We show that the apolynomial of the torus knot can be derived from the difference equation. A knot is a simple closed curve obtained by embedding of. In general proofs will be discussed rather than given as it is easy to. On the colored jones polynomials of certain cable of the torus knots. As wellknown, the colored jones polynomial is the quantum sl2,cinvariant. We will recall the idea of satellite knot and the formula for their alexander polynomial. The coloured jones function and alexander polynomial for. Rosso and jones gave a formula for the colored jones polynomial of a torus knot, colored by an irreducible representation of a simple lie algebra. A formula for the jones polynomial of pretzel knots and links is constructed using kauffmans state model of the jones polynomial. We suggest a new construction for the quantum groups jones polynomials of torus knots in terms of the pbw theorem of daha for. Journal of knot theory and its rami cations world scienti. The realzeros of jones polynomial of torus the realzeros of jones polynomial of torus vtp. Outline 3manifold geometric structures geometrization of s3 r k invariants arising from geometry.
The jones polynomial leds to discover more knot polynomials, such as socalled homfly polynomial 5,11. Given a torus knot, we apply the corresponding element of the projective to the macdonald polynomial representing the color and then take the daha evaluation coinvariant. We prove that the ncolored jones polynomial for the torus knot satisfies the second order difference equation, which reduces to the first order difference equation for a case of. Our interpretation of the qgjones colored polynomials of torus knots is based directly on the pbw theorem for daha. The coloured jones function and alexander polynomial for torus knots. We reveal an intimate connection between the quantum knot invariant for torus knot ts,t and the character of the minimal model ms,t, where s and t are relatively prime integers. Compared with the jones polynomial, the colored jones polynomial reveals much stronger connections between quantum algebra and 3dimensional topology, for example, the volume conjecture, which relates the asymptotic behavior of the colored jones polynomial of a. They showed thistheorem by usingsome propertiesofthe quantum groupuqsl2,c. Pdf the coloured jones function and alexander polynomial.
For any knot k, the colored jones polynomial jnk has a recursion relation in terms of n. I using jones polynomial and relations to graph theory, tait conjectures from 100 years were resolved 1987. The jones polynomial is a celebrated invariant of a knot or link in ordinary threedimensional space, originally discovered by v. Iterated torus knots and double affine hecke algebras. 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. In this case, the asymptotic expansion of the colored jones polynomial splits into sums and each summand is related to the chernsimons invariant and the reidemeister torsion associated with.
In 2 it was conjectured that the coloured jones function of a framed knot k. The coloured jones function and alexander polynomial for torus knots by h. In this paper i show that the explicit formula does give the alexander polynomial when k is any torus knot. Links can be represented by diagrams in the plane and the jones polynomials of. The ajconjecture holds for each r,scabled knot c over each p,qtorus knot t if r is not an integer between 0 and pqs. A particular aim of the course will be to obtain the jones polynomial for torus knots. On the asymptotic expansion of the colored jones polynomial for torus knots jer ome dubois and rinat kashaev abstract. Outline 1 linear skein theory 2 colored jones polynomial of a torus knot 3 chernsimons invariant 4 cs of torus knots 5 reidemeister torsion 6 reidemeister torsion of 7 cj, cs, and reidemeister for torus knots 8 parametrized vc for 9 cs of 10 reidemeister torsion of 11 generalizations of the volume conjecture hitoshi murakami tohoku university volume conjecture, iii iiser, pune, 21st.
In this case, the asymptotic expansion of the colored jones polynomial splits into sums, and each summand is related to the chernsimons invariant and the reidemeister torsion associated with a. We show that an asymptotics of the ncolored jones polynomial with q exp 2. We study the asymptotic behaviors of the colored jones polynomials of torus knots. Department of mathematics, university of california at berkeley, berkeley ca 94720, u. This is a series of 8 lectures designed to introduce someone with a certain amount of. A cabling formula for apolynomials of cabled knots in s3 is given in 9. In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted. In the mathematical field of knot theory, the jones polynomial is a knot polynomial discovered by vaughan jones in 1984. In this paper we compute a qhypergeometric expression for the cyclotomic expansion of the colored jones polynomial for the lefthanded torus knot 2. The jones polynomials of knot graphs and their applications were introduced by murasugi murasugi 1991. The colored jones polynomial of a knot k is a collection of laurent polynomials. Kopuzlu found an algorithm for the jones polynomials of torus knots. Hajij, jones polynomial for links in the handlebody, rocky mountain journal of mathematics, 43 3 20, 737753.
The rossojones formula involves a plethysm function, unknown in general. A survey of colored jones polynomials with emphasis on relations to geometry and topology of knot complements. Torus knots and quantum modular forms kazuhiro hikami and jeremy lovejoy abstract. Khovanov homology and torus knots uc davis mathematics. Torus knot polynomials and susy wilson loops springerlink. Asymptotics of the colored jones polynomial and the a. Jones made a connection between the operator algebras that he was researching and knots. The strong aj conjecture for cables of torus knots. We also state a conjecture suggesting a topological interpretation of the linear terms of the degree of the colored jones polynomial conjecture 5. A knot is a smooth closed curve in three dimensional space r3. Vaughan jones2 february 12, 2014 2 supportedbynsfundergrantno. Many descriptions and generalizations of the jones polynomial were discovered in the years immediately after joness work. Hyperbolic volume incompressible surfaces quantum topology colored jones polynomials knot diagrammatic approaches cjp and volume. The focus of this paper is to study the homfly polynomial of 2.
Alexander used the determinant of a matrix to calculate the alexander polynomial of a knot. Atorus knotis a knot which can be embedded on the torus as a simple closed curve. Morton department of pure mathematics, university of liverpool, po box 147, liverpool, z69 sbx, u. In the asymptotic expansion of the hyperbolic speci.
On lassos and the jones polynomial of satellite knots. The aj conjecture, formulated by garoufalidis, relates the apolynomial and the colored jones polynomial of a knot in the 3sphere. On the jones polynomial and its applications alan chang abstract. Pdf representations and the colored jones polynomial of. It is stronger than the alexander polynomial in that it is able to distinguish more knots. The distribution of zeros of jones polynomial is an interesting topic in knots theory of math and physics. Pdf on polynomials of k2,n torus knots researchgate.
After jones discovery, many new of polynomial invariants were discovered. Vaughan jones 2 february 12, 2014 2 supported by nsf under grant no. The combinatorics of knot invariants arising from the study of macdonald polynomials. Received 5 july 1993 abstract in 2 it was conjectured that the coloured jones function of a frame odr knot k, equivalently the jones polynomials of all parallels. The calculating of writhe numbers for some knot examples is presented. Jones polynomials of torus knots via daha international. The euler characteristic for a 1dimensional object is 0 when applied to.
In this article, we study a similar phenomenon when the knot is a twiceiterated torus knot. The aj conjecture relates the apolynomial and the colored jones polynomial of a knot in the 3sphere. It has been confirmed for all torus knots, some classes of twobridge knots and pretzel knots, and most cable knots over torus knots. This paper is a selfcontained introduction to the jones polynomial that assumes no background in knot theory. I will then investigate the kau man bracket and the jones polynomial of knots in the solid torus, and straightaway give explicit formulae for.
It has been veri ed for some classes of knots, including all torus knots, most double twist knots, 2. Representations and the colored jones polynomial of a torus knot. Introduction we will use many standard terminologies and notations knot theory. The colored jones polynomial, the chernsimons invariant. In this article we study a similar phenomenon when the knot is a twiceiterated torus knot. In particular when c is the r,scabled knot over the torus knot tp,qins3,its 15500512. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a laurent polynomial in the variable with integer coefficients. We show here that this relation can be rearranged into a different form theorem 1, probably more natural, from which one can prove that every derivative of a quantum group invariant, evaluated at 1, is a vassiliev invariant. Asymptotic behaviors of the colored jones polynomials of a torus knot hitoshi murakami abstract.
Kau man jones polynomial is preserved by the reidemeister moves and hence depends only on the topological type of a knot. Every knot in s3 is either a torus knot, a satellite knot or a hyperbolic knot. Pdf we introduce general formulas finding the bracket polynomials of torus knots k2,n and the jones polynomials of torus knots k2,n. He created the jones polynomial, a new knot invariant. Furthermore we show that, for these knots, the degree of the colored jones polynomial also determines the topology of a surface that satis es the slope conjecture. Jones knot invariants and vassiliev invariants via a substitution t ex. Alexander, and until the 1980s, it was the only polynomial invariant known.
We study gukovs conjecture, which relates an asymptotics of the colored jones polynomial to the apolynomial, in the case of twist knots. On the jones polynomial in the solid torus bataineh. A computer program in maple, which is given for calculations of these polynomials is also used to show that an infinite class of pretzel knots with trivial alexander polynomial has nontrivial jones polynomial. We show that the strong slope conjecture implies that the degree of the colored jones polynomial detects all torus knots. The coloured jones function and alexander polynomial for torus knots article pdf available in mathematical proceedings of the cambridge philosophical society 11701. Yokota, and the author, they do not seem to give the volumes or the. Belkhirat, the derivatives of the hoste and przytycki polynomial for oriented links in the solid torus, accepted for publication in. Homfly polynomials of torus links as generalized fibonacci. We give, using an explicit expression obtained in jones v, ann math 126. Alexander polynomial thealexander polynomialof a knot was the. The combinatorics of knot invariants arising from the.
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