scholarly journals The Alexander polynomial of a rational link

2019 ◽  
Vol 28 (03) ◽  
pp. 1950004
Author(s):  
Mark E. Kidwell ◽  
Kerry M. Luse
Keyword(s):  
Standard Form ◽  
Newton Polygon ◽  
Laurent Polynomial ◽  
Alexander Polynomial ◽  
Total Degree ◽  
Polynomial Formula ◽  
Positive Coefficients

We relate some terms on the boundary of the Newton polygon of the Alexander polynomial [Formula: see text] of a rational link to the number and length of monochromatic twist sites in a particular diagram that we call the standard form. Normalize [Formula: see text] to be a true polynomial (as opposed to a Laurent polynomial), in such a way that terms of even total degree have positive coefficients and terms of odd total degree have negative coefficients. If the rational link has a reduced alternating diagram with no self-crossings, then [Formula: see text]. If the standard form of the rational link has [Formula: see text] monochromatic twist sites, and the [Formula: see text]th monochromatic twist site has [Formula: see text] crossings, then [Formula: see text]. Our proof employs Kauffman’s clock moves and a lattice for the terms of [Formula: see text] in which the [Formula: see text]-power cannot decrease.

The generalized Alexander polynomial of periodic virtual links

2019 ◽  
Vol 28 (06) ◽  
pp. 1950042
Author(s):  
Joonoh Kim ◽  
Kyoung-Tark Kim ◽  
Mi Hwa Shin
Keyword(s):  
Alexander Polynomial ◽  
State Model ◽  
Quantum Algebras ◽  
Virtual Link ◽  
Linking Numbers ◽  
Virtual Links ◽  
Polynomial Formula

In this paper, we give several simple criteria to detect possible periods and linking numbers for a given virtual link. We investigate the behavior of the generalized Alexander polynomial [Formula: see text] of a periodic virtual link [Formula: see text] via its Yang–Baxter state model given in [L. H. Kauffman and D. E. Radford, Bi-oriented quantum algebras and a generalized Alexander polynomial for virtual links, in Diagrammatic Morphisms and Applications, Contemp. Math. 318 (2003) 113–140, arXiv:math/0112280v2 [math.GT] 31 Dec 2001].

scholarly journals Cn-moves and the difference of Jones polynomials for links

2017 ◽  
Vol 26 (05) ◽  
pp. 1750029 ◽  
Author(s):  
Ryo Nikkuni
Keyword(s):  
Jones Polynomial ◽  
Laurent Polynomial ◽  
Link Invariant ◽  
Oriented Link ◽  
Jones Polynomials ◽  
The Difference ◽  
Polynomial Formula

The Jones polynomial [Formula: see text] for an oriented link [Formula: see text] is a one-variable Laurent polynomial link invariant discovered by Jones. For any integer [Formula: see text], we show that: (1) the difference of Jones polynomials for two oriented links which are [Formula: see text]-equivalent is divisible by [Formula: see text], and (2) there exists a pair of two oriented knots which are [Formula: see text]-equivalent such that the difference of the Jones polynomials for them equals [Formula: see text].

On Newton polygons techniques and factorization of polynomials over Henselian fields

2019 ◽  
Vol 19 (10) ◽  
pp. 2050188
Author(s):  
Lhoussain El Fadil
Keyword(s):  
Newton Polygon ◽  
Monic Polynomial ◽  
Local Fields ◽  
Algebraic Number Theory ◽  
Newton Polygons ◽  
Valued Field ◽  
Rank One ◽  
Difference Polynomials ◽  
Polynomial Formula ◽  
Factorization Of Polynomials

Let [Formula: see text] be a valued field, where [Formula: see text] is a rank-one discrete valuation, with valuation ring [Formula: see text]. The goal of this paper is to investigate some basic concepts of Newton polygon techniques of a monic polynomial [Formula: see text]; namely, theorem of the product, of the polygon, and of the residual polynomial, in such way that improves that given in [D. Cohen, A. Movahhedi and A. Salinier, Factorization over local fields and the irreducibility of generalized difference polynomials, Mathematika 47 (2000) 173–196] and generalizes that given in [J. Guardia, J. Montes and E. Nart, Newton polygons of higher order in algebraic number theory, Trans. Amer. Math. Soc. 364(1) (2012) 361–416] to any rank-one valued field.

On lassos and the Jones polynomial of satellite knots

2016 ◽  
Vol 25 (02) ◽  
pp. 1650011
Author(s):  
Adrián Jiménez Pascual
Keyword(s):  
Jones Polynomial ◽  
Solid Torus ◽  
Alexander Polynomial ◽  
New Family ◽  
Jones Polynomials ◽  
Polynomial Formula ◽  
Satellite Knots

In this paper, I present a new family of knots in the solid torus called lassos, and their properties. Given a knot [Formula: see text] with Alexander polynomial [Formula: see text], I then use these lassos as patterns to construct families of satellite knots that have Alexander polynomial [Formula: see text] where [Formula: see text]. In particular, I prove that if [Formula: see text] these satellite knots have different Jones polynomials.

Twisted Alexander polynomials of 2-bridge knots associated to dihedral representations

2018 ◽  
Vol 27 (02) ◽  
pp. 1850015 ◽  
Author(s):  
Mikami Hirasawa ◽  
Kunio Murasugi
Keyword(s):  
Direct Product ◽  
Cyclic Group ◽  
Alexander Polynomial ◽  
Alexander Polynomials ◽  
Twisted Alexander Polynomial ◽  
Polynomial Formula ◽  
Integer Polynomial

Let [Formula: see text] be a non-abelian semi-direct product of a cyclic group [Formula: see text] and an elementary abelian [Formula: see text]-group [Formula: see text] of order [Formula: see text], [Formula: see text] being a prime and [Formula: see text]. Suppose that the knot group [Formula: see text] of a knot [Formula: see text] in the [Formula: see text]-sphere is represented on [Formula: see text]. Then we conjectured (and later proved) that the twisted Alexander polynomial [Formula: see text] associated to [Formula: see text] is of the form: [Formula: see text], where [Formula: see text] is the Alexander polynomial of [Formula: see text] and [Formula: see text] is an integer polynomial in [Formula: see text]. In this paper, we present a proof of the following. For a [Formula: see text]-bridge knot [Formula: see text] in [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text] is written as [Formula: see text], where [Formula: see text] is the set of [Formula: see text]-bridge knots whose knot groups map on that of [Formula: see text] with [Formula: see text] odd.

scholarly journals The Alexander polynomial for virtual twist knots

2017 ◽  
Vol 26 (01) ◽  
pp. 1750007
Author(s):  
Isaac Benioff ◽  
Blake Mellor
Keyword(s):  
Knot Theory ◽  
Recursive Formula ◽  
Alexander Polynomial ◽  
Polynomial Invariants ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Virtual Links ◽  
Polynomial Formula

We define a family of virtual knots generalizing the classical twist knots. We develop a recursive formula for the Alexander polynomial [Formula: see text] (as defined by Silver and Williams [Polynomial invariants of virtual links, J. Knot Theory Ramifications 12 (2003) 987–1000]) of these virtual twist knots. These results are applied to provide evidence for a conjecture that the odd writhe of a virtual knot can be obtained from [Formula: see text].

scholarly journals Kauffman–Jones polynomial of a curve on a surface

2017 ◽  
Vol 26 (11) ◽  
pp. 1750062
Author(s):  
Shinji Fukuhara ◽  
Yusuke Kuno
Keyword(s):  
Homotopy Class ◽  
Jones Polynomial ◽  
Intersection Number ◽  
Laurent Polynomial ◽  
Computational Results ◽  
Oriented Surface ◽  
Variable Formula ◽  
Polynomial Formula

We introduce a Kauffman–Jones type polynomial [Formula: see text] for a curve [Formula: see text] on an oriented surface, whose endpoints are on the boundary of the surface. The polynomial [Formula: see text] is a Laurent polynomial in one variable [Formula: see text] and is an invariant of the homotopy class of [Formula: see text]. As an application, we obtain an estimate in terms of the span of [Formula: see text] for the minimum self-intersection number of a curve within its homotopy class. We then give a chord diagrammatic description of [Formula: see text] and show some computational results on the span of [Formula: see text].

scholarly journals Alexander and writhe polynomials for virtual knots

2016 ◽  
Vol 25 (08) ◽  
pp. 1650050 ◽  
Author(s):  
Blake Mellor
Keyword(s):  
Knot Theory ◽  
Alexander Polynomial ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Virtual Links ◽  
New Interpretation ◽  
Polynomial Formula ◽  
Knots And Links

We give a new interpretation of the Alexander polynomial [Formula: see text] for virtual knots due to Sawollek [On Alexander–Conway polynomials for virtual knots and Links, preprint (2001), arXiv:math/9912173] and Silver and Williams [Polynomial invariants of virtual links, J. Knot Theory Ramifications 12 (2003) 987–1000], and use it to show that, for any virtual knot, [Formula: see text] determines the writhe polynomial of Cheng and Gao [A polynomial invariant of virtual links, J. Knot Theory Ramifications 22(12) (2013), Article ID: 1341002, 33pp.] (equivalently, Kauffman’s affine index polynomial [An affine index polynomial invariant of virtual knots, J. Knot Theory Ramifications 22(4) (2013), Article ID: 1340007, 30pp.]). We also use it to define a second-order writhe polynomial, and give some applications.

scholarly journals POLYNOMIAL KNOT AND LINK INVARIANTS FROM THE VIRTUAL BIQUANDLE

2013 ◽  
Vol 22 (04) ◽  
pp. 1340004 ◽  
Author(s):  
ALISSA S. CRANS ◽  
ALLISON HENRICH ◽  
SAM NELSON
Keyword(s):  
Polynomial Ring ◽  
Laurent Polynomial ◽  
Alexander Polynomial ◽  
Virtual Knot ◽  
Link Invariants ◽  
Virtual Knots ◽  
Monomial Ordering ◽  
Laurent Polynomial Ring ◽  
Future Work ◽  
Knots And Links

The Alexander biquandle of a virtual knot or link is a module over a 2-variable Laurent polynomial ring which is an invariant of virtual knots and links. The elementary ideals of this module are then invariants of virtual isotopy which determine both the generalized Alexander polynomial (also known as the Sawollek polynomial) for virtual knots and the classical Alexander polynomial for classical knots. For a fixed monomial ordering <, the Gröbner bases for these ideals are computable, comparable invariants which fully determine the elementary ideals and which generalize and unify the classical and generalized Alexander polynomials. We provide examples to illustrate the usefulness of these invariants and propose questions for future work.

Production of New and Revised A.A.V.S.O. Charts

1979 ◽  
Vol 46 ◽  
pp. 368
Author(s):  
Clinton B. Ford
Keyword(s):  
Variable Star ◽  
Standard Form ◽  
Original Data ◽  
University Of Pennsylvania ◽  
Standard Format ◽  
Additional Information ◽  
The Gift ◽  
The University

A “new charts program” for the Americal Association of Variable Star Observers was instigated in 1966 via the gift to the Association of the complete variable star observing records, charts, photographs, etc. of the late Prof. Charles P. Olivier of the University of Pennsylvania (USA). Adequate material covering about 60 variables, not previously charted by the AAVSO, was included in this original data, and was suitably charted in reproducible standard format.Since 1966, much additional information has been assembled from other sources, three Catalogs have been issued which list the new or revised charts produced, and which specify how copies of same may be obtained. The latest such Catalog is dated June 1978, and lists 670 different charts covering a total of 611 variables none of which was charted in reproducible standard form previous to 1966.

Sign in / Sign up

Export Citation Format

Share Document