17,033 research outputs found
Unexpected behaviour of crossing sequences
The n-th crossing number of a graph G, denoted cr_n(G), is the minimum number
of crossings in a drawing of G on an orientable surface of genus n. We prove
that for every a>b>0, there exists a graph G for which cr_0(G) = a, cr_1(G) =
b, and cr_2(G) = 0. This provides support for a conjecture of Archdeacon et al.
and resolves a problem of Salazar.Comment: 21 page
The -genus of Kuratowski minors
A drawing of a graph on a surface is independently even if every pair of
nonadjacent edges in the drawing crosses an even number of times. The
-genus of a graph is the minimum such that has an
independently even drawing on the orientable surface of genus . An
unpublished result by Robertson and Seymour implies that for every , every
graph of sufficiently large genus contains as a minor a projective
grid or one of the following so-called -Kuratowski graphs: , or
copies of or sharing at most common vertices. We show that
the -genus of graphs in these families is unbounded in ; in
fact, equal to their genus. Together, this implies that the genus of a graph is
bounded from above by a function of its -genus, solving a problem
posed by Schaefer and \v{S}tefankovi\v{c}, and giving an approximate version of
the Hanani-Tutte theorem on orientable surfaces. We also obtain an analogous
result for Euler genus and Euler -genus of graphs.Comment: 23 pages, 7 figures; a few references added and correcte
The forbidden number of a knot
Every classical or virtual knot is equivalent to the unknot via a sequence of
extended Reidemeister moves and the so-called forbidden moves. The minimum
number of forbidden moves necessary to unknot a given knot is an invariant we
call the {\it forbidden number}. We relate the forbidden number to several
known invariants, and calculate bounds for some classes of virtual knots.Comment: 14 pages, many figures; v2 improves the upper bounds from the
crossing number, and adds more detail to the data presented in the conclusio
Counterexample to an extension of the Hanani-Tutte theorem on the surface of genus 4
We find a graph of genus and its drawing on the orientable surface of
genus with every pair of independent edges crossing an even number of
times. This shows that the strong Hanani-Tutte theorem cannot be extended to
the orientable surface of genus . As a base step in the construction we use
a counterexample to an extension of the unified Hanani-Tutte theorem on the
torus.Comment: 12 pages, 4 figures; minor revision, new section on open problem
Classification of alternating knots with tunnel number one
This paper gives a complete classification of all alternating knots with
tunnel number one, and all their unknotting tunnels. We prove that the only
such knots are two-bridge knots and certain Montesinos knots.Comment: 38 pages, 26 figures; to appear in Communications in Analysis and
Geometr
Cusp volumes of alternating knots
We show that the cusp volume of a hyperbolic alternating knot can be bounded
above and below in terms of the twist number of an alternating diagram of the
knot. This leads to diagrammatic estimates on lengths of slopes, and has some
applications to Dehn surgery. Another consequence is that there is a universal
lower bound on the cusp density of hyperbolic alternating knots.Comment: 21 pages, 8 figures; v4: revised final version, with corrected
constants throughout the paper; to appear in Geometry & Topolog
Virtual knot groups and almost classical knots
We define a group-valued invariant of virtual knots and relate it to various
other group-valued invariants of virtual knots, including the extended group of
Silver-Williams and the quandle group of Manturov and Bardakov-Bellingeri. A
virtual knot is called almost classical if it admits a diagram with an
Alexander numbering, and in that case we show that the group factors as a free
product of the usual knot group and Z. We establish a similar formula for mod p
almost classical knots, and we use these results to derive obstructions to a
virtual knot K being mod p almost classical. Viewed as knots in thickened
surfaces, almost classical knots correspond to those that are homologically
trivial. We show they admit Seifert surfaces and relate their Alexander
invariants to the homology of the associated infinite cyclic cover. We prove
the first Alexander ideal is principal, recovering a result first proved by
Nakamura et al. using different methods. The resulting Alexander polynomial is
shown to satisfy a skein relation, and its degree gives a lower bound for the
Seifert genus. We tabulate almost classical knots up to 6 crossings and
determine their Alexander polynomials and virtual genus.Comment: 44 page
Flipping Cubical Meshes
We define and examine flip operations for quadrilateral and hexahedral
meshes, similar to the flipping transformations previously used in triangular
and tetrahedral mesh generation.Comment: 20 pages, 24 figures. Expanded journal version of paper from 10th
International Meshing Roundtable. This version removes some unwanted
paragraph breaks from the previous version; the text is unchange
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