Borromean Rings provides knots in the working range. Borromean Rings is a multi-line link of six crossings that is found in the knots and links tables as L6a4. It is the first ordered knot or link that has no twists. For this feature of having no degenerate twists, it carries the Conway knot notation, 6*. As seen in the tables, Borromean Rings is an invariant of knot crossings that can be identified as a projection of a geometry. That geometry is of a triangular antiprism, or equivalently, an octahedron.
As an instance of a geometry, the Triangular Antiprism is the model for a number of practical knots. This is the common attribute of all knots taken from the tables of knots and links as projections of a geometry, that their possible use as practical objects extends from a choice of projection in a tying diagram.
Here by example, simply by choice of projection and choice of working ends, we see Borromean Rings variously as 1) a lashing, 2) a hitch, 3) a bend, 4) a fixed loop. In each case, the practical knot faithfully follows a projection of the invariant geometry of the Triangular Antiprism.
The practicality of this method, of faithfully following a mathematical invariant of a geometry, is made clear when we consider the purposes for making tying diagrams, the purposes for making knots. We see our requirements as being available in a geometry and the only further requirement is to utilize the form with our need in mind. In a bend, the two working ends are directed toward and past each other, keeping to the path within the geometry, until there is sufficient friction in knot crossings to make the knot secure. In a fixed loop, the method is to pick an edge on the projected geometry that is to be called the loop, and to make this chosen edge far away enough in numbers of knot crossings from both the standing end and the working end, for friction to hold the applied forces. in a lashing or hitch, the working end follows the path of one line of a multiline projection, around the other line(s) of the projection where the other line(s) are the object(s) lashed or hitched upon. This is continued until sufficient friction is obtained. If these simple rules of choice in tying are adhered to then all of the dependence on knot settings and knot finishes shall be seen as nonsensible wish making for security.
Further, this aspect of practical knots conforming to a geometry can bring understanding to the primary knot form, the Carrick. The Carrick form, called the Josephine, is a pattern of saturated over-under crossings composing a square and six triangles. This figure as circuitry is completable in two significant ways. One way to complete the circuit is as Carrick Mat or Turkshead 3x4, another way to complete the circuit is as Carrick Bend or Turkshead 4x3. These completed circuits are then identifiable in the tables of knots and links as Rolfsen 8_18 and Rolfsen 9_40, respectively. As such, the Carrick will be seen to represent the second ordered and the third ordered of all knots and links that do not have twists (Conway 8* and 9*). The geometry of Rolfsen 8_18 is that of the Square Antiprism. The geometry of Rolfsen 9_40 is that of the Gyro-elongated Triangular Antiprism. Here we demonstrate, in a paper construction, the form of the Gyro-elongated Triangular Antiprism.
With the vision that a knot form can be the projection of an unvarying geometry, there is then the possibility of choosing different projections, choosing different tying diagrams, for a given geometry, including when using the two simple geometries that share the presence of the Carrick form. With Carrick Mat, there are two possible projections of the Square Antiprism. One projection is centered on the square which is the top or bottom of the Square Antiprism, this square then surrounded by the ring of triangles that gird any antiprism, and then the outer bound is the other square of the Square Antiprism. This is the IGKT logo. This could be called a Polar projection of the Square Antiprism. The other possible projection of the Carrick Mat is centered over both the triangles and the squares of the Square Antiprism with the outer bound being a last remaining triangle that completes the geometry. This could be called a Girder projection for the place that the ring of triangles have in all Antiprisms.
For Carrick Bend or Rolfsen 9_40, there are two similar projections to Carrick Mat and one additional projection. First is a Polar projection bounded on the outside by the top or bottom triangle of the Gyro-elongated Triangular Antiprism. Second is a Girder projection bounded on the outside by any one of the triangles that ring around the geometry. The additional, third projection is one that is bounded on the outside by one of the three squares of the gyro-elongation. This could be called the Elongate projection. Here we show practical knots obtained by various projections of the two geometries. What is remarkable is how the different tying diagrams provide different dynamics within the knot without changing the basic principle that sufficient knot crossings provide sufficient security.
