Triple crossing number and double crossing braid index

Traditionally, knot theorists have considered projections of knots where there are two strands meeting at every crossing. A triple crossing is a crossing where three strands meet at a single point, such that each strand bisects the crossing. In this paper we find a relationship between the triple crossing number and the double crossing braid index for unoriented links, namely [Formula: see text]. This yields a new method for determining braid indices. We find an infinite family of knots that achieve equality, which allows us to determine both the double crossing braid index and the triple crossing number of these knots.

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Braid index bounds ropelength from below

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216520500194 ◽

2020 ◽

Vol 29 (04) ◽

pp. 2050019

Author(s):

Yuanan Diao

Keyword(s):

Lower Bound ◽

General Formula ◽

Crossing Number ◽

Crossing Numbers ◽

Braid Index ◽

Alternating Links

For an unoriented link [Formula: see text], let [Formula: see text] be the ropelength of [Formula: see text]. It is known that in general [Formula: see text] is at least of the order [Formula: see text], and at most of the order [Formula: see text] where [Formula: see text] is the minimum crossing number of [Formula: see text]. Furthermore, it is known that there exist families of (infinitely many) links with the property [Formula: see text]. A long standing open conjecture states that if [Formula: see text] is alternating, then [Formula: see text] is at least of the order [Formula: see text]. In this paper, we show that the braid index of a link also gives a lower bound of its ropelength. More specifically, we show that there exists a constant [Formula: see text] such that [Formula: see text] for any [Formula: see text], where [Formula: see text] is the largest braid index among all braid indices corresponding to all possible orientation assignments of the components of [Formula: see text] (called the maximum braid index of [Formula: see text]). Consequently, [Formula: see text] for any link [Formula: see text] whose maximum braid index is proportional to its crossing number. In the case of alternating links, the maximum braid indices for many of them are proportional to their crossing numbers hence the above conjecture holds for these alternating links.

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On the Minimal Crossing Number and the Braid Index of Links

Canadian Journal of Mathematics ◽

10.4153/cjm-1993-007-x ◽

1993 ◽

Vol 45 (1) ◽

pp. 117-131 ◽

Cited By ~ 17

Author(s):

Yoshiyuki Ohyama

Keyword(s):

Bipartite Graph ◽

Crossing Number ◽

Braid Index

AbstractIn this paper we prove an inequality that involves the minimal crossing number and the braid index of links by estimating Murasugi and Przytycki’s index for a planar bipartite graph.

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AN INFINITE FAMILY OF KNOTS WHOSE MOSAIC NUMBER IS REALIZED IN NON-REDUCED PROJECTIONS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216513500363 ◽

2013 ◽

Vol 22 (07) ◽

pp. 1350036 ◽

Cited By ~ 3

Author(s):

LEWIS D. LUDWIG ◽

ERICA L. EVANS ◽

JOSEPH S. PAAT

Keyword(s):

Infinite Family ◽

Crossing Number

Lomonaco and Kauffman [Quantum knots and mosaics, Quantum Inf. Process. 7(2–3) (2008) 85–115] introduced the notion of knot mosaics in their work on quantum knots. It is conjectured that knot mosaic type is a complete invariant of tame knots. In this paper, we answer a question of C. Adams by constructing an infinite family of knots whose mosaic number can be realized only when the crossing number is not. That is, there is an infinite family of non-minimal knots whose mosaic numbers are known.

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A new method for multi-pass single-point incremental forming of vertical wall square box with small corner radius

Materials Research Innovations ◽

10.1179/1432891714z.0000000001148 ◽

2015 ◽

Vol 19 (sup5) ◽

pp. S5-543-S5-545

Author(s):

Z. Liuru ◽

Z. Yinmei

Keyword(s):

Incremental Forming ◽

Single Point ◽

Vertical Wall ◽

New Method ◽

Single Point Incremental Forming ◽

Corner Radius ◽

Square Box

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Reconstructing the wave speed and the source

10.22541/au.161925842.29764673/v1 ◽

2021 ◽

Author(s):

Amin Boumenir ◽

Vu Kim Tuan

Keyword(s):

Inverse Problem ◽

Wave Equation ◽

Wave Speed ◽

Simple Procedure ◽

Single Point ◽

Spectral Estimation ◽

New Method ◽

Estimation Techniques ◽

Nodal Lines

We are concerned with the inverse problem of recovering the unknown wave speed and also the source in a multidimensional wave equation. We show that the wave speed coefficient can be reconstructed from the observations of the solution taken at a single point. For the source, we may need a sequence of observation points due to the presence of multiple spectrum and nodal lines. This new method, based on spectral estimation techniques, leads to a simple procedure that delivers both uniqueness and reconstruction of the coefficients at the same time.

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Decomposition Strategies for the Synthesis of Single Input-Single Output and Dual Input-Single Output Compliant Mechanisms

Volume 8: 31st Mechanisms and Robotics Conference, Parts A and B ◽

10.1115/detc2007-35449 ◽

2007 ◽

Author(s):

Charles Kim

Keyword(s):

Compliant Mechanisms ◽

Compliant Mechanism ◽

Single Point ◽

Building Blocks ◽

New Method ◽

Single Input Single Output ◽

Single Output ◽

Alternative Method ◽

Decomposition Strategies ◽

Single Input

In this paper a new method for the synthesis of compliant mechanism topologies is presented which involves the decomposition of motion requirements into more easily solved sub-problems. The decomposition strategies are presented and demonstrated for both single input-single output (SISO) and dual input-single output (DISO) planar compliant mechanisms. The methodology makes use of the single point synthesis (SPS) which effectively generates topologies which satisfy motion requirements at one point by assembling compliant building blocks. The SPS utilizes compliance and stiffness ellipsoids to characterize building blocks and to combine them in an intelligent manner. Both the SISO and DISO problems are decomposed into sub-problems which may be addressed by the SPS. The decomposition strategies are demonstrated with illustrative example problems. This paper presents an alternative method for the synthesis of compliant mechanisms which augments designer insight.

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Centre modes in inviscid swirling flows and their application to the stability of the Batchelor vortex

Journal of Fluid Mechanics ◽

10.1017/s0022112006004447 ◽

2007 ◽

Vol 576 ◽

pp. 325-348 ◽

Cited By ~ 30

Author(s):

C. J. HEATON

Keyword(s):

Continuous Spectrum ◽

Single Point ◽

Stability Boundary ◽

Infinite Family ◽

Neutral Curve ◽

Swirling Flows ◽

Mean Flow ◽

Linearized Euler Equations ◽

Numerical Studies ◽

The Stability

We identify a family of centre-mode disturbances to inviscid swirling flows such as jets, wakes and other vortices. The centre modes form an infinite family of modes, increasingly concentrated near to the symmetry axis of the mean flow, and whose frequencies accumulate to a single point in the complex plane. This asymptotic accumulation allows analytical progress to be made, including a theoretical stability boundary, inO(1) parameter regimes. The modes are located close to the continuous spectrum of the linearized Euler equations, and the theory is closely related to that of the continuous spectrum. We illustrate our analysis with the inviscid Batchelor vortex, defined by swirl parameterq. We show that the inviscid instabilities found in previous numerical studies are in fact the first members of an infinite set of centre modes of the type we describe. We investigate the inviscid neutral curve, and find good agreement of the neutral curve predicted by the analysis with the results of numerical computations. We find that the unstable region is larger than previously reported. In particular, the value ofqabove which the inviscid vortex stabilizes is significantly larger than previously reported and in agreement with a long-standing theoretical prediction.

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A NOTE ON THE CROSSING NUMBER AND THE BRAID INDEX FOR VIRTUAL LINKS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s021821651000825x ◽

2010 ◽

Vol 19 (07) ◽

pp. 867-880

Author(s):

YASUSHI TAKEDA

Keyword(s):

Crossing Number ◽

Virtual Link ◽

Braid Index ◽

Classical Link ◽

Virtual Links

It is well known that any virtual link is described as the closure of a virtual braid. Therefore, we can define the virtual braid index. Ohyama proved an inequality for the crossing number and the braid index of a classical link. In this paper, we prove an analogous inequality for the (total) crossing number and the braid index of a virtual link.

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Generating links that are both quasi-alternating and almost alternating

Journal of Knot Theory and Its Ramifications ◽

10.1142/s021821652050090x ◽

2020 ◽

pp. 2050090

Author(s):

Hamid Abchir ◽

Mohammed Sabak

Keyword(s):

Infinite Family ◽

Crossing Number ◽

Alternating Links ◽

Alternating Link

We construct an infinite family of links which are both almost alternating and quasi-alternating from a given either almost alternating diagram representing a quasi-alternating link, or connected and reduced alternating tangle diagram. To do that we use what we call a dealternator extension which consists in replacing the dealternator by a rational tangle extending it. We note that all non-alternating and quasi-alternating Montesinos links can be obtained in that way. We check that all the obtained quasi-alternating links satisfy Conjecture 3.1 of Qazaqzeh et al. (JKTR 22 (6), 2013), that is the crossing number of a quasi-alternating link is less than or equal to its determinant. We also prove that the converse of Theorem 3.3 of Qazaqzeh et al. (JKTR 24 (1), 2015) is false.

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