BLM/Ho Polynomial
المؤلف:
Brandt, R. D.; Lickorish, W. B. R.; and Millett, K. C.
المصدر:
"A Polynomial Invariant for Unoriented Knots and Links." Invent. Math. 84
الجزء والصفحة:
...
12-6-2021
2830
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.
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