Non-Prime Diagrams

Knotilus is designed to accept any gauss code for a knot or link that does not exceed 1,000 characters in length (this restriction is due to html limitations).

The java annealer will allow any knot or link to be sketched and from the sketch, the annealer constructs a gauss code for an alternating diagram constructed from the sketch. This gauss code can then be copied into any editor to enable one to make further modifications, such as to add signs to convert the code into one for a non-alternating link with that diagram.

Knotilus also accepts Dowker codes for knots.

A proper or unambiguous diagram drawn in the plane (or projected onto the plane), representing a knot or link is such that each crossing is represented as a vertex with four edges only, referred to as a vertex of degree four.

A non-prime diagram is one for which it is possible to draw a closed curve that crosses the diagram exactly twice in such a way that the region inside the closed curve and the region outside the closed curve each contain at least one crossing of the diagram. Each of the two regions is referred to as a 2-tangle of the diagram.

Knotilus' annealing forces are set to give pleasant looking diagrams for a prime knot or link. However, Knotilus does surprisingly well at rendering non-prime diagrams, but it is hit and miss. There are a number of ways in which the user can take control in these situations. To start, always enter a gauss code to Knotilus and see what it does and be sure to look at all the face inversions. The first embedding presented is not necessarily the most satisfactory by your aesthetics. If none of the face inversions were to your liking, then click on java annealer (in green on the main page) and play with the settings for the forces between edges and crossings, etcetera. The java annealer gives you complete control when clicking off temperature.

If you prefer that Knotilus do the annealing then follow these next suggestions and note that these suggestions can be applied even if the java annealer is used.

Taming The Nugatories

A nugatory crossing (coined by P.G. Tait) is a crossing that can be twisted out as if it were never there in the first place. A lot of people like nugatory crossings in their knots and links for decorative purposes (see discipline1 and dsize10 for example), and for others it has phenomenological importance. If a diagram is non-prime due to nugatories then the nugatory crossings could end up in one of two adjacent faces which may not be the face you wanted the nugatory in. Consider figure 1 and figure 2. The desire was to have the six nugatory towers (or a tail of three crossings) to be in one face. Knotilus rendered figure 1 where we see only two of the six nugatory tails ended up in one face. Figure 2 which is the same link with the same number of three crossing nugatory towers but all six tails are in the same face, namely the outside face. This was achieved by connecting all six tails with a single ring then removing the ring which we call a control ring. See figure 3. Knotilus has the ability to remove components one at a time always starting with the last component in the gauss code provided. Observe with the control ring (figure 3) the diagram is prime hence the annealing by Knotilus provided a nice rendering. It's not always necessary to make sure the diagram is prime for a better rendering of nugatories. Look at figure 4, figure 5, and figure 6 where 4 and 6 are the same link. Figure 4 has no control rings and figure 5 has three control rings (coloured orange, purple and yellow) and figure 6 has the control rings removed giving the desired outcome. One can continue to experiment with the notion of adding more components to alter the outcome of non-prime diagrams and we encourage your feedback with reference to interesting results.

Composite Knots and Links

The other type of non-prime diagrams is due to composite knots and links. Have a look at the composite knots and links in the two galleries. Knotilus in general does a pretty good job rendering these type of non-prime diagrams but sometimes it may be the case that some part of the diagram was not in the face you had wanted it to be. Here adding more components as control components will guarantee the placement you desire. As an example look at figure 7 and figure 9. The subtle difference is the blue and red trefoils were flipped in figure 9 which required a control ring as shown in figure 8. In principle, one could start with a grid of diagrams then put more diagrams on top. Altogether they tack each other in place.

Although Knotilus does not provide 2-tangle inversion to place each 2-tangle in the desired face in the diagram, the java annealer does thus extra components are not needed to arrange 2-tangles as desired. Control rings can still provide exciting outcomes that altering the forces alone could not achieve without some investment in time and a good acquaintance with annealing.