BLM/Ho Polynomial

A 1-variable unoriented knot polynomial . It satisfies

(1)

and the skein relationship

(2)

It also satisfies

(3)

where  is the knot sum and

(4)

where  is the mirror image of . The BLM/Ho polynomials of mutant knots are also identical. Brandt et al. (1986) give a number of interesting properties. For any link  with  components,  is divisible by . If  has  components, then the lowest power of  in  is , and

(5)

where  is the HOMFLY polynomial. Also, the degree of  is less than the link crossing number of . If  is a 2-bridge knot, then

(6)

where  (Kanenobu and Sumi 1993).

The polynomial was subsequently extended to the 2-variable Kauffman polynomial F, which satisfies

(7)

Brandt et al. (1986) give a listing of  polynomials for knots up to 8 crossings and links up to 6 crossings.

REFERENCES:

Brandt, R. D.; Lickorish, W. B. R.; and Millett, K. C. "A Polynomial Invariant for Unoriented Knots and Links." Invent. Math. 84, 563-573, 1986.

Ho, C. F. "A New Polynomial for Knots and Links--Preliminary Report." Abstracts Amer. Math. Soc. 6, 300, 1985.

Kanenobu, T. and Sumi, T. "Polynomial Invariants of 2-Bridge Knots through 22-Crossings." Math. Comput. 60, 771-778 and S17-S28, 1993.

Stoimenow, A. "Brandt-Lickorish-Millett-Ho Polynomials." http://www.ms.u-tokyo.ac.jp/~stoimeno/ptab/blmh10.html.