Sharp Bounds on Some Classical Knot Invariants

2003 ◽  
Vol 12 (06) ◽  
pp. 805-817
Author(s):  
C. Kearton ◽  
S. M. J. Wilson
Keyword(s):  
Alexander Polynomial ◽  
Sharp Bounds ◽  
Knot Invariants ◽  
Bridge Number

There are obvious inequalities relating the Nakanishi index of a knot, the bridge number, the degree 2n of the Alexander polynomial and the length of the chain of Alexander ideals. We give examples for every positive value of n to show that these bounds are sharp.

MANY CLASSICAL KNOT INVARIANTS ARE NOT VASSILIEV INVARIANTS

1994 ◽  
Vol 03 (01) ◽  
pp. 7-10 ◽  
Author(s):  
JOHN DEAN
Keyword(s):  
Crossing Number ◽  
Torus Knots ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Vassiliev Invariant ◽  
Bridge Number

We show that under twisting, a Vassiliev invariant of order n behaves like a polynomial of degree at most n. This greatly restricts the values that a Vassiliev invariant can take, for example, on the (2, m) torus knots. In particular, this implies that many classical numerical knot invariants such as the signature, genus, bridge number, crossing number, and unknotting number are not Vassiliev invariants.

RECURSION FORMULAS FOR SOME ABELIAN KNOT INVARIANTS

2000 ◽  
Vol 09 (03) ◽  
pp. 413-422 ◽  
Author(s):  
WAYNE H. STEVENS
Keyword(s):  
Recursion Formula ◽  
Alexander Polynomial ◽  
Square Roots ◽  
Knot Invariants ◽  
Homology Groups ◽  
Cyclic Covers ◽  
Recursion Formulas ◽  
Branched Cyclic Covers ◽  
Linear Recursion ◽  
Fold Cover

Let K be a tame knot in S3. We show that the sequence of cyclic resultants of the Alexander polynomial of K satisfies a linear recursion formula with integral coefficients. This means that the orders of the first homology groups of the branched cyclic covers of K can be computed recursively. We further establish the existence of a recursion formula that generates sequences which contain the square roots of the orders for the odd-fold covers that contain the square roots of the orders for the even-fold covers quotiented by the order for the two-fold cover. (That these square roots are all integers follows from a theorem of Plans.)

CASSON KNOT INVARIANTS OF PERIODIC KNOTS WITH RATIONAL QUOTIENTS

2007 ◽  
Vol 16 (04) ◽  
pp. 439-460 ◽  
Author(s):  
HEE JEONG JANG ◽  
SANG YOUL LEE ◽  
MYOUNGSOO SEO
Keyword(s):  
Normal Form ◽  
Recurrence Formula ◽  
Alexander Polynomial ◽  
Knot Invariants ◽  
Knot Invariant

We give a formula for the Casson knot invariant of a p-periodic knot in S3 whose quotient link is a 2-bridge link with Conway's normal form C(2, 2n1, -2, 2n2, …, 2n2m, 2) via the integers p, n1, n2, …, n2m(p ≥ 2 and m ≥ 1). As an application, for any integers n1, n2, ≥, n2m with the same sign, we determine the Δ-unknotting number of a p-periodic knot in S3 whose quotient is a 2-bridge link C(2, 2n1, -2, 2n2, ≥, 2n2m, 2) in terms of p, n1, n2, ≥, n2m. In addition, a recurrence formula for calculating the Alexander polynomial of the 2-bridge knot with Conway's normal form C(2n1, 2n2, ≥, 2nm) via the integers n1,n2, ≥, nm is included.

Classical Invariants for Holomic Knots

2003 ◽  
Vol 12 (06) ◽  
pp. 751-765
Author(s):  
Magnus Bergqvist
Keyword(s):  
Knot Invariants ◽  
Braid Index ◽  
Bridge Number

There are some well known indices of knots that give rise to classical knot invariants when minimized for a knot type. By classical knot invariants I mean the minimal number of crossing points, bridge number, braid index, genus and unknotting number. Considering these indices when restricting the minimum over holonomic knots, some of the minima stay the same while others differ. This supplies us with a new set of knot invariants: the holonomic counterparts of the classical knot invariants.

scholarly journals Regional knot invariants

2017 ◽  
Vol 26 (06) ◽  
pp. 1742006
Author(s):  
Zhiqing Yang
Keyword(s):  
Alexander Polynomial ◽  
Link Diagram ◽  
Polynomial Invariant ◽  
Knot Invariants ◽  
Planar Region ◽  
Knot Invariant ◽  
The Matrix

In this paper, a regional knot invariant is constructed. Like the Wirtinger presentation of a knot group, each planar region contributes a generator, and each crossing contributes a relation. The invariant is called a tridle of the link. As in the quandle theory, one can define Alexander quandle and get Alexander polynomial from it. For link diagram, one can also define a linear tridle and its presentation matrix. A polynomial invariant can be derived from the matrix just like the Alexander polynomial case.

scholarly journals Embedding spheres in knot traces

2021 ◽  
Vol 157 (10) ◽  
pp. 2242-2279
Author(s):  
Peter Feller ◽  
Allison N. Miller ◽  
Matthias Nagel ◽  
Patrick Orson ◽  
Mark Powell ◽  
...  
Keyword(s):  
Fundamental Group ◽  
Homotopy Group ◽  
Alexander Polynomial ◽  
Homotopy Equivalent ◽  
Knot Invariants

Abstract The trace of the $n$ -framed surgery on a knot in $S^{3}$ is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded $2$ -sphere whose complement has abelian fundamental group. Our characterisation is in terms of classical and computable $3$ -dimensional knot invariants. For each $n$ , this provides conditions that imply a knot is topologically $n$ -shake slice, directly analogous to the result of Freedman and Quinn that a knot with trivial Alexander polynomial is topologically slice.

scholarly journals Sharp Bounds on Davenport-Schinzel Sequences of Every Order

2015 ◽  
Vol 62 (5) ◽  
pp. 1-40 ◽  
Author(s):  
Seth Pettie
Keyword(s):  
Sharp Bounds

The sharp bounds of the second and third Hankel determinants for the class 𝓢𝓛*

2020 ◽  
Vol 70 (4) ◽  
pp. 849-862
Author(s):  
Shagun Banga ◽  
S. Sivaprasad Kumar
Keyword(s):  
Triangle Inequality ◽  
Starlike Functions ◽  
Sharp Bound ◽  
The Novel ◽  
Sharp Bounds ◽  
Hankel Determinants ◽  
Lemniscate Of Bernoulli ◽  
Carathéodory Functions ◽  
The Right

AbstractIn this paper, we use the novel idea of incorporating the recently derived formula for the fourth coefficient of Carathéodory functions, in place of the routine triangle inequality to achieve the sharp bounds of the Hankel determinants H3(1) and H2(3) for the well known class 𝓢𝓛* of starlike functions associated with the right lemniscate of Bernoulli. Apart from that the sharp bound of the Zalcman functional: $\begin{array}{} |a_3^2-a_5| \end{array}$ for the class 𝓢𝓛* is also estimated. Further, a couple of interesting results of 𝓢𝓛* are also discussed.

scholarly journals Coefficient inequalities for a subclass of Bazilevič functions

2020 ◽  
Vol 53 (1) ◽  
pp. 27-37
Author(s):  
Sa’adatul Fitri ◽  
Derek K. Thomas ◽  
Ratno Bagus Edy Wibowo ◽  
Keyword(s):  
Recent Work ◽  
Sharp Bounds ◽  
Coefficient Problems ◽  
Coefficient Inequalities ◽  
Bazilevic Functions

AbstractLet f be analytic in {\mathbb{D}}=\{z:|z\mathrm{|\hspace{0.17em}\lt \hspace{0.17em}1\}} with f(z)=z+{\sum }_{n\mathrm{=2}}^{\infty }{a}_{n}{z}^{n}, and for α ≥ 0 and 0 < λ ≤ 1, let { {\mathcal B} }_{1}(\alpha ,\lambda ) denote the subclass of Bazilevič functions satisfying \left|f^{\prime} (z){\left(\frac{z}{f(z)}\right)}^{1-\alpha }-1\right|\lt \lambda for 0 < λ ≤ 1. We give sharp bounds for various coefficient problems when f\in { {\mathcal B} }_{1}(\alpha ,\lambda ), thus extending recent work in the case λ = 1.

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