On the degree of the Brandt-Lickorish-Millett-Ho polynomial of a link

Author:
Mark E. Kidwell

Journal:
Proc. Amer. Math. Soc. **100** (1987), 755-762

MSC:
Primary 57M25

MathSciNet review:
894450

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be the link polynomial defined by Brandt, Lickorish, Millett, and Ho. Let be the maximum degree of a nonzero term. If is any regular link projection and is any bridge (maximal connected component after undercrossing points are deleted), define the *length* of as the number of crossings in which the overcrossing segment is a part of .

Theorem 1. *Let* *be a connected, regular link projection with* *crossing points. Let* *be the maximal length of any bridge in* . *Then* .

Theorem 2. *If* *is a prime, connected alternating projection with* *crossing points, then the coefficient of* *is a positive number*.

**[BLM]**Robert D. Brandt, W. B. R. Lickorish, and Kenneth C. Millett,*A polynomial invariant for unoriented knots and links*, Invent. Math.**84**(1986), no. 3, 563–573. MR**837528**, 10.1007/BF01388747**[C]**J. H. Conway,*An enumeration of knots and links, and some of their algebraic properties*, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), Pergamon, Oxford, 1970, pp. 329–358. MR**0258014****[H]**C. F. Ho,*A new polynomial invariant for knots and links*, Abstracts Amer. Math. Soc.**6**(1985), 300.**[K]**Louis H. Kauffman,*State models and the Jones polynomial*, Topology**26**(1987), no. 3, 395–407. MR**899057**, 10.1016/0040-9383(87)90009-7**[M]**Kunio Murasugi,*Jones polynomials and classical conjectures in knot theory*, Topology**26**(1987), no. 2, 187–194. MR**895570**, 10.1016/0040-9383(87)90058-9**[T]**Morwen B. Thistlethwaite,*A spanning tree expansion of the Jones polynomial*, Topology**26**(1987), no. 3, 297–309. MR**899051**, 10.1016/0040-9383(87)90003-6

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1987-0894450-0

Keywords:
Brandt-Lickorish-Millett-Ho polynomial,
alternating link projection,
bridge length,
arborescent link

Article copyright:
© Copyright 1987
American Mathematical Society