Then we prove the volume conjecture for whitehead doubles of a family of torus knots and show some interesting observations. Before proving this result, some commentary on terminology. A fundamental open problem in knot theory speci cally the theory of jones polynomial is as to. Alexanderconway and jones polynomials abstract before the 1920s, there were a few scattered papers. A re ned upper bound for the volume of links and the colored jones polynomial 6 29. Abstract there is a remarkable connection between the tutte and the jones polynomials. Pdf the jones polynomial as a new invariant of topological.
The cases of left and righthanded trefoil knots mirror knots, and the case of the whitehead link via the. Now consider the link called h in the following picture sometimes called the hopflink. It is known that the q polynomial of a 2bridge knot or link can be obtained from the jones polynomial. Kashaev conjectured that the asymptotic behavior of his link invariant, which equals the colored jones polynomial evaluated at a root of unity, determines the hyperbolic volume of any hyperbolic link complement. Thus the problem of understanding knots and links is one of geometry and topology, and within those disciplines the subject has received considerable study during the last hundred or more years. Infinite families of links with trivial jones polynomial. For a link l, the jones polynomial of lis a laurent polynomial in t12. I using jones polynomial and relations to graph theory, tait conjectures from 100 years were resolved 1987. How hard is it to approximate the jones polynomial. Using the notion of a seifert surface of a knot, we define a knots genus, an additive invariant which allows to prove the existence of prime knots. Department of mathematics, university of california at berkeley, berkeley ca 94720, u. An algorithm to approximate the jones polynomial is only directly useful for topology if the approximation is valuedistinguishing. Explicit computations of the jones polynomial for some standard configurations, including the whitehead link and the borromean rings whose linking numbers are zero, are presented for illustration. Computation of the jones polynomial for the whitehead link and borromean links it is well known that there are many link types that are inseparably linked, even though their linking number is zero as for two unlinked loops.
Kashaevs conjecture and the chernsimons invariants of. The prime link 050201, illustrated above, with braid word. The architecture and the jones polynomial of polyhedral links xianan jin fuji zhang school of mathematical sciences xiamen university, xiamen, fujian 361005 p. Explicit calculations of the jones polynomial for the left and righthanded trefoil knots and for the whitehead link via the figureofeight knot are presented for illustration. A knot is called prime if it can not be represented as a connected sum of two knots such that both of these are knotted. The rst step is to assign a bracket to every knot or link diagram k.
The geometry of knot complements city university of new. Introduction and background the colored jones polynomial is a knot invariant that assigns to each knot a sequence of laurent polynomials indexed by n 2, the number of colors. A 1 be the laurent polynomial ring in one indeterminate a with coe cients in the ring of integers. A technique to calculate the colored jones polynomials of satellite knots, illustrated by the whitehead doubles of knots, is presented. The unlink, hopf link and whitehead link all have two components, and the borromean rings has three. 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. Vaughan jones2 february 12, 2014 2 supportedbynsfundergrantno. Also, i suppose you mean the skein relation at the top equals 0. By \laurent polynomial, we mean that both positive and negative integral powers of t12 are allowed. The borromean rings are a 3component example of a brunnian link, which is. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Liam watson generalized a construction of kanenobu to produce infinitely many pairs of knots with the same jones polynomial and khovanov homology but distinct homflypt polynomials, so they are not mutants. Cohen dedicated to professor yoko tao on her sixtieth birthday abstract. There are several ways to look at these invariants, some of these use the knot group previously defined by max dehn, but there are also various combinatorial methods derived.
The prime link 050201, illustrated above, with braid word or and jones polynomial the whitehead link has linking number 0. Various approaches to the alexander polynomial are described in introductory texts such as n. Figure 4 illustrates the calculation of the bracket for the hopf link and for the. Jones and q polynomials for 2bridge knots and links taizo kanenobu communicated by frederick r. Cromwell, some infinite families of satellite knots with given alexander polynomial, mathematika 381 1991 156169. A note on jones polynomial and cosmetic surgery kazuhiro ichihara and zhongtao wu we show that two dehn surgeries on a knot knever yield manifolds that are homeomorphic as oriented manifolds if v00 k1 6 0 or v000 k 1 6 0. The resulting polynomials are thus function of the topology of the knot type and vortex circulation and we provide several examples of those. The jones polynomial as a new invariant of topological.
On the colored jones polynomial of some simple links. John stallings, constructions of fibered knots and links, proceedings of symposia in pure mathematics, volume 32,1987 pdf. The jones polynomial as a new invariant of topological fluid. In this paper i construct a state model for the original jones polynomial 5. Up to change of vari able and multiplication by some power of a the bracket polynomial is the jones polynomial. A refined upper bound for the volume of links and the. The hyperbolic volume of the complement of the whitehead link is 4 times catalans constant, approximately 3. The jones polynomial as an advantage over the alexander is able to distinguish between a knot or link and its mirror image i. The jones polynomial for fluid knots from helicity. In particular, we show a technique to calculate the colored jones polynomial of satellite knots by cutting and gluing method.
We will go through the construction of this polynomial that is due to louis kau man. We observe numerically that for knots 63, 89 and 820 and for the whitehead link, the colored jones polynomials are related. As the name suggests, the jones polynomial is a polynomial invariant. On lassos and the jones polynomial of satellite knots. Mutation and the colored jones polynomial alexander stoimenow and toshifumi tanaka with appendices by daniel matei and the. Cs e6204 lectures 9b and 10 alexanderconway and jones. Kashaevs conjecture and the chernsimons invariants of knots. Mutation and the colored jones polynomial 3 the whitehead link, and t the complement of the other component, then we call k0 a whitehead double of k. Jones lq computes the jones polynomial of a knot or link l as a function of the variable q.
The new polynomial invariants of knots and links w. As an application, we verify the cosmetic surgery conjecture for all knots with no more than 11 crossings except for. This is a series of 8 lectures designed to introduce someone with a certain amount of mathematical knowledge to the jones polynomial of knots and links in 3 dimensions. The jones polynomial for fluid knots from helicity iopscience. Any tangle extends to nonmutant knots with the same jones polynomial. 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.
Vaughan jones 2 february 12, 2014 2 supported by nsf under grant no. Notably, is the jones polynomial for the mirror image of a link having jones polynomial. Tackling fluid structures complexity by the jones polynomial. We observe numerically that for knots 63,89 and 820 and for the whitehead link, the colored jones polynomials are related to the hyperbolic volumes and the chernsimons invariants and. May 26, 2011 abstract in this paper, we first recall some known architectures of polyhedral links 1016. Whitehead links, borromean double of the figureeight knot. We will go through the construction of this polynomial that is due to louis kau. The whitehead group of a polynomial extension springerlink. Joneslq computes the jones polynomial of a knot or link l as a function of the variable q. 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. Nothing is known about other in nite families and the conjecture even a class of 2bridge links remains a mystery in this regard. In the mathematical field of knot theory, the jones polynomial is a knot polynomial discovered by vaughan jones in 1984.
A refined upper bound for the volume of links and the colored. Then, after defining an equivalence relation on all possible ways of factoring a knot, we will show that there is only one. In the case of a homogeneous, isotropic tangle of vortex. Gt 4 mar 2006 proof of the volume conjecture for whitehead. The jones polynomial is an invariant1 whose discovery in 1985 brought on major advances in knot theory. Alexander published a paper topological invariants of knots and links in which he defined a polynomial invariant of knots and developed new insights including the braid relations. We give the general form of the tutte polynomial of a family of positivesigned. Colored jones polynomials michigan state university. Here is the kauffman bracket calculation for the hopf link. This polynomial is a remarkably good method of distinguishing knots. 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. It was discovered by whitehead in 1934 whitehead 1962, pp.
It is known that the colored jones polynomials, various 2cable link polynomials, the hyperbolic volume, and the fundamental group of the double branched cover coincide on mutant knots. I khovanov homology 1999 categori ed the jones polynomial. Ideas originated in physics and in representation theory led to vast families invariants of knots and 3manifolds. Anastasiia tsvietkova university of california, davis joint work with oliver dasbach. The jones polynomial and its limitations masters thesis. The classical alexander polynomial kt of a knot k is the determinant of its alexander matrix. This novel approach establishes a topological foundation of classical field theory in general, and of mathematical fluid dynamics in particular, by opening up new. The geometry of knot complements city university of new york. We put q a2 a4 so that q 1t for the variable t of the jones polynomial standardly.
Explicit computations of the jones polynomial for some standard configurations, including the whitehead link and the borromean rings whose linking numbers are zero, are. I jones polynomial 1984, discovered via representations of braid groups, led to many new quantum invariants, which can be computed diagrammatically, e. A whitehead double of a knot k is a knot obtained as follows. The degree of the leading term tells you the degree of the whole polynomial.
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