Welcome to The Polyknot and Polylink Library

This library currently includes over 300 structures, with ongoing additions. It is split between the Escher and the Roelofs polyknots and polylinks. It serves as a foundational resource for researchers, mathematicians, and scientists with a shared interest in topological structures constructed from geometric components—including points, lines, polygons, and polyhedra.

The encyclopedic material below provides essential background before exploring the six catalog categories. Additional resources such as 3D printable models and kits, polylink diagrams, as well as topological nets are provided for teaching purposes. A Bibliography and Glossary are included for reference and further research. 

Table of Contents

Definitions: 

Polyknots and polylinks are three dimensional analogues of knots and links with polyhedral symmetries and asymmetries depending on the base polyhedron from which they are derived (Figure 1). For example, the Roelofs polyknots and polylinks are derived from  polyhedra such as the Platonic and Archimedeans among others. While the Escher polyknots and polylinks are derived from similar pyramidal augmented star polyhedra. 

(Figure 1)

History:

The origins of polyknots can be traced to the world of mathematical art. The earliest precursor appears in M. C. Escher’s 1952 lithograph Gravity, which depicts a small stellated dodecahedron with one vertical half removed from each triangular face of its pentagonal pyramids, revealing the dodecahedral core within. Although visually suggestive of later developments, Escher’s construction is not a true polyknot, as it lacks the clearly defined under‑and‑over crossings characteristic of knot‑like structures.

A major step forward came in 1972 with chemist Alan Holden’s book Orderly Tangles. Holden introduced polylinks exhibiting polyhedral symmetries and described examples constructed from polygon rings and skeletal polyhedra. His work established the idea that polyhedral frameworks could support linked or interwoven structures, laying conceptual groundwork for later developments.

The first publication to explicitly describe and develop polyknots appeared in 2008, when mathematical artist Rinus Roelofs introduced what he called double‑layered polyhedra in his paper Connected Holes. Roelofs presented an extensive collection of sculptural forms derived from Platonic, Archimedean, and related polyhedra, many of which exhibit the essential features of modern polyknots.

More recently, these structures have been renamed polyknots to emphasize their dual nature: they possess both topological crossing numbers and the symmetries of polyhedra. Their diversity has expanded significantly through constructions based on geometric components such as polygons, with future possibilities involving points, lines, and polyhedra yet to be fully explored.

Nomenclature: Trinomial System

Example: Escher cubic polyknot

1. Naming conventions for polyknots and polylinks begin with the individuals who first described them or the sources that inspired their construction. Roelofs polyknots and polylinks are named in honor of artist Rinus Roelofs, who first documented and developed these structures in 2008. Although M. C. Escher never formally described polyknots or polylinks, his geometric explorations and visual motifs have inspired a distinct family that bears his name.

In cases where a structure draws from multiple influences, the naming reflects this combined heritage. For example, a form inspired by both Roelofs’ constructions and Platonic solids may be designated a Roelofs–Platonic polyknot or polylink. This approach ensures that both conceptual and geometric origins are acknowledged within the classification system.

2. The base polyhedron from which the polyknot or polylink is derived is incorporated following the attribution.  

3. The last term indicates the category either a polyknot or polylink that the structure belongs to.   

STEAM CAD Taxonomy:

The STEAM CAD classification system is hierarchical and modeled on both Linnaean biological taxonomy and the four‑level nested structure of academic organization: departments, disciplines, categories, and objects. Within this framework, polyknots and polylinks are classified across four levels, from their specific geometric construction to their broader disciplinary context.

Object Level (Level 1)

At the object level, polyknots and polylinks may be classified in two complementary ways:

1. By structure and structural properties, such as the specific polyhedron from which they are derived (e.g., a cube, tetrahedron, or other uniform solid) and associated characteristics like crossing numbers.

2. By components and component properties, based on the geometric elements used in their construction—points, line segments, specific polygons (e.g., triangles, squares), and specific polyhedra (e.g., cubes, tetrahedra)—along with the properties of those components.

Category Level (Level 2)

At this level, objects are grouped according to the broader categories their structures or components belong to. For example, Roelofs polyknots and polylinks may be classified by the structural families from which they arise, such as the Archimedean polyhedra, or by the component categories of points, lines, polygons, and polyhedra. This level organizes individual objects into coherent structural or compositional families.

Discipline Level (Level 3)

The categories of Level 2 are combined into a broader discipline. In the case of polyknots and polylinks, this discipline is structural geometric topology, reflecting the fact that these objects are topological in nature but constructed from geometric components. 

Department Level (Level 4)

Because structural geometric topology is a composite discipline—arising from the combination of geometry and topology—it is situated within the Metamathematics Department, where composite mathematical disciplines are constructed and studied. 

For more information about the STEAM CAD system, see: Structural Geometric Topology 

Topological Properties of Structures:

Crossing Numbers

A. Local Crossing Number

The local crossing number of a polyknot or polylink is determined by the vertex configuration of the base polyhedron from which the structure is derived. The parity of this configuration provides a natural distinction:

• Odd vertex configuration → polyknot

• Even vertex configuration → polylink

This correspondence arises because the number of faces meeting at each vertex maps directly to the number of crossings in the resulting structure when the polyhedral framework is converted into a knot‑like or link‑like form.

For example:

• A cube has a vertex configuration of three squares meeting at each vertex. When this configuration is mapped into a cubic polyknot, it corresponds to the trefoil knot, which has a crossing number of three.

• An icosahedron has a vertex configuration of five triangles around each vertex. When mapped into an icosahedral polyknot, this configuration corresponds to the cinquefoil knot, which has a crossing number of five (Figure 2).

This method provides a direct and intuitive way to associate classical knot invariants with polyhedral‑derived structures, linking geometric configuration to topological complexity.

(Figure 2)

B. Global Crossing Number: 

The global crossing number of a polyknot or polylink is calculated by multiplying the vertex crossing number by the number of vertex knots or links present in the structure. This provides a total measure of topological complexity based on the repeated local crossing pattern at each vertex of the base polyhedron.

For example:

• A icosahedral polyknot has a vertex crossing number of five, corresponding to the five triangles meeting at each vertex of the icosahedron. Since the icosahedron has 12 vertices, the total crossing number is:

5x12=60 (Figure 2).

• Similarly, an octahedral polylink has a link vertex crossing number of four, reflecting the four triangles meeting at each vertex of the octahedron. With six vertices, the total linking number is:

4×6=24

This method provides a consistent way to extend classical knot and link invariants to polyhedral‑derived structures by scaling local crossing behavior across the full symmetry of the base polyhedron.

C. Crossing Numbers and Euler's Characteristic: 

A convenient way to calculate the total crossing number of a polyknot or polylink is to use a varient of the Euler characteristic of the base polyhedron from which the structure is derived. For any convex polyhedron, the Euler characteristic is:

χ=V-E+F=2

where

• V = number of vertices

• E = number of edges

• F = number of faces

For polyknots and polylinks, this expression is 

V+E+F-2 

and when summed gives the total crossing number.

For example, the Roelofs Cubic Polyknot—a polyknot derived from a base cube and constructed from cubes—has:

• V=8 vertices

• E=12 edges

• F=6 faces

Applying the Euler‑based formula:

(8+12+6) −2=24

Thus, the total crossing number of the Roelofs Cubic Polyknot is 24.

This method provides a fast, generalizable way to compute total crossing numbers for any polyhedral‑derived structure, linking classical topological invariants to the combinatorial geometry of the underlying polyhedron.

Geometric Properties of Components: 

Because the polyknots and polylinks in this database are constructed from polygons, their geometric properties are straightforward to compute. The areas and perimeters of these polygons can be summed directly, providing simple aggregate measures of the structure’s two‑dimensional components.

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