4,598 research outputs found
A Survey of Quandle Ideas
This article surveys many aspects of the theory of quandles which
algebraically encode the Reidemeister moves. In addition to knot theory,
quandles have found applications in other areas which are only mentioned in
passing here. The main purpose is to give a short introduction to the subject
and a guide to the applications that have been found thus far for quandle
cocycle invariants.Comment: Submitted to conference proceedings; embarrassing misspellings of
various names corrected. Many apologies and thanks to readers who pointed out
correction
Reidemeister/Roseman-type Moves to Embedded Foams in 4-dimensional Space
The dual to a tetrahedron consists of a single vertex at which four edges and
six faces are incident. Along each edge, three faces converge. A 2-foam is a
compact topological space such that each point has a neighborhood homeomorphic
to a neighborhood of that complex. Knotted foams in 4-dimensional space are to
knotted surfaces, as knotted trivalent graphs are to classical knots. The
diagram of a knotted foam consists of a generic projection into 4-space with
crossing information indicated via a broken surface. In this paper, a finite
set of moves to foams are presented that are analogous to the Reidemeister-type
moves for knotted graphs. These moves include the Roseman moves for knotted
surfaces. Given a pair of diagrams of isotopic knotted foams there is a finite
sequence of moves taken from this set that, when applied to one diagram
sequentially, produces the other diagram.Comment: 18 pages, 29 figures, Be aware: the figure on page 3 takes some time
to load. A higher resolution version is found at
http://www.southalabama.edu/mathstat/personal_pages/carter/Moves2Foams.pdf .
If you want to use to any drawings, please contact m
Algebraic Structures Derived from Foams
Foams are surfaces with branch lines at which three sheets merge. They have
been used in the categorification of sl(3) quantum knot invariants and also in
physics. The 2D-TQFT of surfaces, on the other hand, is classified by means of
commutative Frobenius algebras, where saddle points correspond to
multiplication and comultiplication. In this paper, we explore algebraic
operations that branch lines derive under TQFT. In particular, we investigate
Lie bracket and bialgebra structures. Relations to the original Frobenius
algebra structures are discussed both algebraically and diagrammatically.Comment: 11 pages; 14 figure
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