The t3, moves conjecture for oriented links with matched diagrams

1990 ◽  
Vol 108 (1) ◽  
pp. 55-61 ◽  
Author(s):  
Józef H. Przytycki
Keyword(s):  
Local Change ◽  
Link Diagram ◽  
Oriented Link ◽  
Positive Half

The local change in an oriented link diagram which replaces by k positive half-twists is called a tk move. For k even, the local change replacing by is called a tk move. For an unoriented diagram define a k-move, replacing by for any k. The following conjecture was stated in [14] and [10].

The self-smoothing number of knots and links

2018 ◽  
Vol 27 (12) ◽  
pp. 1850070
Author(s):  
Hideo Takioka
Keyword(s):  
The Self ◽  
Link Diagram ◽  
Oriented Link ◽  
Two Component ◽  
Crossing Point ◽  
Knots And Links

We call smoothing a self-crossing point of an oriented link diagram self-smoothing. By self-smoothing repeatedly, we obtain an oriented link diagram without self-crossing points. In this paper, we show that every knot has an oriented diagram which becomes a two-component oriented link diagram without self-crossing points by a single self-smoothing.

scholarly journals Quantum (sln, ∧V n ) link invariant and matrix factorizations

2011 ◽  
Vol 204 ◽  
pp. 69-123
Author(s):  
Yasuyoshi Yonezawa
Keyword(s):  
Matrix Factorizations ◽  
Link Diagram ◽  
Link Invariant ◽  
Poincaré Polynomial ◽  
Oriented Link

AbstractIn this paper, we give a generalization of Khovanov-Rozansky homology. We define a homology associated to the quantum (sln, ∧Vn) link invariant, where∧Vnis the set of fundamental representations ofUq(sln). In the case of an oriented link diagram composed of[k, 1]-crossings, we define a homology and prove that the homology is invariant under Reidemeister II and III moves. In the case of an oriented link diagram composed of general[i,j]-crossings, we define a normalized Poincaré polynomial of homology and prove that the normalized Poincaré polynomial is a link invariant.

GENERALIZED SPIN MODELS

1994 ◽  
Vol 03 (04) ◽  
pp. 465-475 ◽  
Author(s):  
KENICHI KAWAGOE ◽  
AKIHIRO MUNEMASA ◽  
YASUO WATATANI
Keyword(s):  
Partition Function ◽  
Spin Model ◽  
Symmetry Condition ◽  
Type Ii ◽  
Link Diagram ◽  
Spin Models ◽  
Oriented Link ◽  
Reidemeister Moves

We introduce a generalization of spin models by dropping the symmetry condition. The partition function of a generalized spin model on a connected oriented link diagram is invariant under Reidemeister moves of type II and III, giving an invariant for oriented links.

scholarly journals Rectangular Seifert circles and arcs system

2014 ◽  
Vol 23 (08) ◽  
pp. 1450041
Author(s):  
Tatsuo Ando ◽  
Chuichiro Hayashi ◽  
Miwa Hayashi
Keyword(s):  
Negative Integer ◽  
Vertical Line ◽  
Horizontal Line ◽  
Link Diagram ◽  
Line Segments ◽  
Oriented Link ◽  
Link Diagrams ◽  
Ambient Isotopy

Rectangular diagrams of links are link diagrams in the plane ℝ2 such that they are composed of vertical line segments and horizontal line segments and vertical segments go over horizontal segments at all crossings. Cromwell and Dynnikov showed that rectangular diagrams of links are useful for deciding whether a given link is split or not, and whether a given knot is trivial or not. We show in this paper that an oriented link diagram D with c(D) crossings and s(D) Seifert circles can be deformed by an ambient isotopy of ℝ2 into a rectangular diagram with at most c(D) + 2s(D) vertical segments, and that, if D is connected, at most 2c(D) + 2 - w(D) vertical segments, where w(D) is a certain non-negative integer. In order to obtain these results, we show that the system of Seifert circles and arcs substituting for crossings can be deformed by an ambient isotopy of ℝ2 so that Seifert circles are rectangles composed of two vertical line segments and two horizontal line segments and arcs are vertical line segments, and that we can obtain a single circle from a connected link diagram by smoothing operations at the crossings regardless of orientation.

scholarly journals Quantum (sln, ∧V n) link invariant and matrix factorizations

2011 ◽  
Vol 204 ◽  
pp. 69-123 ◽  
Author(s):  
Yasuyoshi Yonezawa
Keyword(s):  
Matrix Factorizations ◽  
Link Diagram ◽  
Link Invariant ◽  
Poincaré Polynomial ◽  
Oriented Link

AbstractIn this paper, we give a generalization of Khovanov-Rozansky homology. We define a homology associated to the quantum (sln, ∧Vn) link invariant, where ∧Vn is the set of fundamental representations of Uq(sln). In the case of an oriented link diagram composed of [k, 1]-crossings, we define a homology and prove that the homology is invariant under Reidemeister II and III moves. In the case of an oriented link diagram composed of general [i,j]-crossings, we define a normalized Poincaré polynomial of homology and prove that the normalized Poincaré polynomial is a link invariant.

scholarly journals The Rasmussen Invariant, Four-genus, and Three-genus of an Almost Positive Knot Are Equal

2014 ◽  
Vol 57 (2) ◽  
pp. 431-438 ◽  
Author(s):  
Keiji Tagami
Keyword(s):  
Link Diagram ◽  
Oriented Link ◽  
Genus One

AbstractAn oriented link is positive if it has a link diagram whose crossings are all positive. An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing. It is known that the Rasmussen invariant, 4-genus, and 3-genus of a positive knot are equal. In this paper, we prove that the Rasmussen invariant, 4-genus, and 3-genus of an almost positive knot are equal. Moreover, we determine the Rasmussen invariant of an almost positive knot in terms of its almost positive knot diagram. As corollaries, we prove that all almost positive knots are not homogeneous, and there is no almost positive knot of 4-genus one.

Knot invariants with multiple skein relations

2018 ◽  
Vol 27 (02) ◽  
pp. 1850017 ◽  
Author(s):  
Zhiqing Yang
Keyword(s):  
Link Diagram ◽  
Oriented Link ◽  
Knot Invariants ◽  
Link Invariants ◽  
Kauffman Polynomial ◽  
Skein Relation

Given any oriented link diagram, one can construct knot invariants using skein relations. Usually such a skein relation contains three or four terms. In this paper, the author introduces several new ways to smooth a crossings, and uses a system of skein equations to construct link invariants. This invariant can also be modified by writhe to get a more powerful invariant. The modified invariant is a generalization of both the HOMFLYPT polynomial and the two-variable Kauffman polynomial. Using the diamond lemma, a simplified version of the modified invariant is given. It is easy to compute and is a generalization of the two-variable Kauffman polynomial.

Ce(III)/Ce(IV) in methanesulfonic acid as the positive half cell of a redox flow battery

2011 ◽  
Vol 56 (5) ◽  
pp. 2145-2153 ◽  
Author(s):  
P.K. Leung ◽  
C. Ponce de León ◽  
C.T.J. Low ◽  
F.C. Walsh
Keyword(s):  
Methanesulfonic Acid ◽  
Redox Flow Battery ◽  
Flow Battery ◽  
Half Cell ◽  
Positive Half

WAVELET PACKETS ASSOCIATED WITH NONUNIFORM MULTIRESOLUTION ANALYSIS ON POSITIVE HALF LINE

2013 ◽  
Vol 06 (01) ◽  
pp. 1350007 ◽  
Author(s):  
Vikram Sharma ◽  
P. Manchanda
Keyword(s):  
Multiresolution Analysis ◽  
Wavelet Packets ◽  
Half Line ◽  
State University ◽  
Spectral Pairs ◽  
Funct Anal ◽  
Positive Half

Gabardo and Nashed [Nonuniform multiresolution analysis and spectral pairs, J. Funct. Anal.158 (1998) 209–241] introduced the Nonuniform multiresolution analysis (NUMRA) whose translation set is not a group. Farkov [Orthogonal p-wavelets on ℝ+, in Proc. Int. Conf. Wavelets and Splines (St. Petersburg State University, St. Petersburg, 2005), pp. 4–26] studied multiresolution analysis (MRA) on positive half line and constructed associated wavelets. Meenakshi et al. [Wavelets associated with Nonuniform multiresolution analysis on positive half line, Int. J. Wavelets, Multiresolut. Inf. Process.10(2) (2011) 1250018, 27pp.] studied NUMRA on positive half line and proved the analogue of Cohen's condition for the NUMRA on positive half line. We construct the associated wavelet packets for such an MRA and study its properties.

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