Ideas originated in physics and in representation theory led to vast families invariants of knots and 3manifolds. 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 resulting polynomials are thus function of the topology of the knot type and vortex circulation and we provide several examples of those. 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. Vaughan jones2 february 12, 2014 2 supportedbynsfundergrantno. Colored jones polynomials michigan state university. The geometry of knot complements city university of new york. Tackling fluid structures complexity by the jones polynomial. The prime link 050201, illustrated above, with braid word or and jones polynomial the whitehead link has linking number 0. The classical alexander polynomial kt of a knot k is the determinant of its alexander matrix. Mutation and the colored jones polynomial alexander stoimenow and toshifumi tanaka with appendices by daniel matei and the.
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. Infinite families of links with trivial jones polynomial. The jones polynomial for fluid knots from helicity iopscience. The jones polynomial and its limitations masters thesis.
The jones polynomial as an advantage over the alexander is able to distinguish between a knot or link and its mirror image i. Abstract there is a remarkable connection between the tutte and the jones polynomials. The hyperbolic volume of the complement of the whitehead link is 4 times catalans constant, approximately 3. A technique to calculate the colored jones polynomials of satellite knots, illustrated by the whitehead doubles of knots, is presented. On lassos and the jones polynomial of satellite knots. I khovanov homology 1999 categori ed the jones polynomial. A refined upper bound for the volume of links and the colored. 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. Now consider the link called h in the following picture sometimes called the hopflink. The degree of the leading term tells you the degree of the whole polynomial. 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. Vaughan jones 2 february 12, 2014 2 supported by nsf under grant no. Whitehead links, borromean double of the figureeight knot.
The jones polynomial as a new invariant of topological. Nothing is known about other in nite families and the conjecture even a class of 2bridge links remains a mystery in this regard. The borromean rings are a 3component example of a brunnian link, which is. 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. For a link l, the jones polynomial of lis a laurent polynomial in t12. Then, after defining an equivalence relation on all possible ways of factoring a knot, we will show that there is only one. Notably, is the jones polynomial for the mirror image of a link having jones polynomial. The jones polynomial as a new invariant of topological fluid. Before proving this result, some commentary on terminology. 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.
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. A whitehead double of a knot k is a knot obtained as follows. In particular, we show a technique to calculate the colored jones polynomial of satellite knots by cutting and gluing method. 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. Pdf the jones polynomial as a new invariant of topological. As an application, we verify the cosmetic surgery conjecture for all knots with no more than 11 crossings except for. The new polynomial invariants of knots and links w. Figure 4 illustrates the calculation of the bracket for the hopf link and for the.
The whitehead group of a polynomial extension springerlink. 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. Gt 4 mar 2006 proof of the volume conjecture for whitehead. May 26, 2011 abstract in this paper, we first recall some known architectures of polyhedral links 1016. In this paper i construct a state model for the original jones polynomial 5. We observe numerically that for knots 63, 89 and 820 and for the whitehead link, the colored jones polynomials are related. We will go through the construction of this polynomial that is due to louis kau man. I using jones polynomial and relations to graph theory, tait conjectures from 100 years were resolved 1987. Kashaevs conjecture and the chernsimons invariants of.
Also, i suppose you mean the skein relation at the top equals 0. The rst step is to assign a bracket to every knot or link diagram k. 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. An algorithm to approximate the jones polynomial is only directly useful for topology if the approximation is valuedistinguishing.
The colored jones polynomial of a knot or link is a generalization of the jones polynomial. This novel approach establishes a topological foundation of classical field theory in general, and of mathematical fluid dynamics in particular, by opening up new. Kashaevs conjecture and the chernsimons invariants of knots. Any tangle extends to nonmutant knots with the same jones polynomial. Jones and q polynomials for 2bridge knots and links taizo kanenobu communicated by frederick r. 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. The prime link 050201, illustrated above, with braid word. A 1 be the laurent polynomial ring in one indeterminate a with coe cients in the ring of integers. John stallings, constructions of fibered knots and links, proceedings of symposia in pure mathematics, volume 32,1987 pdf. Cs e6204 lectures 9b and 10 alexanderconway and jones.
I jones polynomial 1984, discovered via representations of braid groups, led to many new quantum invariants, which can be computed diagrammatically, e. 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. Alexanderconway and jones polynomials abstract before the 1920s, there were a few scattered papers. 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. Cohen dedicated to professor yoko tao on her sixtieth birthday abstract. Then we prove the volume conjecture for whitehead doubles of a family of torus knots and show some interesting observations. The cases of left and righthanded trefoil knots mirror knots, and the case of the whitehead link via the. The architecture and the jones polynomial of polyhedral links xianan jin fuji zhang school of mathematical sciences xiamen university, xiamen, fujian 361005 p. We put q a2 a4 so that q 1t for the variable t of the jones polynomial standardly. A refined upper bound for the volume of links and the. In the mathematical field of knot theory, the jones polynomial is a knot polynomial discovered by vaughan jones in 1984.
In the case of a homogeneous, isotropic tangle of vortex. Department of mathematics, university of california at berkeley, berkeley ca 94720, u. 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. The unlink, hopf link and whitehead link all have two components, and the borromean rings has three. 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. The geometry of knot complements city university of new. Various approaches to the alexander polynomial are described in introductory texts such as n. It is known that the q polynomial of a 2bridge knot or link can be obtained from the jones polynomial. Joneslq computes the jones polynomial of a knot or link l as a function of the variable q. 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 jones polynomial is an invariant1 whose discovery in 1985 brought on major advances in knot theory. The jones polynomial for fluid knots from helicity.
Explicit computations of the jones polynomial for some standard configurations, including the whitehead link and the borromean rings whose linking numbers are zero, are. This polynomial is a remarkably good method of distinguishing knots. 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. Cromwell, some infinite families of satellite knots with given alexander polynomial, mathematika 381 1991 156169. A re ned upper bound for the volume of links and the colored jones polynomial 6 29. 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. Here is the kauffman bracket calculation for the hopf link. A fundamental open problem in knot theory speci cally the theory of jones polynomial is as to. Jones lq computes the jones polynomial of a knot or link l as a function of the variable q. 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. 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.