Infinitely many knots with the same polynomial invariant

Taizo Kanenobu
1986 Proceedings of the American Mathematical Society  
We give infinitely many examples of infinitely many knots in S3 with the same recently discovered two-variable and Iones polynomials, but distinct Alexander module structures, which are hyperbolic, fibered, ribbon, of genus 2, and 3-bridge. Two knots Kx and K2 in S3 belong to the same isotopy type if there exists an orientation preserving homeomorphism of S3 which maps Kx onto K2. We denote it by Kx ~ K2. In 1984, V. Jones [9] discovered a very powerful polynomial invariant of the isotopy type
more » ... f the isotopy type of an oriented knot or link. Subsequently, the Jones polynomial was generalized to the two-variable polynomial invariant simultaneously and independently by Ocneanu [13], Lickorish and Millett [12], Hoste [8], and Freyd and Yetter. In this note we follow Lickorish and Millett. For a link L, the polynomial L(l,m) is defined recursively by the following two conditions: (I) If L + , L_ and L0 are three links with completely identical projections except at one crossing, where they are related as shown in Figure 1, then ¡L + (l, m) + rlL_(l, m) + mL0(l, m) = 0. (II) If AT is a trivial knot, then K(l, m) = 1. Let A,(/), V,(z) and V,(t) be the Alexander polynomial, the Conway polynomial [4] and the Jones polynomial of a link L, respectively. They can be recovered from L(l, m) by the formulas A/(í) = L(/,/(í1/2-í-1/2)), VL(t) = L(i,iz), VL(t) = L(it,i(t^-r^)), where ; = J -1 . Received bv the editors lanuary 4, 1985 and, in revised form, March 1, 1985. 19X0 Mathematics Subject Classification. Primary 57M25.
doi:10.1090/s0002-9939-1986-0831406-7 fatcat:egpjolzeu5hzfjlljnxvmw4yjq