RECURSIVE CALCULATION FOR AN INVARIANT OF A RIBBON KNOT

1998 ◽  
Vol 07 (08) ◽  
pp. 1093-1105 ◽  
Author(s):  
TAIZO KANENOBU
Keyword(s):  
Finite Type ◽  
Alexander Polynomial ◽  
Second Derivative ◽  
Conway Polynomial ◽  
Type Invariant ◽  
Recursive Calculation ◽  
Second Degree

We give an algorithm for calculating the second degree coefficient of the Conway polynomial of a ribbon 1-knot. This naturally yields a recursive calculation for the second derivative at t = 1, Δ′′(1), of the normalized Alexander polynomial of a ribbon 2-knot in R4, which is the first nontrivial finite type invariant of a ribbon 2-knot defined by Habiro, Kanenobu, and Shima.

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 On a Realization of Prime Tangles and Knots

1983 ◽  
Vol 35 (2) ◽  
pp. 311-323
Author(s):  
Quach Thi Cam Van
Keyword(s):  
Alexander Polynomial ◽  
Conway Polynomial ◽  
Natural Means ◽  
Natural Way

The notion of a prime tangle is introduced by Kirby and Lickorish [7]. It is related deeply to the notion of a prime knot by the following result in [8]: summing together two prime tangles gives always a prime knot.The purpose of this paper is to exploit this above mentioned result of Lickorish in creating or detecting prime knots which satisfy certain properties. First, we shall express certain knots (two-bridge knots and Terasaka slice knots [14]) as a sum of a prime tangle and an untangle (the existence of such a sum is proven to every knot in [7] and is not unique) in a natural way (natural means here depending on certain specific geometrical characters of the class of knots). Second, every Alexander polynomial (or Conway polynomial) is shown to be realized by a prime algebraic knot (algebraic in the sense of Conway [3], Bonahon-Siebenmann [2]) which can be expressed as the sum of two prime algebraic tangles.

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.

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.

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 Explicit formulae for two-bridge knot polynomials

2005 ◽  
Vol 78 (2) ◽  
pp. 149-166 ◽  
Author(s):  
Shinji Fukuhara
Keyword(s):  
Normal Form ◽  
Open Problem ◽  
Alexander Polynomial ◽  
Alexander Polynomials ◽  
Elementary Number ◽  
Conway Polynomial ◽  
Coprime Integers ◽  
Explicit Formulae ◽  
Knots And Links

AbstractA two-bridge knot (or link) can be characterized by the so-called Schubert normal formKp, qwherepandqare positive coprime integers. Associated toKp, qthere are the Conway polynomial ▽kp, q(z)and the normalized Alexander polynomial Δkp, q(t). However, it has been open problem how ▽kp, q(z) and Δkp, q(t) are expressed in terms ofpandq. In this note, we will give explicit formulae for the Conway polynomials and the normalized Alexander polynomials in the case of two-bridge knots and links. This is done using elementary number theoretical functions inpandq.

Equivalence of Cables Of Mutants of Knots

1989 ◽  
Vol 41 (2) ◽  
pp. 250-273 ◽  
Author(s):  
Józef H. Przytycki
Keyword(s):  
Mirror Image ◽  
Alexander Polynomial ◽  
The Other ◽  
Homfly Polynomial ◽  
Similar Formula ◽  
Conway Polynomial ◽  
Other Hand ◽  
Limited Possibility

There is the nice formula which links the Alexander polynomial of (m, k)-cable of a link with the Alexander polynomial of the link [5] [36] [38]. H. Morton and H. Short investigated whether a similar formula holds for the Jones-Conway (Homfly) polynomial and they found that it is very unlikely. Morton and Short made many calculations of the Jones-Conway polynomial of (2, q)-cables along knots (2 was chosen because of limited possibility of computers) and they get very interesting experimental material [24], [25]. In particular they found that using their method they were able to distinguish some Birman [4] and Lozano-Morton [22] examples (all which they tried) and the 942 knot (in the Rolfsen [37] notation) from its mirror image. On the other hand they were unable to distinguish the Conway knot and the Kinoshita-Terasaka knot.

Sign in / Sign up

Export Citation Format

Share Document