BLM/Ho Polynomial

A 1-variable unoriented Knot Polynomial $Q(x)$. It satisfies

$$Q_{\rm unknot}=1$$ (1)

and the Skein Relationship

$$Q_{L_+}+Q_{L_-}=x(Q_{L_0}+Q_{L_\infty}).$$ (2)

It also satisfies

$$Q_{L_1\char93 L_2}=Q_{L_1}Q_{L_2},$$ (3)

where $\char93$ is the Knot Sum and

$$Q_{L^*}=Q_{L},$$ (4)

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 $\geq 2$ 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

$$\lim_{x\to 0}x^{c-1}Q_L(x)=\lim_{(\ell,m)\to(1,0)} (-m)^{c-1}P_L(\ell,m),$$ (5)

where $P_L$ is the HOMFLY Polynomial. Also, the degree of $Q_L$ is less than the Crossing Number of $L$. If $L$ is a 2-Bridge Knot, then

$$Q_L(z)=2z^{-1}V_L(t)V_L(t^{-1}+1-2z^{-1}),$$ (6)

where $z\equiv -t-t^{-1}$ (Kanenobu and Sumi 1993).

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

$$Q(x)=F(1,x).$$ (7)

Brandt et al. (1986) give a listing of $Q$ 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.informatik.hu-berlin.de/~stoimeno/ptab/blmh10.html.

Weisstein, E. W. ``Knots.'' Mathematica notebook Knots.m.