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Minimal Embeddings of Knots in the Cubic Lattice
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Minimal Embeddings of Knots in the Cubic Lattice
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Description
Identifier
Thesis
1984
Author
Wysong, Kimberly Ann, 1979
Title
Minimal
Embeddings
of
Knots
in the
Cubic
Lattice
Publisher
Central Connecticut State University
Date of Publication
2008
Resource Type
Master's Thesis
Abstract
steps
required
to
represent
the
knot
as a
polygon
in the
cubic
lattice
.
Several
lower
bounds
for the
lattice
step
numbers
of
different
knots
have been
obtained
by
computer
simulations
but the
exact
value
of the
lattice
step
number
seems
to be
known
only
for the
trefoil
knot
which
is
the
simplest
nontrivial
knot
. In this
thesis
we
analyze
three
different
characteristics
of
minimal
embeddings
in the
cubic
lattice
. In
Chapter
2
,
we
study
the
work
of
Diao
that
proves
that the
minimum
number
of
steps
required
to
produce
a
nontrivial
knot
in the
cubic
lattice
is
24
. In
Chapter
3
,
we
study
two
works
.
First
, the
proof
of
van
Rensburg
and
Promislov
regarding
the
curvature
of
lattice
knots
which
is
essentially
the
number
of
turns
of the
polygon
. For their
investigations
, these
authors
develop
a
technique
that
uses
weighted
sequences
and
functions
.
We
elaborate
on their
theorems
by
providing
examples
and
diagrams
that
we
hope
help
to
clarify
their
work
.
Second
,
we
look
at the
proof
of
Huh
and
Oh
who
develop
methods
to
study
the
minimal
stick
number
of the
trefoil
and the
gure
eight
knots
. In
Chapter
4
,
we
use
the
techniques
of
Diao
to
investigate
the
lattice
step
number
of the
gure
eight
knot
.
We
prove
that
several
of
Diao's
results
can
be
adapted
and
applied
to the
four
crossing
knot
.
Our
results
support
the
conjecture
that the
lattice
step
number
of the
gure
eight
knot
is
30
.
Subject
Knot theory
Department
Department of Mathematical Sciences
Advisor
Castañeda, Nelson
Type
Text
Digital Format
application/pdf
Language
eng
OCLC number
713734310
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