Remarks on the A-Polynomial of a Knot (original) (raw)

2011

This article provides an overview of relative strengths of polynomial invariants of knots and links, such as the Alexander, Jones, Homflypt, and Kaufman two-variable polynomial, Khovanov homology, factorizability of the polynomials, and knot primeness detection.

1 0 Ju l 2 01 1 KNOT POLYNOMIALS : MYTHS AND REALITY

2019

A New Polynomial Invariant of Knots and LINKS1

1985

The purpose of this note is to announce a new isotopy invariant of oriented links of tamely embedded circles in 3-space. We represent links by plane projections, using the customary conventions that the image of the link is a union of transversely intersecting immersed curves, each provided with an orientation, and undercrossings are indicated by broken lines. Following Conway [6], we use the symbols L+, Lo, L_ to denote links having plane projections which agree except in a small disk, and inside that disk are represented by the pictures of Figure 1. Conway showed that the one-variable Alexander polynomials of L+, Lo, L_ (when suitably normalized) satisfy the relation

A new polynomial invariant of knots and links

Bulletin of the American Mathematical Society, 1985

Knot polynomials and generalized mutation

Topology and its Applications, 1989

The motivation for this work was to construct a nontrivial knot with trivial Jones polynomial. Although that open problem has not yielded, the methods are useful for other problems in the theory of knot polynomials. The subject of the present paper is a generalization of Conway's mutation of knots and links. Instead of flipping a 2-strand tangle, one flips a many-string tangle to produce a generalized mutant. In the presence of rotational symmetry in that tangle, the result is called a "rotant". We show that if a rotant is sufficiently simple, then its Jones polynomial agrees with that of the original link. As an application, this provides a method of generating many examples of links with the same Jones polynomial, but different Alexander polynomials. Various other knot polynomials, as well as signature, are also invariant under such moves, if one imposes more stringent conditions upon the symmetries. Applications are also given to polynomials of satellites and symmetric knots.

On the twisted Alexander polynomial and the A-polynomial of 2-bridge knots

We show,that the A-polynomial A(L; M) of a 2-bridge knot b(p; q) is irreducible if p is prime, and if (p 1)=2 is also prime and q , 1 then the L-degree of A(L; M) is (p,1)=2. This shows that the AJ conjecture relating the A-polynomial and the colored Jones polynomial holds true for these knots, according to work of the second author. We also study relationships between the A-polynomial of a 2-bridge knot and a twisted Alexander polynomial associated with the adjoint representation of the fundamental,group of the knot complement. We show that for twist knots the A-polynomial is a factor of the twisted Alexander polynomial. 1. Background,and conventions

Knot Invariants and Their Implications for Closed Plane Cur Ves

2005

A dominating feature of knot theory is the problem of knot classification. In this paper, we hope to simplify the task of classification through strong connections between Arnold's work with knot invariants and that of Xiao-Song Lin and Zhenghan Wang. We show that the defect of a plane curve associated with a flattened alternating knot may be considered an invariant for alternating knots. We also give simple method of computing a certain Vassiliev knot invariant from Arnold's plane curve invariants and signed crossings.

Knot Theory and the Jones Polynomial

2018

In this paper we introduce the notion of a knot, knot equivalence, and their related terms. In turn, we establish that these terms correspond to the much simpler knot diagram, Reidemeister moves, and their related terms. It is then shown that working with these latter terms, one can construct a number of invariants that can be used to distinguish inequivalent knots. Lastly, the use of these invariants is applied in setting of molecular biology to study DNA supercoiling and recombination.

Knots are determined by their complements

Bulletin of The American Mathematical Society, 1989

Two (smooth or PL) knots K, K' in S 3 are equivalent if there exists a homeomorphism h: S 3 -• S 3 such that h(K) = K'. This implies that their complements S 3 -K and S 3 -K' are homeomorphic. Here we announce the converse implication. THEOREM 1. If two knots have homeomorphic complements then they are equivalent. This answers a question apparently first raised by Tietze [T, p. 83]. It was previously known that there were at most two knots with a given complement [CGLS, Corollary 3]. Whitten [W] has shown that prime knots with isomorphic groups have homeomorphic complements. Hence we have COROLLARY 1.1. If two prime knots have isomorphic groups then they are equivalent.

Compositions of Knots Using Alexander Polynomial

arXiv (Cornell University), 2023

Knot theory is the Mathematical study of knots. In this paper we have studied the Composition of two knots. Knot theory belongs to Mathematical field of Topology, where the topological concepts such as topological spaces, homeomorphisms, and homology are considered. We have studied the basics of knot theory, with special focus on Composition of knots, and knot determinants using Alexander Polynomials. And we have introduced the techniques to generalize the solution of composition of knots to present how knot determinants behave when we compose two knots.

Remarks on the A-Polynomial of a Knot (original) (raw)

Related papers