IS THE JONES POLYNOMIAL OF A KNOT REALLY A POLYNOMIAL?
نویسندگان
چکیده
منابع مشابه
Is the Jones Polynomial of a Knot Really a Polynomial?
The Jones polynomial of a knot in 3-space is a Laurent polynomial in q, with integer coefficients. Many people have pondered why is this so, and what is a proper generalization of the Jones polynomial for knots in other closed 3-manifolds. Our paper centers around this question. After reviewing several existing definitions of the Jones polynomial, we show that the Jones polynomial is really an ...
متن کاملThe Knot Group and The Jones Polynomial
In this thesis, basic knot theory is introduced, along with concepts from topology, algebra and algebraic topology, as they relate to knot theory. In the first chapter, basic definitions concerning knots are presented. In the second chapter, the fundamental group is applied as a method of distinguishing knots. In particular the torus knots are classified using the fundamental group, and a gener...
متن کاملDoes the Jones Polynomial Determine the Signature of a Knot?
The signature function of a knot is an integer valued step function defined on the unit circle. The jumps (i.e., the discontinuities) of the signature function can occur only at the roots of the Alexander polynomial on the unit circle. The latter are important in deforming U(1) representations of knot groups to irreducible SU(2) representations. Under the assumption that these roots are simple,...
متن کاملThe Jones Polynomial, Genus and Weak Genus of a Knot
In his book [Ad, p. 105 bottom], C. Adams mentions a result of Morton that there exist knots, whose genus g is strictly less than their weak genus g̃, the minimal genus of (the surface of Seifert’s algorithm applied on) all their diagrams. This observation appears just as a remark in [Mo], but was very striking to the author. Motivated by Morton’s example, the author started in a series of paper...
متن کاملIs the third coefficient of the Jones knot polynomial a quantum state of gravity?
Some time ago it was conjectured that the coefficients of an expansion of the Jones polynomial in terms of the cosmological constant could provide an infinite string of knot invariants that are solutions of the vacuum Hamiltonian constraint of quantum gravity in the loop representation. Here we discuss the status of this conjecture at third order in the cosmological constant. The calculation is...
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ژورنال
عنوان ژورنال: Journal of Knot Theory and Its Ramifications
سال: 2006
ISSN: 0218-2165,1793-6527
DOI: 10.1142/s0218216506004919