Year: 1987
Type: article
Abstract: In 1984, V. Jones [8] introduced a new polynomial invariant of link isotopy types which is now called the (one variable) Jones polynomial, which was subsequently generalized to the two variable Jones polynomial [4], [13].These invariants are closely related to the traces of irreducible representations of Iwahori algebras (or Hecke algebras) [2], [7] associated with the symmetric groups (see [5], [9]).The purpose of this paper is to show that the Kauffman polynomial [12] can also be interpreted as a function F on a certain associative algebra.We define a knit semi-group D n of degree n which is generated by the generators of the braid group B n on n strings and elements e l9 e 2y •••, e n ^ in Figure 4. We call an element of D n an n-knit.We get a link d^ in the 3-sphere by closing an w-knit d.We call d^ a closed n-knit coming from d.We also define an algebra E n (a, β) for non-zero complex numbers α, β&C-{0} as a quotient of a semi-group algebra C [D n ] of D n over C. Then the Kauffman polynomial of a closed w-knit is obtained through E n (a, β).From Section 7 on, we treat the case n=Z.Then we can show that the function F is a sum of traces of irreducible representations of the algebra E z (a, β) (Theorem 10.1).The author expects the same is true for general n.In Sections 12-16, we apply our formula to closed 3-braids and 2-bridge links, since they are special types of closed 3-knits.For example, if two closed 3braids have the same writhe (or twist number) and the same Jones polynomial, they also have the same ^-polynomial (Theorem 13.1).Thus for the closed 3-braid, the Alexnader polynomial and the writhe determine the two variable Jones, Jones, and ^-polynomials.In the actual claculation of the examples in Sections 14 b), c) and 16, the author used a personal computer (NEC PC-9801) with muMATH-83 (Symbolic Mathematics Package) for MS-DOS.
DOI: 10.18910/4549
Author: Murakami, Jun
Institution: The University of Osaka
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1513992157