Welcome to The Polyknot and Polylink Atlas

This atlas currently includes over 300 structures, with ongoing additions. It is organized into two major groups: the Escher tradition and the Roelofs tradition of polyknots and polylinks. The collection serves as a foundational resource for researchers, mathematicians, and scientists interested in topological structures constructed from geometric components—including points, lines, polygons, and polyhedra.

The encyclopedic material below provides essential background for exploring the six catalog categories. Additional resources—such as 3D‑printable models and kits, and topological nets—are included for instructional use. A bibliography and glossary are provided for reference and further study.

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 star polyhedra (Figure 2). 

(Figure 1)

(Figure 2)

History:

In artist Rinus Roelofs' 2008 paper, Connected Holes, the conceptual origins of polyknots and polylinks can be traced to the mathematical art of M. C. Escher, whose work contains early examples of polyhedral structures with interwoven or interpenetrating components. Two pieces in particular — Double Planetoid (1949) and Gravity (1952) — anticipate the essential ideas behind polyknots and polylinks.

In Double Planetoid, Escher depicts two interlocking polyhedral worlds, each composed of creatures inhabiting a distinct geometric domain. Although not a true polylink, the image introduces the idea of interpenetrating polyhedral frameworks. In Gravity, Escher presents a small stellated dodecahedron with portions removed, revealing a dodecahedral core. This construction suggests a layered, interwoven polyhedral structure — a conceptual precursor to later polyknots and polylinks.

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 extended the idea that polyhedral frameworks could support linked or interwoven structures, opening new avenues of future work.

In Rinus Roelofs' same paper of 2008, he develops what he called entangled double‑layered polyhedra. His first step was to open up Escher's dodecahedron which enabled him to identify the presence of a trefoil knot at each vertex of a dodecahedral polyknot( Figures 3 and 4). 

(Figure 3): Rinus Roelofs, from Connected Holes (2008): the opening of Escher’s dodecahedron and the discovery of the trefoil knot at the vertex of the sculpture.

(Figure 4): Rinus Roelofs, from Connected Holes (2008): trefoil knot derived from the tetrahedral and cubic polyknots.

This subsequently led to their renaming as polyknots since they have the symmetries of polyhedra and the crossing numbers characteristic of knots. In addition, his paper and website include an extensive collection of sculptural forms derived from Platonic, Archimedean, and related polyhedra, many of which exhibit the essential features of the polyknots on this site.

More recently, these structures have been standardized with a uniform design system based on triangulation, enabling their organization into six major polyhedral categories and a significant expansion of known varieties. The extension of polyknots and polylinks to constructions based on points, lines, and polyhedra represents a promising frontier for future research—a prime example being the forthcoming addition of Holden radial polylinks composed of prisms. 

A further development is the emergence of edge hinged modular polyknots, whose articulated components allow the structures to expand, contract, or retract across layers. This discovery introduces a mechanical dimension to polyknots and suggests new possibilities for transformable or kinetic designs, several of which will be shown on this site in the near future. 

Nomenclature: Trinomial System

Naming conventions for these structures follow a strict, three-part trinomial system to ensure clarity across the database: 

Example: Escher cubic polyknot

1. The naming of polyknots and polylinks begins 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. 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 convention 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 to which the structure belongs.   

Taxonomy: The STEAM CAD System

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 departmental 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. 

More information is available at STEAM CAD  

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 5).

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

(Figure 5)

B. Global Crossing Number: 

Since polyknot and polylink crossings are located at the edges that correspond to those of their derived polyhedra, the global crossing number is the sum of all the crossings of any given polyknot or polylink and is equal to the edge number 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 30 edges, the global crossing number is:

E=30 (Figure 2).

• Similarly, an octahedral polylink has a local link number of four, reflecting the four triangles meeting at each vertex of the octahedron. With 12 edges, the global linking number is:

E=12.

This is 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 variant 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]/2=C where C is the global crossing number.  

For example, the base cube of a Roelofs cubic polyknot has:

• V=8 vertices

• E=12 edges

• F=6 faces

Applying the Euler‑based formula:

[(8+12+6) −2]/2=12

Thus, the total crossing number of the Roelofs cubic polyknot is 12.

A third expression for the global crossing number follows directly from combining Methods 1 and 2. Since polyknots satisfy E=C, substituting into Method 2 yields the simplified identity V+F−2=C. This provides a compact way to compute C while remaining fully consistent with the earlier formulas. 

These methods provide 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.

D. Chirality of Polyknots and Polylinks

Polyknots and polylinks all exhibit chirality, meaning they occur in distinct right‑handed and left‑handed forms that cannot be superimposed by any rigid motion in three‑dimensional space. Chirality arises whenever the construction process breaks mirror symmetry, either through the edge‑based knotting pattern or through the directional layering of components.

In this atlas, chirality is determined by the orientation of the local knot or link structure assigned to each edge of the base polyhedron. This orientation propagates to the knots located at each vertex. Because these local structures are repeated uniformly across the polyhedron, the entire polyknot or polylink inherits a consistent global handedness. As a result, all polyknots and polylinks appear in chiral pairs, analogous to classical knot chirality such as the right‑ and left‑handed trefoil.

Chirality is a topological property: a right‑handed polyknotted structure cannot be continuously deformed into its left‑handed counterpart without passing through a self‑intersection. In the interest of economy, only the right‑handed versions of the standardized models are depicted on this site.

Geometric Properties of Components: 

Because the polyknots and polylinks in this database are constructed from geometric components, their geometric properties are straightforward to compute. The lengths, areas, perimeters, surface areas, and volumes can be calculated without need of advanced mathematical training. 

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