scholarly journals Braid group and leveling of a knot

2020 ◽  
pp. 1-24
Author(s):  
Sangbum Cho ◽  
Yuya Koda ◽  
Arim Seo
Keyword(s):  
Braid Group ◽  
Upper Bounds ◽  
Numerical Invariant ◽  
Level Number ◽  
Text Length ◽  
Genus Formula ◽  
Minimal Formula ◽  
Bridge Position

Any knot [Formula: see text] in genus-[Formula: see text] [Formula: see text]-bridge position can be moved by isotopy to lie in a union of [Formula: see text] parallel tori tubed by [Formula: see text] tubes so that [Formula: see text] intersects each tube in two spanning arcs, which we call a leveling of the position. The minimal [Formula: see text] for which this is possible is an invariant of the position, called the level number. In this work, we describe the leveling by the braid group on two points in the torus, which yields a numerical invariant of the position, called the [Formula: see text]-length. We show that the [Formula: see text]-length equals the level number. We then find braid descriptions for [Formula: see text]-positions of all [Formula: see text]-bridge knots providing upper bounds for their level numbers and also show that the [Formula: see text]-pretzel knot has level number two.

On cofactors of subnormal subgroups

2016 ◽  
Vol 15 (09) ◽  
pp. 1650169
Author(s):  
Victor Monakhov ◽  
Irina Sokhor
Keyword(s):  
Finite Group ◽  
Soluble Group ◽  
Upper Bounds ◽  
Finite Soluble Group ◽  
Nilpotent Length ◽  
Derived Length ◽  
Text Length ◽  
Subnormal Subgroups

For a soluble finite group [Formula: see text] and a prime [Formula: see text] we let [Formula: see text], [Formula: see text]. We obtain upper bounds for the rank, the nilpotent length, the derived length, and the [Formula: see text]-length of a finite soluble group [Formula: see text] in terms of [Formula: see text] and [Formula: see text].

scholarly journals Determining the doubly slice genera of prime knots with up to 12 crossings

2021 ◽  
Author(s):  
Lucia P. Karageorghis ◽  
Frank Swenton
Keyword(s):  
Crossing Number ◽  
Genus Formula ◽  
Minimal Formula

For a knot [Formula: see text], the doubly slice genus [Formula: see text] is the minimal [Formula: see text] such that [Formula: see text] divides a closed, orientable, and unknotted surface of genus [Formula: see text] embedded in [Formula: see text]. In this paper, we identify the doubly slice genera of 2909 of the 2977 prime knots which have a crossing number of 12 or fewer.

scholarly journals Upper bounds for some Brill–Noether loci over a finite field

2018 ◽  
Vol 14 (03) ◽  
pp. 739-749 ◽  
Author(s):  
Kamal Khuri-Makdisi
Keyword(s):  
Line Bundle ◽  
Finite Field ◽  
Classical Result ◽  
Base Point ◽  
Upper Bounds ◽  
Line Bundles ◽  
Effective Version ◽  
General Line ◽  
Genus Formula ◽  
Explicit Constant

Let [Formula: see text] be a smooth projective algebraic curve of genus [Formula: see text], over the finite field [Formula: see text]. A classical result of H. Martens states that the Brill–Noether locus of line bundles [Formula: see text] in [Formula: see text] with [Formula: see text] and [Formula: see text] is of dimension at most [Formula: see text], under conditions that hold when such an [Formula: see text] is both effective and special. We show that the number of such [Formula: see text] that are rational over [Formula: see text] is bounded above by [Formula: see text], with an explicit constant [Formula: see text] that grows exponentially with [Formula: see text]. Our proof uses the Weil estimates for function fields, and is independent of Martens’ theorem. We apply this bound to give a precise lower bound of the form [Formula: see text] for the probability that a line bundle in [Formula: see text] is base point free. This gives an effective version over finite fields of the usual statement that a general line bundle of degree [Formula: see text] is base point free. This is applicable to the author’s work on fast Jacobian group arithmetic for typical divisors on curves.

scholarly journals ON WORD REVERSING IN BRAID GROUPS

2006 ◽  
Vol 16 (05) ◽  
pp. 941-957 ◽  
Author(s):  
PATRICK DEHORNOY ◽  
BERT WIEST
Keyword(s):  
Braid Group ◽  
Upper Bounds ◽  
Braid Groups ◽  
Space Complexity ◽  
The Other ◽  
Reduction Algorithm ◽  
Group Presentation ◽  
Linear Factor ◽  
Counter Example ◽  
Other Hand

It has been conjectured that in a braid group, or more generally in a Garside group, applying any sequence of monotone equivalences and word reversings can increase the length of a word by at most a linear factor depending on the group presentation only. We give a counter-example to this conjecture, but, on the other hand, we establish length upper bounds for the case when only right reversing is involved. We also state a new conjecture which would, like the above one, imply that the space complexity of the handle reduction algorithm is linear.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

Energy of spherical fuzzy graphs

2020 ◽  
pp. 321-332
Author(s):  
S. Yahya Mohamed ◽  
A. Mohamed Ali
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Fuzzy Graph ◽  
Lower And Upper Bounds ◽  
Fuzzy Graphs

In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.

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