Structural Geometric Topology by Albert P. Carpenter

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Glossary for Structural Geometric Topology

Complimentary: two objects that are heteromorphs yet share a common structural component

example: the two Heteromorphic Monocycles below are complimentaryDSC00473

Composition: the objects used to construct a structure

example: the Heteromorphic Monocycle below is composed of polygons (squares and regular pentagons)DSC00590

Chromoticity number: number of colors necessary to reveal the structural elements of a given topological structure

example: the Heteromorphic Monocycle below has a chromoticity number of fiveDSC00590

Cycle: a ring topological structure that is either a knot, link, torus or Mobius strip

Discrete Annulus: a closed cyclic topological structure with a positive annulus number

example: DSC00600

example: the torus below is a Heterogenous MonocycleDSC00573

Heteromorph: two or more objects that have a different shape or morphology

example: the first structure (a) is a heteromorph to the second (b)DSC00600

(a)

DSC00594

(b)

Homomorph: two or more objects that have the same shape or morphology

example: the sieve below is homomorphic to the Knotted Mobius strip next to it

DSC00595

Knot: an open or closed interwoven topological structure with a positive knot number

example:

Knot number: the number of crossing in a knot

example:

Link: two or more interwoven (i.e. with a positive link number) though unattached tori

example: the link belowDSC00523

Link number: the number of crossings in a link

example: the link below has a link number of 6DSC00523

Mathematical property: a quantitative or qualitative feature of a topological structure that contributes to its identity

example:

Mobius strip: a closed topological structure with a positive Mobius number

example: the Mobius strip below is single sided i.e. odd numbered  with a twist number or Mobius number of 5DSC00590

Mobius number: the number of twists a topological structure has such as in a Mobius strip

example: In the example below the Mobius strip has five twists that corresponds to its chromoticity numberDSC00590

Nomenclature: In Geometric Topology a binary system based on composition and structure

example: Heteromorphic MonocycleDSC00600

Ontology: a catalog of objects

example: this web page

Physical analog: the physical object that corresponds to the mathematical object

example:

or the Homohexagonal Monocycle:

DSC00536

Diagram of Coronene and physical analog of 6^6 above

https://upload.wikimedia.org/wikipedia/commons/d/d8/Coronene_structure.png

Polycycle: a multiringed topological structure

example: a sieve with a sieve number of 4DSC00576

Polygonal Knot: an open or closed topological structure with a positive knot number (greater than 0) and whose composition is that of polygons

example:

Polygonal Link: an open or closed topological structure with positive link number (greater than 0) and whose composition is that of polygons.

example:

Polygonal Mobius Strip: an open or closed topological structure with a positive Mobius number (greater than 0) and whose composition is that of polygons.

example:

Polygonal Annulus: a closed topological structure with a positive torus number (genus) (greater than 0) and whose composition is that of polygons.

example:

Scale Invariance: two or more objects that are homomorphs independently  of scale

example:

Scale Variance: two or more objects that are heteromorphs with scale

example:

Structural component: a symmetry operation on topological structures

example:

Structure:

example:

Symmetry:

example:

Symmetry group:

example:

Symmetry number:

example:

Taxonomy: the classification of objects based on shared properties

example:

Topological analog: The corresponding mathematical structure to it physical counterpart

example:

 

 

 

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