15 research outputs found
Twisted Link Theory
We introduce stable equivalence classes of oriented links in orientable
three-manifolds that are orientation -bundles over closed but not
necessarily orientable surfaces. We call these twisted links, and show that
they subsume the virtual knots introduced by L. Kauffman, and the projective
links introduced by Yu. Drobotukhina. We show that these links have unique
minimal genus three-manifolds. We use link diagrams to define an extension of
the Jones polynomial for these links, and show that this polynomial fails to
distinguish two-colorable links over non-orientable surfaces from
non-two-colorable virtual links.Comment: 33 pages and 35 figure
Real algebraic knots of low degree
In this paper we study rational real algebraic knots in . We show
that two real algebraic knots of degree are rigidly isotopic if and
only if their degrees and encomplexed writhes are equal. We also show that any
irreducible smooth knot which admits a plane projection with less than or equal
to four crossings has a rational parametrization of degree .
Furthermore an explicit construction of rational knots of a given degree with
arbitrary encomplexed writhe (subject to natural restrictions) is presented.Comment: 28 page
On the ribbon graphs of links in real projective space
Every link diagram can be represented as a signed ribbon graph. However,
different link diagrams can be represented by the same ribbon graphs. We
determine how checkerboard colourable diagrams of links in real projective
space, and virtual link diagrams, that are represented by the same ribbon
graphs are related to each other. We also find moves that relate the diagrams
of links in real projective space that give rise to (all-A) ribbon graphs with
exactly one vertex
Intrinsically triple-linked graphs in RP^3
Flapan--Naimi--Pommersheim showed that every spatial embedding of ,
the complete graph on ten vertices, contains a non-split three-component link;
that is, is intrinsically triple-linked in . The work of
Bowlin--Foisy and Flapan--Foisy--Naimi--Pommersheim extended the list of known
intrinsically triple-linked graphs in to include several other
families of graphs. In this paper, we will show that while some of these graphs
can be embedded 3-linklessly in , is intrinsically
triple-linked in .Comment: 23 pages, 6 figures; v2: revised introduction, minor corrections, new
outlines to longer proof
Biquandles with structures related to virtual links and twisted links
We introduce two kinds of structures, called v-structures and t-structures,
on biquandles. These structures are used for colorings of diagrams of virtual
links and twisted links such that the numbers of colorings are invariants.
Given a biquandle or a quandle, we give a method of constructing a biquandle
with these structures. Using the numbers of colorings, we show that Bourgoin's
twofoil and non-orientable virtual -foils do not represent virtual links
Spindle configurations of skew lines
We prove a conjecture of Crapo and Penne which characterizes isotopy classes
of skew configurations with spindle-structure. We use this result in order to
define an invariant, spindle-genus, for spindle-configurations.
We also slightly simplify the exposition of some known invariants for
configurations of skew lines and use them to define a natural partition of the
lines in a skew configuration.
Finally, we describe an algorithm which constructs a spindle in a given
switching class, or proves non-existence of such a spindle.Comment: 42 pages, many figures. A new corrected proof of a conjecture of
Crapo and Penne is added. More new material is also adde