Synthesising Topological Links

Abstract

We discuss the chemical synthesis of topological links, in particular higher order links which have the Brunnian property (namely that removal of any one component unlinks the entire system). Furthermore, we suggest how to obtain both two dimensional and three dimensional objects (surfaces and solids, respectively) which also have this Brunnian property.

Keywords: Brunnian links; Hopf links; Synthesizing double stranded DNA nodes; Synthetic DNA topology of links; carpets and solids.

Fig. 1

The Hopf Family

Fig. 2

The Brunnian Family

Fig. 3

The Hopf Link

Fig. 4

Hopf Links to Hopf Chains

Fig. 5

A Hopf Ring

Fig. 6

A Brunnian Ring

Fig. 7

The Borromean Rings and the Brunnian Ring of Length 3

Fig. 8

First Order Hopf Ring of Length 3

Fig. 9

Brunnian Chain

Fig. 10

Schematic Brunnian chain with eight components.

Fig. 11

Schematic Brunnian chain with shifted components.

Fig. 12

Segment of a Brunnian carpet.

Fig. 13

Segment of a Brunnian carpet with closed running edges.

Fig. 14

Building a torus from a square.

Fig. 15

A Brunnian carpet.

Fig. 16

Layered carpets

Fig. 17

The Brunnian surface components.

Fig. 18

Brunnian cube of four components.

Fig. 19

Segment of a Brunnian carpet with unlinkings marked.

Fig. 20. The Relationship Between Nodes and Antiparallel B-DNA Illustrated on a Trefoil Knot

A trefoil knot is drawn with negative nodes. The path is indicated by the arrows and the very thick curved lines connecting them. The nodes formed by the individual arrows are drawn at right angles to each other. Each pair of arrows forming a node defines a quadrilateral (a square in this figure), which is drawn in dotted lines. Each square is divided by the arrows into four of domains, two between parallel arrows and two between antiparallel arrows. The domains between antiparallel arrows contain lines that correspond to base pairing between antiparallel DNA (or RNA) strands. Dotted double-arrowheaded helix axes are shown perpendicular to these lines. The amount of DNA shown corresponds to about half a helical turn. It can be seen that three helical segments of this length could assemble to form a trefoil knot. The DNA shown could be in the form of a 3-arm DNA branched junction. A trefoil of the opposite sense would need to be made from Z-DNA, in order to generate positive nodes.

Fig. 21. PX DNA

PX-DNA is drawn with strands of two colors, red and dark blue. The four strands form a structure with a central dyad axis (indicated by green arrows), and consisting of two double helical domains that are linked every half-turn. The double helical domains are indicated by small black arrows above and below them. Half-turns near the dyad axis corresponding to minor (narrow) groove spacings are indicated by ‘N’ and half-turns corresponding to major (wide) groove spacings are indicated by ‘W’. The helical pitch consists of four of these half-turns. Each duplex contains a repeat of a red-red half turn above a red-blue half turn, below which is a blue-blue half-turn, and then a blue-red half turn, before the sequence repeats. Note that the central portion of the PX structure consists of a right-handed (negative) node. at each point.