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BLM/Ho Polynomial

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

Q_{\rm unknot}=1
\end{displaymath} (1)

and the Skein Relationship
\end{displaymath} (2)

It also satisfies
Q_{L_1\char93 L_2}=Q_{L_1}Q_{L_2},
\end{displaymath} (3)

where $\char93 $ is the Knot Sum and
\end{displaymath} (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),
\end{displaymath} (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
\end{displaymath} (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

\end{displaymath} (7)

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


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.''

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

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© 1996-9 Eric W. Weisstein