Alternating polyhedral KLs with s = 2 are given in the following table, each with its BFR. KLs in this table are given in "standard" Conway notation (that is "standardized" for knots with n £ 10 and links with n £ 9 crossings according to Rolfsen's book (1976)). This table can be extended to an infinite list of antiprismatic basic polyhedra (2n)* described by the BFRs (Ab)n, n ³ 3 and BFRs with s = 2 obtained as their extensions. 

Table 1

Basic polyhedron .1 = 6*
 
 
ApbAbAb  .p  (1)  ApbAbAqbr r:p 0:q 0  (7) 
ApbAbAbq .p.q  (2)  ApbAbqArbs p.s.r.q (8) 
ApbAqbAb  .p.q 0  (3)  ApbAqbArbs q 0.p.r 0.s 0  (9) 
ApbAbqAb  .p:q 0  (4)  ApbAqbrAbs .p.s.r 0.q 0  (10) 
ApbAqbAbr .r.p.q 0  (5)  ApbAqbrAsbt p.t.s.r.q  (11) 
ApbAqbAr p:q:r  (6)  ApbqArbsAtbu p.q.r.s.t.u  (12) 

If we apply minimum braid criteria (Gittings, 2004), we need to add ten braids for the basic polyhedron .1 = 6*: (1') ApbAbqAbr, (2') ApbAqbrAb, (3') ApbqAbAbr, (4') ApbAqbrAsb, (5') ApbqAbArbs, (6') ApbqAbrAbs, (7') ApbqArbAbs, (8') ApbqAbrAsbt, (9') ApbqArbAsbt, (10') ApbqArbsAbt. Applying BFR criteria, according to the minimum source braid criterion all KLs obtained from the braids (1') and (2') will be obtained from BFR (5), KLs obtained from (3') will be obtained from (7), KLs obtained from (4') and (6') will be obtained from (9), KLs obtained from (5') and (7') will be obtained from (8), and KLs obtained from (8'), (9') and (10') will be obtained from (11). Using minimum braid criteria (Gittings, 2004), we need to make analogous additions to all classes of BFRs considered. 

PreviousContentsNext