2.6 Basic polyhedra and non-algebraic tangles

Basic polyhedra are well known from works by T.S. Kirkman (who called them "solid knots") (1885a), J. Conway (1970), and A. Caudron (1982). In the first part of his paper (June 2, 1984) T.S. Kirkman wrote:

"Of solid knots we are not treating. If the apparent dignity of knots so maintains itself as to make a treatise on these n-acra desirable, it will be no difficult thing to show in a future
memoir how to enumerate and construct them to any value of n without omission or repetition.The beginner can amuse himself with the regular 8-hedron, which is trifilar, or with the unifilar of eight crossings made by drawing within a square askew, and filling up with eight triangles."

and in its Postcript (September 1, 1984):

"As it is is a brief matter, it may be worth the wile to show how all solid knots can be constructed without omission and repetition."

A century after, the "missing" basic polyhedron 12E with n=12 crossings was discovered by A. Caudron (1982). A. Caudron derived basic polyhedra with n £ 12 crossings by combining
non-algebraic (hyperbolic) tangles.

Every basic polyhedron is a 4-regular, at least three edge-connected graph without bigons. Polyhedral KLs can be derived from basic polyhedra by substituting their vertices by algebraic tangles. In order to work with basic polyhedra and KLs derived from them we need the tables of basic polyhedra with their drawings. The information tables provide for every basic polyhedron are the order of vertices and the orientation of tangles, so we have again a data base, very similar to the classical Alexander-Briggs notation of KLs, but now for basic polyhedra. In the program the list (data base) of basic polyhedra is extended to n £ 20 crossings.

Basic polyhedra with n £ 11 crossings and source links derived from them permit no flypes. Each basic polyhedron has only one projection, so there is one-to-one correspondence between them and their alternating diagrams, i.e., between them and their corresponding alternating KLs.

Unfortunately, this does not hold for the basic polyhedra with 12 £ n crossings. This probably explains the "mystery of the missing basic polyhedron" 12E. Among the basic polyhedra with
£ 12 crossings, this is the only basic polyhedron that is a two vertex-connected graph and the first basic polyhedron permitting flypes. It has two non-isomorphic alternating diagrams- its other projection is the link 11***2.

A non-algebraic tangle is a tangle that cannot be obtained from elementary tangles 0, 1 and -1 by using two operations, sum and product. Two infinite series of non-algebraic tangles: first series 5* , 81* , 111* , 141* , 171* ,... with n=3k+2 crossings, giving as the numerator closures the link 2 1 2 and the basic polyhedra 8*, 11**, 141*, 171225*,... The second is  7*, 91*, 112*, 131*,... with n=2k+5 crossings. They are are illustrated in the corresponding figure. Let 5* and 7* denote non-algebraic tangles with n=5 and n=7 crossings, respectively. The numerator closures of the products 5* 1, 7* 1, 91* 1 are the basic polyhedra 6*, 8*, 10*, respectively. We can distinguish elementary basic polyhedra containing at most one non-algebraic tangle, and composite basic polyhedra containing at least two non-algebraic tangles. In this way, the basic polyhedron 10*** can be represented as 5*  5* , 11*** as 5*  1 5* . Applying flypes, we obtain nothing new: they have only one minimal alternating diagram. The first exception will be the basic polyhedron 12E, that can be denoted by 5* ,1,5* ,1. If we apply a flype, we obtain its other projection 5*  2 5* , corresponding to the link 11***2. There is a complete analogy between the first rational link 2 2 2, that has two projections, and the first basic polyhedron 12E, expressed as 5*  2 5*, with the same property. In the same way, we can obtain other basic polyhedra having more then one minimal alternating diagram, e.g., 1318* and 136* that are two non-isomorphic projections of 5*  1 1 1 5*, where 1318* corresponds to the projection 5* 1 1 1 5* . By substituting tangles for their vertices, we can obtain the same KLs. However, there are KLs that can be derived from one, but not from some other projection. For example, the link 125*2 0:::2 0=5*,2,5*,2 cannot be obtained from the basic polyhedron 11***

Among the basic polyhedra with n=12 crossings, tree are composite: 12E=5* ,1,5* ,1 ~ 5*  2 5* = 11***2, 12I=7*  5* , and 12J=5* 1 1 5* . For n=13 composite basic polyhedra are: 131*=81*  5* , 135*=82* 5* , 1318*=5*  1 1 1 5*   ~ 136*, 139*=7*,1,5* , 1313*=7*  1 5*   ~ 1311*, 1319* = 5*  1,1,5* ,1  ~ (5* ,1) 2 5* = 1210*2. From the given results follows that the links 12E  ~ 11***2, 1318*  ~ 136*, 1313*  ~ 1311*, and 1319*   ~ 1210*2 have two different minimal diagrams each. All other KLs corresponding to the basic polyhedra with n £ 13 crossings have a single alternating diagram.