The function Symm (webMathematica Symm) calculates all automorphisms of an alternating KL projection given by its Conway symbol, Dowker code, or P-data. The first part of the result is the list of automorphisms given by permutations, and the other is the list of the corresponding cycles.
Working with all non-isomorphic projections of an alternating KL given in Conway notation, the function MaxSymmProjAltKL (webMathematica MaxSymmProjAltKL) finds its maximum symmetrical projection, giving as an output the number of automorphisms and the maximum symmetrical projection found. For example, among six non-isomorphic projections of the knot 2 1 1 1 1 1 2, the function finds the most symmetrical projection 2 1 1 1 1 1 2, with the order of automorphism group 2.
In the case of links, we can continue with link surgery and obtain two more link invariants. The first of them is the splitting number defined as a minimum number of crossing changes over all projections of a link required to obtain a split link, that is, a link with split components, not necessarily unknotted. The idea of splitting number was proposed by C. Adams (1996), who described an interesting example of a link, where changing a specified crossing turns the link into a splittable link, while at the same time changing one of the trivial components into a trefoil knot. Instead of Adams' example we are proposing a simpler example of a link with splitting number 1, and with unlinking number 2, that will be split by one crossing change that will knot one of its trivial components- the link .2 (or 762 in classical notation). The next link of that kind will be .2.2.2 0, etc. Certainly, without any difficulty it is possible to find an infinite family with the same property: all links of the family .p 1, (p ³2) have the splitting number 1 and unlinking number 2. After one crossing change, one of their trivial components (a component that it is not a circle surrounding the critical crossing) will be knotted, resulting in a knot p 2. If you are searching for link families with an unlinking number greater then the splitting number, one of them is the family .(2k), (k³ 1). In general, we could define a splitting gap- difference between unlinking number and splitting number of an alternating link L. Because for every member of the family .(2k), (k ³1) the unlinking number is k+1 and the splitting number is always 1 (that is clearly visible from the corresponding figures), the splitting gap is k, so it can be arbitrarily large. That holds even for all links of the form .(2k) (2l), (k,l ³1). The first property (that the splitting number is equal to 1 for all such links) is possible to recognize directly. At first glance, it seems that the second property (that the unlinking number of a link is still k+1) is very complicated to show, but just recognize that the best strategy to unlink a link .(2k) (2l) is to get rid of the "circle", and then to unknot the remaining part. In this case we need one crossing change for the first step, and then k crossing changes for the second. Hence, for all links of the form .(2k) (2l), ( k,l ³ 1) the unlinking number is again k+1, the splitting number is always 1, so the same property still holds. Even more general: for every link of the form .(2a1) (2a2)...(2an), (a1,...,an³ 1), the splitting number is always 1, and the unlinking number is åi = 1[[(n+1)/ 2]]a2i-1. This suggests an interesting question: do there exist 3-component links with a splitting number 1? The answer is: probably not!
The LinKnot function SplittNo (webMathematica SplittNo) calculates the splitting number of a link projection given by its Conway symbol, Dowker code, or P-data. In the same way as before, we may extend this function to all minimal different (non-isomorphic) projections of an alternating link.
For the next link invariant we will use even more drastic surgery: we will cut one component of a link L, but without gluing it again. Then we will look at what remains from L. We will continue with this procedure till we obtain a split link. The minimum number of component breaks is the cutting number of a link. Working with rational links, we cannot obtain any attractive or surprising result: every rational link is 2-component, so for all rational links the cutting number will be 1.