The Lehmer Polynomial and Pretzel Links

2001 ◽  
Vol 44 (4) ◽  
pp. 440-451 ◽  
Author(s):  
Eriko Hironaka
Keyword(s):  
Alexander Polynomial ◽  
Mahler Measure ◽  
Integer Polynomials ◽  
Pretzel Knots ◽  
Positive Integers ◽  
Knots And Links

AbstractIn this paper we find a formula for the Alexander polynomial Δp1,…,pk (x) of pretzel knots and links with (p1,…,pk,−1) twists, where k is odd and p1,…, pk are positive integers. The polynomial Δ2,3,7(x) is the well-known Lehmer polynomial, which is conjectured to have the smallest Mahler measure among all monic integer polynomials. We confirm that Δ2,3,7(x) has the smallest Mahler measure among the polynomials arising as Δp1,…,pk (x).

scholarly journals BIRACK SHADOW MODULES AND THEIR LINK INVARIANTS

2013 ◽  
Vol 22 (10) ◽  
pp. 1350056 ◽  
Author(s):  
SAM NELSON ◽  
KATIE PELLAND
Keyword(s):  
Associative Algebra ◽  
Alexander Polynomial ◽  
Link Invariants ◽  
Knots And Links

We introduce an associative algebra ℤ[X, S] associated to a birack shadow and define enhancements of the birack counting invariant for classical knots and links via representations of ℤ[X, S] known as shadow modules. We provide examples which demonstrate that the shadow module enhanced invariants are not determined by the Alexander polynomial or the unenhanced birack counting invariants.

On pretzel knots and Conjecture ℤ

2016 ◽  
Vol 25 (02) ◽  
pp. 1650012 ◽  
Author(s):  
Jesús Rodríguez-Viorato ◽  
Francisco Gonzaléz Acuña
Keyword(s):  
Equivalent Form ◽  
Pretzel Knots ◽  
Positive Integers

Conjecture [Formula: see text] is a knot theoretical equivalent form of the Kervaire conjecture. We show that Conjecture [Formula: see text] is true for all the pretzel knots of the form [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are odd positive integers.

scholarly journals Explicit formulae for two-bridge knot polynomials

2005 ◽  
Vol 78 (2) ◽  
pp. 149-166 ◽  
Author(s):  
Shinji Fukuhara
Keyword(s):  
Normal Form ◽  
Open Problem ◽  
Alexander Polynomial ◽  
Alexander Polynomials ◽  
Elementary Number ◽  
Conway Polynomial ◽  
Coprime Integers ◽  
Explicit Formulae ◽  
Knots And Links

AbstractA two-bridge knot (or link) can be characterized by the so-called Schubert normal formKp, qwherepandqare positive coprime integers. Associated toKp, qthere are the Conway polynomial ▽kp, q(z)and the normalized Alexander polynomial Δkp, q(t). However, it has been open problem how ▽kp, q(z) and Δkp, q(t) are expressed in terms ofpandq. In this note, we will give explicit formulae for the Conway polynomials and the normalized Alexander polynomials in the case of two-bridge knots and links. This is done using elementary number theoretical functions inpandq.

Sliceness of alternating pretzel knots and links

2020 ◽  
Vol 282 ◽  
pp. 107317
Author(s):  
Kengo Kishimoto ◽  
Tetsuo Shibuya ◽  
Tatsuya Tsukamoto
Keyword(s):  
Pretzel Knots ◽  
Knots And Links

scholarly journals Colored HOMFLY polynomials for the pretzel knots and links

2015 ◽  
Vol 2015 (7) ◽  
Author(s):  
A. Mironov ◽  
A. Morozov ◽  
A. Sleptsov
Keyword(s):  
Pretzel Knots ◽  
Homfly Polynomials ◽  
Knots And Links

Identifying non-pseudo-alternating knots by using the free factor property of minimal genus Seifert surfaces

2021 ◽  
Author(s):  
Keisuke Himeno ◽  
Masakazu Teragaito
Keyword(s):  
Seifert Surface ◽  
Major Difficulty ◽  
Flat Surfaces ◽  
Minimal Genus ◽  
Pretzel Knots ◽  
Seifert Surfaces ◽  
Manifold Theory ◽  
Knots And Links

Pseudo-alternating knots and links are defined constructively via their Seifert surfaces. By performing Murasugi sums of primitive flat surfaces, such a knot or link is obtained as the boundary of the resulting surface. Conversely, it is hard to determine whether a given knot or link is pseudo-alternating or not. A major difficulty is the lack of criteria to recognize whether a given Seifert surface is decomposable as a Murasugi sum. In this paper, we propose a new idea to identify non-pseudo-alternating knots. Combining with the uniqueness of minimal genus Seifert surface obtained through sutured manifold theory, we demonstrate that two infinite classes of pretzel knots are not pseudo-alternating.

scholarly journals Colored knot polynomials for arbitrary pretzel knots and links

2015 ◽  
Vol 743 ◽  
pp. 71-74 ◽  
Author(s):  
D. Galakhov ◽  
D. Melnikov ◽  
A. Mironov ◽  
A. Morozov ◽  
A. Sleptsov
Keyword(s):  
Pretzel Knots ◽  
Knots And Links

On divisibility between Alexander polynomials of Turk’s head links

2018 ◽  
Vol 27 (06) ◽  
pp. 1850040 ◽  
Author(s):  
Atsushi Takemura
Keyword(s):  
Alexander Polynomial ◽  
Alexander Polynomials ◽  
Positive Integers

We show that for any positive integers [Formula: see text] and [Formula: see text], the Alexander polynomial of the [Formula: see text]-Turk’s head link is divisible by that of the [Formula: see text]-Turk’s head link.

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.

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