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.

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 Writhe-like invariants of alternating links

2021 ◽  
Vol 30 (01) ◽  
pp. 2150004
Author(s):  
Yuanan Diao ◽  
Van Pham
Keyword(s):  
Jones Polynomial ◽  
Link Diagram ◽  
Link Invariant ◽  
Link Invariants ◽  
Link Diagrams ◽  
Alternating Links ◽  
Alternating Link

It is known that the writhe calculated from any reduced alternating link diagram of the same (alternating) link has the same value. That is, it is a link invariant if we restrict ourselves to reduced alternating link diagrams. This is due to the fact that reduced alternating link diagrams of the same link are obtainable from each other via flypes and flypes do not change writhe. In this paper, we introduce several quantities that are derived from Seifert graphs of reduced alternating link diagrams. We prove that they are “writhe-like” invariants, namely they are not general link invariants, but are invariants when restricted to reduced alternating link diagrams. The determination of these invariants are elementary and non-recursive so they are easy to calculate. We demonstrate that many different alternating links can be easily distinguished by these new invariants, even for large, complicated knots for which other invariants such as the Jones polynomial are hard to compute. As an application, we also derive an if and only if condition for a strongly invertible rational link.

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 Cn-moves and the difference of Jones polynomials for links

2017 ◽  
Vol 26 (05) ◽  
pp. 1750029 ◽  
Author(s):  
Ryo Nikkuni
Keyword(s):  
Jones Polynomial ◽  
Laurent Polynomial ◽  
Link Invariant ◽  
Oriented Link ◽  
Jones Polynomials ◽  
The Difference ◽  
Polynomial Formula

The Jones polynomial [Formula: see text] for an oriented link [Formula: see text] is a one-variable Laurent polynomial link invariant discovered by Jones. For any integer [Formula: see text], we show that: (1) the difference of Jones polynomials for two oriented links which are [Formula: see text]-equivalent is divisible by [Formula: see text], and (2) there exists a pair of two oriented knots which are [Formula: see text]-equivalent such that the difference of the Jones polynomials for them equals [Formula: see text].

scholarly journals The Singular Temperley-Lieb Category

ISRN Geometry ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Carmen Caprau ◽  
Joel Smith
Keyword(s):  
Category Theory ◽  
Point Of View ◽  
State Model ◽  
Link Invariant ◽  
Polynomial Invariants ◽  
Oriented Link ◽  
Theory Point

We introduce and study the singular Temperley-Lieb category over ℤ[q,q-1], which is a free pivotal category over two self-dual generators and is an extension of the (classical) Temperley-Lieb category. Our construction is motivated by a state model for the sl(2)  polynomial of an oriented link and provides a categorical perspective to this link invariant. We also construct a couple of polynomial invariants for oriented tangles from category theory point of view.

scholarly journals A twisted link invariant derived from a virtual link invariant

2017 ◽  
Vol 26 (09) ◽  
pp. 1743007
Author(s):  
Naoko Kamada
Keyword(s):  
Knot Theory ◽  
Orientable Surface ◽  
Link Diagram ◽  
Link Invariant ◽  
Virtual Knot ◽  
Chord Diagrams ◽  
Link Diagrams ◽  
Virtual Links ◽  
Virtual Knot Theory ◽  
Double Coverings

Virtual knot theory is a generalization of knot theory which is based on Gauss chord diagrams and link diagrams on closed oriented surfaces. A twisted knot is a generalization of a virtual knot, which corresponds to a link diagram on a possibly non-orientable surface. In this paper, we discuss an invariant of twisted links which is obtained from the JKSS invariant of virtual links by use of double coverings. We also discuss some properties of double covering diagrams.

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.

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].

scholarly journals Identifying the Invariants for Classical Knots and Links from the Yokonuma–Hecke Algebras

2018 ◽  
Vol 2020 (1) ◽  
pp. 214-286 ◽  
Author(s):  
Maria Chlouveraki ◽  
Jesús Juyumaya ◽  
Konstantinos Karvounis ◽  
Sofia Lambropoulou
Keyword(s):  
Hecke Algebras ◽  
Hecke Algebra ◽  
Type A ◽  
Closed Formula ◽  
Link Invariant ◽  
Polynomial Invariants ◽  
Oriented Link ◽  
Linking Numbers ◽  
Markov Trace ◽  
Knots And Links

Abstract We announce the existence of a family of new 2-variable polynomial invariants for oriented classical links defined via a Markov trace on the Yokonuma–Hecke algebra of type A. Yokonuma–Hecke algebras are generalizations of Iwahori–Hecke algebras, and this family contains the HOMFLYPT polynomial, the famous 2-variable invariant for classical links arising from the Iwahori–Hecke algebra of type A. We show that these invariants are topologically equivalent to the HOMFLYPT polynomial on knots, but not on links, by providing pairs of HOMFLYPT-equivalent links that are distinguished by our invariants. In order to do this, we prove that our invariants can be defined diagrammatically via a special skein relation involving only crossings between different components. We further generalize this family of invariants to a new 3-variable skein link invariant that is stronger than the HOMFLYPT polynomial. Finally, we present a closed formula for this invariant, by W. B. R. Lickorish, that uses HOMFLYPT polynomials of sublinks and linking numbers of a given oriented link.

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