scholarly journals Free groups and finite-type invariants of pure braids

2002 ◽  
Vol 132 (1) ◽  
pp. 117-130 ◽  
Author(s):  
JACOB MOSTOVOY ◽  
SIMON WILLERTON
Keyword(s):  
Finite Type ◽  
Free Group ◽  
Free Groups ◽  
Braid Groups ◽  
Point Of View ◽  
Magnus Expansion ◽  
Vassiliev Invariants ◽  
Integer Coefficients ◽  
Finite Type Invariants ◽  
Theoretic Point

In this paper finite type invariants (also known as Vassiliev invariants) of pure braids are considered from a group-theoretic point of view. New results include a construction of a universal invariant with integer coefficients based on the Magnus expansion of a free group and a calculation of numbers of independent invariants of each type for all pure braid groups.

scholarly journals EXTENDING REPRESENTATIONS OF BRAID GROUPS TO THE AUTOMORPHISM GROUPS OF FREE GROUPS

2005 ◽  
Vol 14 (08) ◽  
pp. 1087-1098 ◽  
Author(s):  
VALERIJ G. BARDAKOV
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Automorphism Groups ◽  
Free Groups ◽  
Linear Representation ◽  
Braid Groups ◽  
Burau Representation ◽  
Pure Braid Group

We construct a linear representation of the group IA (Fn) of IA-automorphisms of a free group Fn, an extension of the Gassner representation of the pure braid group Pn. Although the problem of faithfulness of the Gassner representation is still open for n > 3, we prove that the restriction of our representation to the group of basis conjugating automorphisms Cbn contains a non-trivial kernel even if n = 2. We construct also an extension of the Burau representation to the group of conjugating automorphisms Cn. This representation is not faithful for n ≥ 2.

TWIST SEQUENCES AND VASSILIEV INVARIANTS

1994 ◽  
Vol 03 (03) ◽  
pp. 391-405 ◽  
Author(s):  
ROLLAND TRAPP
Keyword(s):  
Finite Type ◽  
Polynomial Growth ◽  
Difference Sequence ◽  
Uniform Limit ◽  
Topological Information ◽  
Torus Knots ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Finite Type Invariants ◽  
Vassiliev Invariant

In this paper we describe a difference sequence technique, hereafter referred to as the twist sequence technique, for studying Vassiliev invariants. This technique is used to show that Vassiliev invariants have polynomial growth on certain sequences of knots. Restrictions of Vassiliev invariants to the sequence of (2, 2i + 1) torus knots are characterized. As a corollary it is shown that genus, crossing number, signature, and unknotting number are not Vassiliev invariants. This characterization also determines the topological information about (2, 2i + 1) torus knots encoded in finite-type invariants. The main result obtained is that the complement of the space of Vassiliev invariants is dense in the space of all numeric knot invariants. Finally, we show that the uniform limit of a sequence of Vassiliev invariants must be a Vassiliev invariant.

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.

scholarly journals ON VASSILIEV INVARIANTS FOR LINKS IN HANDLEBODIES

1998 ◽  
Vol 07 (05) ◽  
pp. 701-712 ◽  
Author(s):  
VLADIMIR V. VERSHININ
Keyword(s):  
Finite Type ◽  
Braid Group ◽  
Special Class ◽  
Vassiliev Invariants ◽  
Finite Type Invariants ◽  
Relative Version ◽  
One To One

The notion of Vassiliev algebra in case of handlebodies is developed. Analogues of the results of J. Baez for links in handlebodies are proved. This implies that there is a one-to-one correspondence between the special class of finite type invariants of links in hanlebodies and the homogeneous Markov traces on Vassiliev algebras. This approach uses the singular braid monoid and braid group in a handlebody and the generalizations of the theorem of J. Alexander and the theorem of A. A. Markov for singular links and braids and the relative version of Markov's theorem.

SOME REMARKS ON TABACHNIKOV’S INVARIANTS OF PLANE CURVES

1996 ◽  
Vol 05 (06) ◽  
pp. 849-857
Author(s):  
MIKHAIL KHOVANOV
Keyword(s):  
Finite Type ◽  
Free Group ◽  
Plane Curves ◽  
Polynomial Invariant ◽  
Finite Type Invariants

We construct a generalization of the “free group valued” Tabachnikov’s invariant of long curves. We characterize Tabachnikov’s polynomial invariant of plane curves via a representation of the twin group. We also show that these invariants are combinations of finite type invariants.

scholarly journals Stallings graphs, algebraic extensions and primitive elements in F2

2014 ◽  
Vol 157 (1) ◽  
pp. 1-11 ◽  
Author(s):  
ORI PARZANCHEVSKI ◽  
DORON PUDER
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Point Of View ◽  
Primitive Elements

AbstractWe study the free group of rank two from the point of view of Stallings core graphs. The first half of the paper examines primitive elements in this group, giving new and self-contained proofs for various known results about them. In particular, this includes the classification of bases of this group. The second half of the paper is devoted to constructing a counterexample to a conjecture by Miasnikov, Ventura and Weil, which seeks to characterize algebraic extensions in free groups in terms of Stallings graphs.

scholarly journals FINITE TYPE INVARIANTS OF LINKS WITH A FIXED LINKING MATRIX

2002 ◽  
Vol 11 (07) ◽  
pp. 1017-1041 ◽  
Author(s):  
ELI APPLEBOIM
Keyword(s):  
Finite Type ◽  
Vassiliev Invariants ◽  
Finite Type Invariants ◽  
Chord Diagrams ◽  
Knots And Links

In this paper we introduce two theories of finite type invariants for framed links with a fixed linking matrix. We show that these theories are different from, but related to, the theory of Vassiliev invariants of knots and links. We will take special note of the case of zero linking matrix. i.e., zero-framed algebraically split links. We also study the corresponding spaces of "chord diagrams".

KNOTS WITH GIVEN FINITE TYPE INVARIANTS AND Ck-DISTANCES

2001 ◽  
Vol 10 (07) ◽  
pp. 1041-1046 ◽  
Author(s):  
YASUTAKA NAKANISHI ◽  
YOSHIYUKI OHYAMA
Keyword(s):  
Natural Number ◽  
Finite Type ◽  
The Other ◽  
Vassiliev Invariants ◽  
Finite Type Invariants ◽  
Minimum Number

We show that for any given pair of a natural number n and a knot K, there are infinitely many knots Jm (m=1,2,…) such that their Vassiliev invariants of order less than or equal to n coincide with those of K and that each Jm has Ck-distance 1 (k≠q 2, k=1, …, n) and C2-distance 2 from the knot K. The Ck-distance means the minimum number of Ck-moves which transform one knot into the other.

scholarly journals An Ordering for Groups of Pure Braids and Fibre-Type Hyperplane Arrangements

2003 ◽  
Vol 55 (4) ◽  
pp. 822-838 ◽  
Author(s):  
Djun Maximilian Kim ◽  
Dale Rolfsen
Keyword(s):  
Fibre Type ◽  
Free Groups ◽  
Braid Groups ◽  
Hyperplane Arrangements ◽  
The Other ◽  
Fundamental Groups ◽  
Magnus Expansion ◽  
Direct Generalization ◽  
Complex Hyperplane

AbstractWe define a total ordering of the pure braid groups which is invariant under multiplication on both sides. This ordering is natural in several respects. Moreover, it well-orders the pure braids which are positive in the sense of Garside. The ordering is defined using a combination of Artin's combing technique and the Magnus expansion of free groups, and is explicit and algorithmic.By contrast, the full braid groups (on 3 or more strings) can be ordered in such a way as to be invariant on one side or the other, but not both simultaneously. Finally, we remark that the same type of ordering can be applied to the fundamental groups of certain complex hyperplane arrangements, a direct generalization of the pure braid groups.

scholarly journals FINITE-TYPE KNOT INVARIANTS BASED ON THE BAND-PASS AND DOUBLED-DELTA MOVES

2010 ◽  
Vol 19 (03) ◽  
pp. 355-384 ◽  
Author(s):  
JAMES CONANT ◽  
JACOB MOSTOVOY ◽  
TED STANFORD
Keyword(s):  
Abelian Group ◽  
Large Class ◽  
Finite Type ◽  
Equivalence Classes ◽  
Connected Sum ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Finite Type Invariants ◽  
Conway Polynomial ◽  
Band Pass

We study generalizations of finite-type knot invariants obtained by replacing the crossing change in the Vassiliev skein relation by some other local move, analyzing in detail the band-pass and doubled-delta moves. Using braid-theoretic techniques, we show that, for a large class of local moves, generalized Goussarov's n-equivalence classes of knots form groups under connected sum. (Similar results, but with a different approach, have been obtained before by Taniyama and Yasuhara.) It turns out that primitive band-pass finite-type invariants essentially coincide with standard primitive finite-type invariants, but things are more interesting for the doubled-delta move. The complete degree 0 doubled-delta invariant is the S-equivalence class of the knot. In this context, we generalize a result of Murakami and Ohtsuki to show that the only primitive Vassiliev invariants of S-equivalence taking values in an abelian group with no 2-torsion arise from the Alexander–Conway polynomial. We start analyzing degree one doubled-delta invariants by considering which Vassiliev invariants are of doubled-delta degree one, finding that there is exactly one such invariant in each odd Vassiliev degree, and at most one (which is ℤ2-valued) in each even Vassiliev degree. Analyzing higher doubled-delta degrees, we observe that the Euler degree n + 1 part of Garoufalidis and Kricker's rational lift of the Kontsevich integral is a doubled-delta degree 2n invariant.

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