A Partial Order in the Knot Table

Remarks on Suzuki’s knot epimorphism number

“…If it were always the case that the crossing number of J is at least 3 times the crossing number of K whenever J > K, then this would provide a proof of Simon's Conjecture, that a knot group can only map onto finitely many other non-trivial knot groups. While Simon's Conjecture is known to be true [3], it is not true that the bigger knot must always have 3 times as many crossings as the smaller knot, for Kitano and Suzuki have shown that the 8-crossings knots 8 5 , 8 10 , 8 15 , 8 18 , 8 19 , 8 20 and 8 21 are all greater than or equal to the trefoil knot 3 1 [8]. However, these 8-crossing knots are all 3-bridge knots, and in [11], Suzuki shows that if one restricts to the class of 2-bridge knots then the (strictly) bigger knot does indeed always have 3 times as many crossings as the smaller knot.…”

On minimality of two-bridge knots

“…Ohtsuki, Riley, and Sakuma in [15] studied a systematic construction of epimorphisms between two-bridge knot groups. In [8] and [7] all the pairs of prime knots with up to 11 crossings which admit meridional epimorphisms between their knot groups were determined. However, it is not easy to determine whether there exists an epimorphism between knot groups in general.…”

Twisted Alexander polynomials of periodic knots

“…Most recently, Friedl and Kim [13] have demonstrated that the twisted polynomial is sufficient to determine the genus and fibering properties of all prime knots of 12 crossings and less. Other literature on this invariant includes Cogolludo-Florens [8], Kitano [24], Kitano-Suzuki [25], Li-Xu [27] and Morifuji [30; 31]. In this paper we extend the application of the twisted polynomial to the study of knot periodicity.…”