Post on 01-Mar-2022
Faces
Euler formula: v-e+f = 2
Open problem: write a program for finding faces of a given planar (multi)graph.
Planarity criterion (Kuratowski theorem)
Kuratowski theorem: A graph G is planar iff it contains no
subgraph which has K5 or K3,3 as a contraction.
Question: how far is a non-planar graph from becoming planar
Definition: graph crossing number k(G) is the minimal number of edge
crossings among all possible diagrammatic representations of the
graph in a plane
k(K5)=1
Definition: the girth c of a graph is the length of a shortest cycle contained in
the graph.
Open problem: find a better approximation for k(G)
k(K3,3)=1
Fullerene C60
Gauss (1794)
Are atoms knots?
Vortex theory: Kelvin (1867)
Tait, Kirkman, Little (1875-1895)
Graph of a link:
1) color every other region of the KL shadow black or white, so that the infinite
outermost region is black;
2) In the chess-board coloring (or Tait coloring) of the plane obtained, put a vertex at
the center of each white region;
3) connect any two vertices that are in regions that share a crossing with an edge
containing that crossing
Mid-edge graph
Every link shadow is a 4-valent graph. If we have any polyhedral graph G, we can obtain
its corresponding mid-edge graph M(G) defined by mid-edge points of G by connecting
mid-edge points belonging to adjacent edges of G. Clearly, the result M(G) is always a
4-valent graph.
Octahedron graph as the mid-edge graph
obtained from tetrahedron graph
DT (Dowker-Thistlethwaite) codes
1 2 3 4
6 5 8 7
+1 +1 -1 -1
1 3 5 7
6 8 2 4
+ - + -
6 -8 2 -4
6 8 2 4
Dowker codes = permutations of n even numbers 2,4,6,H,2n
Non-realizable Gauss and Dowker codes:
Gauss codes = permutations of n numbers 1,2,H,n, where every number
is used twice.
Potential Dowker code {{5},{8,10,2,4,6}}{1,2,1,2}
Alternating diagram
Definition: A link L that possesses at least one alternating diagram
is called an alternating link.
Examples of (Examples of (un)knotsun)knots that cannot be minimized that cannot be minimized
without increasing the number of crossingswithout increasing the number of crossings
Goeritz's unknot
Nasty unknot
Monster unknot
Minimization
Writhe
Definition: Writhe is the sum of the signs of the crossings in a knot diagram.
For alternating knots, writhe is an invariant of minimal reduced diagrams. In the case
of non-alternating knots, two different minimal projections of the same knot can have
a different writhe (example: Perko pair).
• Flype
Tait Flyping Theorem (Menasco, Thislethwaite, 1991,1993)
Every minimal projection of an alternating link can be obtained from another
minimal projection by a series of flypes.
Knot and link tables
P.G. Tait
T. Kirkman
C.N.. Little
M. Thistlethwaite
J. Hoste
K. Millett
J. Weeks
Knotscape
The first 1 701 936 knots, Math. Intelligencer 20, 4 (1998), 33-48
1890-1900
n
6 1
8 1
9 1
10 3
11 3
12 12
13 19
14 64
15 155
16 510
17 1 514
18 5 146
19 16 966
20 58 782
Kirkman, Caudron
Brendan McKay (LinKnot)
Data base of basic polyhedra
6*
8*
9*
10*-10***
11*-11***
Basic polyhedra
Definition: the basic polyhedra are 4-regular 4-edge-connected, at least 2-vertex
connected plane graphs.
Let 5* and 7* denote non-algebraic tangles with n=5 and n=7 crossings, respectively.
The numerator closures of the products 5* 1, 7* 1, 91* 1 are the basic polyhedra
6*, 8*, 10*, respectively.
5* 7*
5* 1 = 6* 7* 1 = 8*
We can distinguish elementary basic polyhedra containing at most one non-algebraic tangle,
and composite basic polyhedra containing at least two non-algebraic tangles. In this way,
the basic polyhedron 10*** can be represented as 5* 5* , 11***as 5* 1 5* . Applying flypes,
we obtain nothing new: they have only one minimal alternating diagram.
The first exception will be the basic polyhedron 12E,
that can be denoted by 5* ,1,5* ,1. If we apply a flype,
we obtain its other projection 5* 2 5*,
corresponding to the link 11***2.
10*** = 5* 5* 11*** = 5* 1 5*
12E = 5*,1,5*,1 = 5* 2 5* = 11***2
An elementary n-tangle with n-1 vertices is denoted by |n-1| or |1 1H 1|, where 1 occurs
n-1 times. As the basic position of elementary tangle we take the one where one strand
is horizontal and remaining n-1 strands are vertical. An elementary n-tangle |n-1| induces
a coordinate system of concentric regular 2n-gons and corresponding regions, where the
first lower middle or right region with two vertices is denoted by 1, and other regions
(from 1 to 2n) are given in a clockwise order.
Elementary non-algebraic
3-tangles |2| and |3|. Coordinate system of the tangle |2|.
In the following code we are giving the symbol of the elementary n tangle |n|,
and a sequence of regions to which crossings belong, given in a clockwise order.
Tangle |2| 1 2 3 2 5.
Every open region can contain 1, 2, or 3 or more vertices. According to that every region
will be of the type: 1 if it contains 1 vertex; 2 if it contains 2 vertices; 3 if it contains 3 or
more vertices
Change of the types of regions.
Our goal is to obtain basic polyhedra, i.e., 4-valent graphs without bigons. Placing a new
1-tangle (crossing) in an open region changes its type and the types of adjacent regions.
If its original type was 1, the addition of new 1-tangle is forbidden, because a bigon will be
obtained. If the type of a region is 2 or 3, it will be changed to 1, and types of its adjacent
regions increase by 1.
A closure of n-tangle is a basic polyhedron, if connecting of emerging arcs does not result
in the appearance of bigons of loops. By joining free arcs one region can be closed, or two
regions become one. This means that the region type of a closed region must be greater
then 2, and the sum of region types of the two joined regions must be greater then 2.
In the case of 3-tangles and A-closures we need three non-adjacent regions of the type 3,
and for an O-closure two opposite regions of the type 3 and a pair of opposite regions with
the sum of region types greater then 2. In both cases we close regions of the type 3 by
connecting emerging arcs. The closure giving a basic polyhedron is unique (up to symmetry).
Basic polyhedron 9* given by
the minimal code |2| 1 2 1 3 2 1 2.