Teruaki Kitano

A Partial Order in the Knot Table

We write K 1 ≥ K 2 for two prime knots K 1,K 2 if there exists a surjective group homomorphism from G(K 1) onto G(K 2) where G(K 1),G(K 2) are the knot groups of K 1,K 2, respectively. In this paper, we determine this partial order for the knots in Rolfsens knot table.

ERRATA: "A PARTIAL ORDER ON THE SET OF PRIME KNOTS WITH UP TO 11 CROSSINGS"

Let K be a prime knot in S3 and G(K) = π1(S3 - K) the knot group. We write K1 ≥ K2 if there exists a surjective homomorphism from G(K1) onto G(K2). In this paper, we determine this partial order on the set of prime knots with up to 11 crossings. There exist such 801 prime knots and then 640, 800 should be considered. The existence of a surjective homomorphism can be proved by constructing it explicitly. On the other hand, the non-existence of a surjective homomorphism can be proved by the Alexander polynomial and the twisted Alexander polynomial.

Johnson's homomorphisms of subgroups of the mapping class group, the Magnus expansion and Massey higher products of mapping tori

Abstract Johnsons homomorphism τk of the subgroup of the mapping class group of surfaces is defined via the action on the lower central series of the fundamental group. We give some descriptions of τk by using the Magnus expansion and thereby give a geometric meaning to it in terms of the Massey products on mapping tori of the corresponding mapping classes.

Twisted Alexander polynomials and a partial order on the set of prime knots

We give a survey of some recent papers by the authors and Masaaki Wada relating the twisted Alexander polynomial with a partial order on the set of prime knots. We also give examples and pose open problems.

ON THE NUMBER OF SL(2;ℤ/pℤ)-REPRESENTATIONS OF KNOT GROUPS

The number of representations of a knot group is an invariant of knots. In this paper, we calculate these numbers associated to SL(2;ℤ/pℤ)-representations for all the knots in Rolfsens knot table. Moreover, we show some properties of these numbers.