Knot wormholes and the dimensional invariant of exceptional Lie groups and Stein space hierarchies

The present short note points out a most interesting and quite unexpected connection between the number of distinct knot as a function of their crossing number and exceptional Lie groups and Stein space hierarchies. It is found that the crossing number 7 plays the role of threshold similar to 4 and 5 in E-infinity theory and for the 11 crossing the number of distinct knots is very close to 4 a0 þ 1 ¼ 548 þ 1 ¼ 549; where a0 ¼ 137 is the inverse integer electromagnetic fine structure constant. This is particularly intriguing in view of a similar relation pertinent to the 17 two and three Stein spaces where the total dimension is X 17 1 Stein ¼ 5 a0 þ 1 ¼ 685 þ 1 ¼ 686; as well as the sum of the eight exceptional Lie symmetry groups X 8 i¼1 jEij ¼ 4 a0 ¼ 548: The slight discrepancy of one is explained in both cases by the inclusion of El Naschie’s transfinite corrections leading to X 8 i¼1 jEij ¼ ð4Þð137 þ k0Þ ¼ 548:328157 and X 17 i¼1 Stein ¼ ð5Þð137 þ k0Þ ¼ 685:41097; where ko = /5(1 u5) and u ¼ ffi5ffiffi p 1 =2.