996 2. In the mathematical field of knot theory, the jones polynomial is a knot polynomial discovered by vaughan jones in 1984.
We will go through the construction of this polynomial that is due to louis kau man. From the skien relations, and previous theorem, we can calculate jones polynomial for any link with enough patient the idea is any link can be unknotted by. Definition the kauffman bracket polynomial is a function from unoriented link diagrams the oriented plane to laurent polynomials with integer coefficients tn an indeterminate a. Up to change of vari able and multiplication by some power of a the bracket polynomial is the jones polynomial. Kauffman introduced a statesum model construction of the. The colored jones polynomial of a knot or link is a generalization of the jones polynomial. Now consider the link called h in the following picture sometimes called the hopflink. Infinite families of links with trivial jones polynomial. Vaughan jones2 february 12, 2014 2 supportedbynsfundergrantno. For the lefthanded trefoil, the relation yields x4vunknot x 4vltrefoil x 2 x2v hopf.
It has also long been used outside mathematics, for instance as the crest of buzanha, a japanese buddhist sect founded in the 16th. The jones polynomial as a new invariant of topological fluid dynamics. We compute the jones polynomial of the hopf link oriented as in picture 1. A fundamental open problem in knot theory speci cally the theory of jones polynomial is as to. Department of mathematics, university of california at berkeley, berkeley ca 94720, u. In this paper i construct a state model for the original jones polynomial 5. The hopf link is named after topologist heinz hopf, who considered it in 1931 as part of his research on the hopf fibration. The jones polynomial v vkis a topological invariant of the knot k6, function of a dummy variable. A polygonal link is a finite union of polygonal knots, such that the inter section of any two knots in the union is empty. Tackling fluid structures complexity by the jones polynomial. Each link can be represented as the closure of a braid. T denotesthereectionof t intheprojectionplane,andndenotes.
The jones polynomial as an advantage over the alexander is able to distinguish between a knot or link and its mirror image i. Then, no matter if the crossing is positive or negative, the result is an unknot, so it follows from the skein relation that. In it was shown that the jones polynomial as a polynomial in q q is equivalently the partition function of su 2 su2chernsimons theory with a wilson loop specified by the given knot as a function of the exponentiated. May 01, 2020 the prime link 020201 which has jones polynomial. However, in mathematics, it was known to carl friedrich gauss before the work of hopf. Actually, up to a signed multiplication of a power of t the jones polynomial v lt of an alternating link l is equal to the tutte polynomial t g. The jones polynomial is an invariant1 whose discovery in 1985 brought on major advances in knot theory. We give the general form of the tutte polynomial of a family of positivesigned connected planar graphs, and specialize it to the jones polynomial of the alternating. Pdf the jones polynomial as a new invariant of topological. The kauffman bracket and the jones polynomial in 1985 v.
It is known that the q polynomial of a 2bridge knot or link can be obtained from the jones polynomial. The jones polynomial v l of a link l is obtained by making the. Also, i suppose you mean the skein relation at the top equals 0. A useful way to visualize and manipulate a knot is to project it. Thus, the jones polynomial of a kcomponent link represented by a diagram d is trivial if and onlyifthebracketpolynomialofd isequalto k. Morphisms of hopf algebras are just bialgebra maps \preserving the antipode exercise 3. This yields a formula which gives the twist number of any link. Knot theory and the jones polynomial lakehead university. When we specify t, the value vl of each oriented link is unique. By \laurent polynomial, we mean that both positive and negative integral powers of t12 are allowed. The jones polynomial and its limitations masters thesis. On some moves on links and the hopf crossing number.
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