POLYHEDRAL KNOTS AND LINKS

S. Jablan, Lj. Radovic, R. Sazdanovic

Kostenki,

10 000 B.C.

Peruvian quipu

Chinese decorative knots

Celtic knots

Ur, Mesopotamia (2600-2500 B.C.)

Hagfish practicing knot-theory

Octahedron graph = Sghlegel diagram

Faces

Euler formula: v-e+f = 2

Open problem: write a program for finding faces of a given planar (multi)graph.

Complete graphs

Vertex and edge connectivity

2-vertex connected 2-edge connected

Planar and non-planar graphs

Planar graph K4 Planar graph

Non-planar graphs

Complete graph K5 Complete bipartite graph K3,3

Embeddings of graphs on different surfaces

Embedding of K5 on torus

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

Moebius Cantor graph with k=4

Open question: Can be this crossing number improved?

Bigon collapse

Fullerene C60

Gauss (1794)

Are atoms knots?

Vortex theory: Kelvin (1867)

Tait, Kirkman, Little (1875-1895)

“Real” and “mathematical” knots

Knot

Link

Ambient isotopy

KL equality

• KL shadows

Number of components

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

Gauss code

O1 U2 O3 U1 O2 U3

Trefoil

The Gauss code of Borromean

rings 623

{{1,2,4,5},{1,6,4,3},{2,6,5,3}}

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}

Knot tables

Alternating diagram

Definition: A link L that possesses at least one alternating diagram

is called an alternating link.

• Reidemeister moves

Moves I and II are decreasing number of crossings

I II III

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).

Perko pair of knots 3:-2 0:-2 0 = 2 1:-2 0:-2 0

• 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

Classification of knots and links

Chinese nested spheres

RationalStellar Arborescent

By bigon collapse every algebraic link reduces to a point.

Non-algebraic

tangle

Bigon collapse

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

Definition: Crazy spider is a spider weaving only basic polyhedra.

Crazy Spider Algorithm

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.

Knots in chemistry and biology

Catenane, Moebius ladders

DNA

N. Seeman

Architecture

Temari balls

Tensegrity

B. Fuller, K. Snelson

Fulerene C60

H.W. Kroto

R.F. Curl

R.E. Smalley

1985

Mid-edge construction

Cross-curve construction

Jaeger construction

Augmented Dodecahedron

Mid-edge construction

Truncated Augmented Dodecahedron

Stellated Augmented Dodecahedron

Geodesated Augmented Dodecahedron

Augmented Dodecahedron

Cross-curve construction

Jeager construction

Augmented Dodecahedron

Graph 3D

Iterative mid-edge construction

Seifert surfaces

Polyhedral Seifert surfaces

Augmented Dodecahedron

Fullerenes C60

C120

“fullgen” by Gunnar Brinkmann

n=120 10774

Virtual knots and links (Louis Kauffman)

Non-realizable Dowker codes

Knots and links on different surfaces

Hopf link Borromean rings (K5)

K5 (crossing number, Borromean rings with a virtual crossing, Simmons-Paquette molecule)

Non-planar graphs derived from Gauss codes of virtual knots and links

6*(i,2)

"1599*i.2.2.i.i.2 0.2 0.2 0.2.i.i.2.2 0.2 0.2"

FlowerSnark J5