1,776 research outputs found
Link invariants from finite biracks
A birack is an algebraic structure with axioms encoding the blackboard-framed
Reidemeister moves, incorporating quandles, racks, strong biquandles and
semiquandles as special cases. In this paper we extend the counting invariant
for finite racks to the case of finite biracks. We introduce a family of
biracks generalizing Alexander quandles, -racks, Alexander biquandles
and Silver-Williams switches, known as -biracks. We
consider enhancements of the counting invariant using writhe vectors, image
subbiracks, and birack polynomials.Comment: 14 page
Virtual Crossing Realization
We study virtual isotopy sequences with classical initial and final diagrams,
asking when such a sequence can be changed into a classical isotopy sequence by
replacing virtual crossings with classical crossings. An example of a sequence
for which no such virtual crossing realization exists is given. A conjecture on
conditions for realizability of virtual isotopy sequences is proposed, and a
sufficient condition for realizability is found. The conjecture is reformulated
in terms of 2-knots and knots in thickened surfaces.Comment: 21 pages, 10 figures. To appear in J. Knot Theory Ramifications.
Version 6 includes revisions suggested by the reviewe
Classification of Finite Alexander Quandles
Two finite Alexander quandles with the same number of elements are isomorphic
iff their Z[t,t^-1]-submodules Im(1-t) are isomorphic as modules. This yields
specific conditions on when Alexander quandles of the form Z_n[t,t^-1]/(t-a)
where gcd(n,a)=1 (called linear quandles) are isomorphic, as well as specific
conditions on when two linear quandles are dual and which linear quandles are
connected. We apply this result to obtain a procedure for classifying Alexander
quandles of any finite order and as an application we list the numbers of
distinct and connected Alexander quandles with up to fifteen elements.Comment: 10 pages, LaTeX. Typos corrected, proof of Theorem 2.1 fixed. To
appear in Topology Proceeding
Finite Type Enhancements
We enhance the biquandle counting invariant using elements of truncated
biquandle-labeled Polyak algebras. These finite type enhancements reduce to the
finite type enhancements defined by Goussarov, Polyak and Viro for the trivial
biquandle of one element and determine (but are not determined by) the
biquandle counting invariant for general biquandles. Unlike the unlabeled case,
biquandle labeled finite type invariants of degree 1 are nontrivial and are
related to biquandle cocycle invariants.Comment: 12 page
Signed ordered knotlike quandle presentations
We define enhanced presentations of quandles via generators and relations
with additional information comprising signed operations and an order structure
on the set of generators. Such a presentation determines a virtual link diagram
up to virtual moves. We list formal Reidemeister moves in which Tietze moves on
the presented quandle are accompanied by corresponding changes to the order
structure. Omitting the order structure corresponds to replacing virtual
isotopy by welded isotopy.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-20.abs.htm
A polynomial invariant of finite quandles
We define a two-variable polynomial invariant of finite quandles. In many
cases this invariant completely determines the algebraic structure of the
quandle up to isomorphism. We use this polynomial to define a family of link
invariants which generalize the quandle counting invariant.Comment: 9 pages. Typos correcte
Link invariants from finite racks
We define ambient isotopy invariants of oriented knots and links using the
counting invariants of framed links defined by finite racks. These invariants
reduce to the usual quandle counting invariant when the rack in question is a
quandle. We are able to further enhance these counting invariants with
2-cocycles from the coloring rack's second rack cohomology satisfying a new
degeneracy condition which reduces to the usual case for quandles.Comment: 13 pages; typos correcte
An isomorphism theorem for Alexander biquandles
We show that two Alexander biquandles M and M' are isomorphic iff there is an
isomorphism of Z[s,1/s,t,1/t]-modules h:(1-st)M --> (1-st)M' and a bijection
g:O_s(A) --> O_s(A') between the s-orbits of sets of coset representatives of
M/(1-st)M and M'/(1-st)M' respectively satisfying certain compatibility
conditions.Comment: 10 pages. Version 2 includes changes suggested by referee, including
a title change. To appear in Intl. J. Mat
Twisted Virtual Bikeigebras and Twisted Virtual Handlebody-Knots
We generalize unoriented handlebody-links to the twisted virtual case,
obtaining Reidemeister moves for handlebody-links in ambient spaces of the form
for a compact closed 2-manifold up to stable
equivalence. We introduce a related algebraic structure known as twisted
virtual bikeigebras whose axioms are motivated by the twisted virtual
handlebody-link Reidemeister moves. We use twisted virtual bikeigebras to
define -colorability for twisted virtual handlebody-links and define an
integer-valued invariant of twisted virtual
handlebody-links. We provide example computations of the new invariants and use
them to distinguish some twisted virtual handlebody-links.Comment: 9 page
Virtual Yang-Baxter cocycle invariants
We extend the Yang-Baxter cocycle invariants for virtual knots by augmenting
Yang-Baxter 2-cocycles with cocycles from a cohomology theory associated to a
virtual biquandle structure. These invariants coincide with the classical
Yang-Baxter cocycle invariants for classical knots but provide extra
information about virtual knots and links. In particular, they provide a method
for detecting non-classicality of virtual knots and links.Comment: 17 pages, many pictures. To appear in Trans. Amer. Math. Soc. Version
2 includes minor typo correction
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