scholarly journals HOMOMORPHIC EXPANSIONS FOR KNOTTED TRIVALENT GRAPHS

2013 ◽  
Vol 22 (01) ◽  
pp. 1250137 ◽  
Author(s):  
DROR BAR-NATAN ◽  
ZSUZSANNA DANCSO
Keyword(s):  
Finite Type ◽  
Simple Proof ◽  
Knot Theory ◽  
Large Set ◽  
Correction Factors ◽  
Connected Sums ◽  
Trivalent Graphs ◽  
Type Invariant ◽  
Knotted Trivalent Graphs ◽  
Standard Operations

It had been known since old times (works of Murakami–Ohtsuki, Cheptea–Le and the second author) that there exists a universal finite type invariant ("an expansion") Z old for knotted trivalent graphs (KTGs), and that it can be chosen to intertwine between some of the standard operations on KTGs and their chord-diagrammatic counterparts (so that relative to those operations, it is "homomorphic"). Yet perhaps the most important operation on KTGs is the "edge unzip" operation, and while the behavior of Z old under edge unzip is well understood, it is not plainly homomorphic as some "correction factors" appear. In this paper we present two equivalent ways of modifying Z old into a new expansion Z, defined on "dotted knotted trivalent graphs" (dKTGs), which is homomorphic with respect to a large set of operations. The first is to replace "edge unzips" by "tree connected sums", and the second involves somewhat restricting the circumstances under which edge unzips are allowed. As we shall explain, the newly defined class dKTG of KTGs retains all the good qualities that KTGs have — it remains firmly connected with the Drinfel'd theory of associators and it is sufficiently rich to serve as a foundation for an "algebraic knot theory". As a further application, we present a simple proof of the good behavior of the LMO invariant under the Kirby II (band-slide) move, first proven by Le, Murakami, Murakami and Ohtsuki.

Knot Theory and Statistical Mechanics

1997 ◽  
Vol 11 (01n02) ◽  
pp. 39-49 ◽  
Author(s):  
Louis H. Kauffman
Keyword(s):  
Statistical Mechanics ◽  
Finite Type ◽  
Simple Proof ◽  
Knot Theory ◽  
Scattering Amplitude ◽  
Basic Structure ◽  
Vassiliev Invariants ◽  
Link Invariants ◽  
Quantum Link

This paper gives a self-contained exposition of the basic structure of quantum link invariants as state summations for a vacuum-vacuum scattering amplitude. Models of Vaughan Jones are expressed in this context. A simple proof is given that an important subset of these invariants are built from Vassiliev invariants of finite type.

ON THE LE-MURAKAMI-OHTSUKI INVARIANT IN DEGREE 2 FOR SEVERAL CLASSES OF 3-MANIFOLDS

2003 ◽  
Vol 12 (01) ◽  
pp. 17-45 ◽  
Author(s):  
CATHERINE GILLE
Keyword(s):  
Finite Type ◽  
Dehn Surgery ◽  
Integral Homology ◽  
Graded Algebra ◽  
Trivalent Graphs ◽  
Type Invariant ◽  
Framed Link

The 3-manifolds invariant of Le, Murakami and Ohtsuki is the universal finite type invariant for integral homology spheres. It takes values in the graded algebra of trivalent graphs and it is known that its degree one part is essentially the Casson-Walker-Lescop invariant. Here we compute the degree two term for several classes of 3-manifolds. In particular, we give an expression of ω (ML) up to order 2 when MLis the 3-manifold obtained by Dehn surgery along a framed link L with one or two components.

scholarly journals A splicing formula for the LMO invariant

2020 ◽  
pp. 1-28
Author(s):  
Gwénaël Massuyeau ◽  
Delphine Moussard
Keyword(s):  
Finite Type ◽  
Rational Homology ◽  
Three Sphere ◽  
Type Invariant ◽  
Low Degrees ◽  
Surgery Formula

Abstract We prove a “splicing formula” for the LMO invariant, which is the universal finite-type invariant of rational homology three-spheres. Specifically, if a rational homology three-sphere M is obtained by gluing the exteriors of two framed knots $K_1 \subset M_1$ and $K_2\subset M_2$ in rational homology three-spheres, our formula expresses the LMO invariant of M in terms of the Kontsevich–LMO invariants of $(M_1,K_1)$ and $(M_2,K_2)$ . The proof uses the techniques that Bar-Natan and Lawrence developed to obtain a rational surgery formula for the LMO invariant. In low degrees, we recover Fujita’s formula for the Casson–Walker invariant, and we observe that the second term of the Ohtsuki series is not additive under “standard” splicing. The splicing formula also works when each $M_i$ comes with a link $L_i$ in addition to the knot $K_i$ , hence we get a “satellite formula” for the Kontsevich–LMO invariant.

scholarly journals A detailed look at chemical abundances in the Magellanic Clouds

2011 ◽  
Vol 7 (S283) ◽  
pp. 502-503
Author(s):  
Richard A. Shaw ◽  
Ting-Hui Lee ◽  
Letizia Stanghellini ◽  
James E. Davies ◽  
D. Anibal García-Hernández ◽  
...  
Keyword(s):  
Direct Methods ◽  
Abundance Ratio ◽  
Type I ◽  
Large Set ◽  
Agb Stars ◽  
Correction Factors ◽  
Chemical Abundances ◽  
Elemental Abundances ◽  
Low Metallicity ◽  
Low Mass

AbstractWe determine elemental abundances of He, N, O, Ne, S, and Ar in Magellanic Cloud planetary nebulae (PNe) using direct methods and a large set of observed ions, minimizing the need for ionization correction factors. In contrast to prior studies, we find a clear separation between Type I and non-Type I PNe in these low-metallicity environments, and no evidence that the O-N nucleosynthesis cycle is active in low-mass progenitors. We find that the S/O abundance ratio is anomalously low compared to H ii regions, confirming the “sulfur anomaly” found for Galactic PNe. We also found that Ne/O is elevated in some cases, raising the possibility that Ne yields in low-mass AGB stars may be enhanced at low metallicity.

2- AND 3-VARIATIONS AND FINITE TYPE INVARIANTS OF DEGREE 2 AND 3

2013 ◽  
Vol 22 (08) ◽  
pp. 1350042 ◽  
Author(s):  
MIGIWA SAKURAI
Keyword(s):  
Finite Type ◽  
Lower Bounds ◽  
Virtual Knots ◽  
Finite Type Invariants ◽  
Type Invariant

Goussarov, Polyak and Viro defined a finite type invariant and a local move called an n-variation for virtual knots. In this paper, we give the differences of the values of the finite type invariants of degree 2 and 3 between two virtual knots which can be transformed into each other by a 2- and 3-variation, respectively. As a result, we obtain lower bounds of the distance between long virtual knots by 2-variations and the distance between virtual knots by 3-variations by using the values of the finite type invariants of degree 2 and 3, respectively.

FINITE TYPE INVARIANTS FOR SINGULAR SURFACE BRAIDS ASSOCIATED WITH SIMPLE 1-HANDLE SURGERIES

2004 ◽  
Vol 13 (01) ◽  
pp. 1-11
Author(s):  
MASAHIDE IWAKIRI
Keyword(s):  
Finite Type ◽  
Singular Surface ◽  
Finite Type Invariants ◽  
Type Invariant

S. Kamada introduced finite type invariants of knotted surfaces in 4-space associated with finger moves and 1-handle surgeries. In this paper, we define finite type invariants of surface braids associated with simple 1-handle surgeries and prove that a certain set of finite type invariants controls all finite type invariants. As a consequence, we see that every finite type invariant is not a complete invariant.

scholarly journals ON CYCLES AND COVERINGS ASSOCIATED TO A KNOT

2013 ◽  
Vol 22 (13) ◽  
pp. 1350074
Author(s):  
LILYA LYUBICH ◽  
MIKHAIL LYUBICH
Keyword(s):  
Dynamical System ◽  
Abelian Group ◽  
Finite Type ◽  
Weak Topology ◽  
Knot Theory ◽  
Commutator Subgroup ◽  
Finite Abelian Group ◽  
Complete Classification ◽  
And Shifts

Let [Formula: see text] be a knot, G be the knot group, K be its commutator subgroup, and x be a distinguished meridian. Let Σ be a finite abelian group. The dynamical system introduced by Silver and Williams in [Augmented group systems and n-knots, Math. Ann.296 (1993) 585–593; Augmented group systems and shifts of finite type, Israel J. Math.95 (1996) 231–251] consisting of the set Hom (K, Σ) of all representations ρ : K → Σ endowed with the weak topology, together with the homeomorphism [Formula: see text] is finite, i.e. it consists of several cycles. In [Periodic orbits of a dynamical system related to a knot, J. Knot Theory Ramifications20(3) (2011) 411–426] we found the lengths of these cycles for Σ = ℤ/p,p is prime, in terms of the roots of the Alexander polynomial of the knot, mod p. In this paper we generalize this result to a general abelian group Σ. This gives a complete classification of depth 2 solvable coverings over [Formula: see text].

scholarly journals On parabolic submonoids of a class of singular Artin monoids

2007 ◽  
Vol 82 (1) ◽  
pp. 29-37
Author(s):  
Noelle Antony
Keyword(s):  
Finite Type ◽  
Braid Group ◽  
Knot Theory ◽  
Braid Groups ◽  
Artin Groups ◽  
Parabolic Subgroups ◽  
Vassiliev Invariants ◽  
Sole Purpose

AbstractThis paper concerns parabolic submonoids of a class of monoids known as singular Artin monoids. The latter class includes the singular braid monoid— a geometric extension of the braid group, which was created for the sole purpose of studying Vassiliev invariants in knot theory. However, those monoids may also be construed (and indeed, are defined) as a formal extension of Artin groups which, in turn, naturally generalise braid groups. It is the case, by van der Lek and Paris, that standard parabolic subgroups of Artin groups are canonically isomorphic to Artin groups. This naturally invites us to consider whether the same holds for parabolic submonoids of singular Artin monoids. We show that it is in fact true when the corresponding Coxeter matrix is of ‘type FC’ hence generalising Corran's result in the ‘finite type’ case.

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