nonconsecutive edges, X and Y , of an n-gon is Umd (X, Y ) =
(X)(Y )
. (X) gives the length of X and md(X, Y ) gives
md(X,Y )2
the minimum distance between X and Y . The Minimum
Distance Energy of a polygon, P is
Umd (P ) =
Umd (X, Y )
Acknowledgments
I would like to thank my advisor Dr. R. Trapp for his support
andlike
insisupport
work
wasfor
completed
I would
to thankand
my insight.
advisor This
Dr. R.
Trapp
his supduring
theinsisupport
2007 REU and
program
in Mathematics
at California
port and
insight.
This work was
completed
State
San Bernardino,
and was jointly
duringUniversity,
the 2007 REU
program in Mathematics
at sponsored
California
by
CSUSB
and NSF-REU
Grant DMS-0453605.
State
University,
San Bernardino,
and was jointly sponsored
by CSUSB and NSF-REU Grant DMS-0453605.
Knots and Minimum Distance Energy
all edges X
.
Y =Xnor adjacent
The
convex hull of an n-gon, P , is the smallest convex
Abstract
set containing all vertices of P , it is denoted H(P ). The
Professor is
Elizabeth
boundary
denoted Denne
∂H(P ).and I continue work I started in
a research program (summer 2007). We aim to find which
polygonal knots have least Minimum Distance Energy. I
previously showed that the energy is minimized for convex
polygons. We hope relating the energy to chords of polygons
will be a helpful step towards showing that regular n-gons
have the least minimum distance energy for all polygonal
Figure 1: The convex hull (green) of a non-convex polygon (blue)
knots.
A pocket is a set of edges of a polygon not in ∂H(P ) between the vertices i and j on ∂H(P ). Its pocket lid is the
Preliminary Definitions and Theorems
line ij.
“Simon’s Minimum Distance Energy” [3, 5] Ror a pair
A flip is the reflection of a pocket across a pocket lid.
of nonconsecutive edges, X and Y , of an n-gon is Umd (X, Y ) =
(X)(Y )
. Here, (X) is the length of X and md(X, Y ) is the
md(X,Y )2
minimum distance between X jand Y . The Minimum Distance Energy of a polygon,
P , is
i
Umd (P ) =
Umd (X, Y ).
all edges X
Erdös-Nagy
Theorem.
[1,
6]
Every
simple
planar
polygon
Knots and Minimum Distance Energy
can be made convex with a finite number of flips.
Erdös-Nagy Theorem. [1, 6] Every simple planar polygon
, 2008
A
stretch
is
made
by
a
change
in
angles.
For
P
and
P
can be madeofconvex
with a and
finite
number Smith
of flips.
Special Studies, Department
Mathematics
Statistics,
College,
Spring
is a stretched verpolygons
with
corresponding
lengths,
P
Rosanna
Speller,
Advisor:
Elizabeth
Denne
A
stretch
is
made
by
a
change
in
angles.
For
P and P ,
,
y
∈
P
n,
sion of P , if ∀ x, y ∈ P and corresponding
x
is a stretched verpolygons
with
lengths,
P
corresponding
|x − y| ≤ |x − y | [4].
sion of P , if ∀ x, y ∈ P and corresponding x , y ∈ P n,
3
,
Sallee’s
Lemma.
[4]
If
P
is
a
non-convex
polygon
in
E
|x − y| ≤ |x − y | [4].
∃ a stretched polygon P , which is planar and convex, such
n
Sallee’s
Lemma.
[4]
If
P
is
a
non-convex
polygon
in
E
that ∀, x, y ∈ P , with x and y not on the same edge, and,
such
∃fora corresponding
stretched polygon,
∈ which
P , |x is
− planar
y| < |xand
− y convex,
|.
x , yP
that ∀, points x, y ∈ P , with x and y not on the same edge
1 Previous
Results
of P , and corresponding x , y ∈ P , |x − y| < |x − y |.
1
Theorem.
If
P
is
a
planar
n-gon
with
minimized
U
md , then
Previous Results
P is convex.
Theorem. If P is a planar n-gon with minimized Umd
, then
P is convex.
Figure 3: Convex polygon made by flipping
5
(a)
(b)
Regular n-gons, Rn .
The minimum distance energy of Rn when n is odd is:
(Rn )
Umd
2
= 2n · sin
When n is even:
A stretch is made by a change in angles. For P and P ,
polygons with corresponding lengths, P is a stretched version of P , if ∀ x, y ∈ P and corresponding x , y ∈ P n,
|x − y| ≤ |x − y | [4].
Sallee’s Lemma. [4] If P is a non-convex polygon in En ,
∃ a stretched polygon, P which is planar and convex, such
Umd (Rn )
= n · sin
2
π n


sin
π
n
Figure 2: Flipping a pocket over its pocket lid, ij
4
4
2
n
2 −1
j=1
1
π(n−2)
2n
P
P
1
r5,2
8
7
(a)
5
(b)
Figure 5: (a) Polygon with edges (ri,1 ) in black, each ri,2 is given in
green. (b) Regular octagon inscribed in circle of radius t; distances
between vertices are the length of a chord (c) = 2t sin( θ2 ).
sin
1
jπ
2
+2·
energyand
of smooth
knots, Polygonal
J. Knot Theory
Ram[3] tion
E. J.and
Rawdon
J. K. Simon,
approximaifications
15:4 (2006),
429-451.
tion and energy
of smooth
knots, J. Knot Theory Ramifications
15:4
(2006),
429-451.
[4] G. T. Sallee, Stretching chords of space curves, Geome-
n
n
−2
2
j=1
sin
[1] P. Erdös, Problem number
3763,
Amer.
Math.
Monthly
7
(1935)
627.
[1] 42
P. Erdös,
Problem
number
Math. Monthly
Figure
3: Convex
Polygon3763,
Made Amer.
by Flipping
42
(1935)
627.
[2] G. Lükő, On the mean length of the chords of a closed
4 (1966)
23-32.
[2] curve,
G. Lükő,
On the
mean length of the chords of a closed
curve,
4
(1966)
23-32.
[3] E. J. Rawdon and J. K. Simon, Polygonal approxima-

1

2 jπ
n
Theorem. If P is a polygon in E there exists a convex pla
(P ) ≥
nar polygon, P , created by stretching such that Umd
Umd (P ).
3
θ
6
plaTheorem. If P 3is a polygon
in E3 there exists a convex
r
4,1
nar polygon, P , created by stretching such that Umd (P ) ≥
(P ).
Umd
References
References
Figure 4: (b) A convex polygon made by stretching the polygon in (a)
Erdös-Nagy Theorem. [1, 6] Every simple planar polygon
can be made convex with a finite number of flips.
only if the n-gon is regu-
c , then
Theorem.
If
P
is
a
planar
n-gon
with
minimized
U
2
md
P is convex.
t
Figure 3: Convex polygon made by flipping
A pocket is a set of edges of a polygon not in ∂H(P ) between the vertices i and j on ∂H(P ). Its pocket lid is the
line segment ij.
i
8
if 8and
2,2
5
j
n
2 lπ
1
2
2 sin ( n )
g(ri,l ) ≤ g a
n i=1
sin2 ( πn )
Previous Results
r
and 2007 REU program, jointly sponsored by CSUSB and NSF-REU Grant DMS0453605
1 R.S. thanks adviser Dr. R. Trapp of California State University, San Bernardino
and 2007 REU program, jointly sponsored by CSUSB and NSF-REU Grant DMS
P
P
0453605
3
1
Theorem. If P is a polygon in E3 there exists a convex pla
(P ) ≥
nar polygon,
2 P , created by stretching such that Umd
1 R.S. thanks adviser Dr. R. Trapp of California State University, San Bernardino
(P ).
Umd
A flip is the reflection of a pocket across a pocket lid.
Lükő’s Theorem II. [2] Let the vertices of a n-gon be
labeled 1, 2, . . . , n and let ri,l denote the distance between
vertices i and i + l. Let a be a constant greater than or
equal to the length of every edge (denoted ri,1 ) of the n-gon.
Let g(t) be an increasing, concave function, then,
When g(t) = t, this theorem implies the average squared
distance between the vertices of an n-gon is maximized by
the regular n-gon.
Knots and Minimum Distance Energy
Figure hull
2: Flipping
pocket over
lid, ij convex
The convex
of ana n-gon,
P , its
is pocket
the smallest
set containing all vertices of P and is denoted H(P ). The
boundary is denoted ∂H(P ).
Figure 1: The convex hull (green) of a non-convex polygon (blue)
Conjecture. Regular n-gons have least minimum distance
energy.
for all n ≥ 4, with equality
lar.
Y =Xnor adjacent
4
New Investigations
2 (1973) 311-315.
[4] triae
G. T.Dedicata
Sallee, Stretching
chords of space curves, Geometriae
Dedicata
2
(1973)
311-315.
[5] J. Simon, Energy functions for polygonal knots, J. Knot
Ramifications
3:3 (1994),
299-320.knots, J. Knot
[5] Theory
J. Simon,
Energy functions
for polygonal
Theory
Ramifications
3:3
(1994),
299-320.
[6] G. Toussaint, The Erdös-Nagy theorem and its ramifiComputational
Geometry
31 (2005)
[6] cations,
G. Toussaint,
The Erdös-Nagy
theorem
and 219-236.
its ramifications, Computational Geometry 31 (2005) 219-236.
1
R.S. thanks adviser Dr. R. Trapp of California State University, San Bernardino
and 2007 REU program, jointly sponsored by CSUSB and NSF-REU Grant DMS0453605