A polygonal link is a finite union of polygonal knots, such that the inter section of any two knots in the union is empty. Here is the kauffman bracket calculation for the hopf link. We extend this result to links, nding that the jones polynomial doesnt detect hopf link factors. In particular, we can readily deduce the result due to yamada17 on the jones polynomial of 2parallels. The kauffman bracket and the jones polynomial in 1985 v. We compute the jones polynomial of the hopf link oriented as in picture 1. Infinite families of links with trivial jones polynomial. The jones polynomial now recall that the jones polynomial s is defined by the identities. Jones polynomial in a kth primitive root of unity is unchanged under a kmove, if k is odd, and changes. Before proving this result, some commentary on terminology. The jones polynomial university of california, berkeley. The jones polynomial as a new invariant of topological fluid dynamics. Up to change of vari able and multiplication by some power of a the bracket polynomial is the jones polynomial.
Pdf the jones polynomial as a new invariant of topological. 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. Kauffman introduced a statesum model construction of the. Then our initial hopf link gives l so we will write l for that link. The jones polynomial v l of a link l is obtained by making the. The jones polynomial of rational links request pdf. On the khovanov homology of 2 and 3strand braid links. A primer of hopf algebras 3 basis, and the multiplication in gis extended to kgby linearity. The jones polynomial and its limitations masters thesis arxiv. For a link l, the jones polynomial of lis a laurent polynomial in t12.
When we specify t, the value vl of each oriented link is unique. 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. For links of more than one component, however, the jones polynomial may change depending on the choice of orientation for each component. Cs e6204 lectures 9b and 10 alexanderconway and jones. This yields a formula which gives the twist number of any link. 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. We will go through the construction of this polynomial that is due to louis kau man.
Kauffman this paper is an introduction to the theory of virtual knots. The colored jones polynomial of a knot or link is a generalization of the jones polynomial. 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. 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. The jones polynomial is an invariant1 whose discovery in 1985 brought on major advances in knot theory. We construct arbitrarily many 2bridge knots or links with the same q polynomial but. This polynomial is a remarkably good method of distinguishing knots. The hopf link on the left and the righthanded trefoil. The jones polynomial of the negative hopf link download. Jones lq computes the jones polynomial of a knot or link l as a function of the variable q. Also, i suppose you mean the skein relation at the top equals 0. For the lefthanded trefoil, the relation yields x4vunknot x 4vltrefoil x 2 x2v hopf.
Knot theory and the jones polynomial lakehead university. In this paper i construct a state model for the original jones polynomial 5. A fundamental open problem in knot theory speci cally the theory of jones polynomial is as to. In the mathematical field of knot theory, the jones polynomial is a knot polynomial discovered by vaughan jones in 1984. By \laurent polynomial, we mean that both positive and negative integral powers of t12 are allowed. Each link can be represented as the closure of a braid. We start with the skein relation for one crossing of the trefoil, which gives. 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. Department of mathematics, university of california at berkeley, berkeley ca 94720, u. T denotesthereectionof t intheprojectionplane,andndenotes. On some moves on links and the hopf crossing number. Then, no matter if the crossing is positive or negative, the result is an unknot, so it follows from the skein relation that.
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. 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. However, in mathematics, it was known to carl friedrich gauss before the work of hopf. Download scientific diagram the jones polynomial of the negative hopf link from. 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. 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. The rst step is to assign a bracket to every knot or link diagram k. Now consider the link called h in the following picture sometimes called the hopflink. May 01, 2020 the prime link 020201 which has jones polynomial. 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. The jones polynomial v vkis a topological invariant of the knot k6, function of a dummy variable. Tackling fluid structures complexity by the jones polynomial.
The jones polynomial as an advantage over the alexander is able to distinguish between a knot or link and its mirror image i. By examining some particular examples we show how numerical. Let us consider the upper crossing point, see picture 1. Jones polynomial invariants for knots and satellites. Vaughan jones2 february 12, 2014 2 supportedbynsfundergrantno. The jones polynomial and its limitations masters thesis. It has also long been used outside mathematics, for instance as the crest of buzanha, a japanese buddhist sect founded in the 16th. The hopf link is named after topologist heinz hopf, who considered it in 1931 as part of his research on the hopf fibration.
Jonesconway polynomial encyclopedia of mathematics. 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. Figure 4 illustrates the calculation of the bracket for the hopf link and for the. It is known that the q polynomial of a 2bridge knot or link can be obtained from the jones polynomial. Morphisms of hopf algebras are just bialgebra maps \preserving the antipode exercise 3. To see this, add a crossing connecting the two components.