BLM/Ho Polynomial

A 1-variable unoriented knot polynomial Q(x). It satisfies


and the skein relationship


It also satisfies


where # is the knot sum and


where L^* is the mirror image of L. The BLM/Ho polynomials of mutant knots are also identical. Brandt et al. (1986) give a number of interesting properties. For any link L with >=2 components, Q_L-1 is divisible by 2(x-1). If L has c components, then the lowest power of x in Q_L(x) is 1-c, and


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


where z=-t-t^(-1) (Kanenobu and Sumi 1993).

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


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

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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."

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BLM/Ho Polynomial

Cite this as:

Weisstein, Eric W. "BLM/Ho Polynomial." From MathWorld--A Wolfram Web Resource.

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