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

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

MathSciNet review:
894450

Full-text PDF

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]**R. D. Brandt, W. B. R Lickorish and K. C. Millett,*A polynomial invariant for unoriented knots and links*, Invent. Math.**84**(1986), 563-573. MR**837528 (87m:57003)****[C]**J. H. Conway,*An enumeration of knots and links and some of their algebraic properties*, Computational Problems in Abstract Algebra (J. Leech, ed.), Pergamon Press, Oxford and Elmsford, N.Y., 1969, pp. 329-358. MR**0258014 (41:2661)****[H]**C. F. Ho,*A new polynomial invariant for knots and links*, Abstracts Amer. Math. Soc.**6**(1985), 300.**[K]**L. Kauffman,*State models and the Jones polynomial*(to appear). MR**899057 (88f:57006)****[M]**K. Murasugi,*Jones polynomials and classical conjectures in knot theory*, Topology (to appear). MR**895570 (88m:57010)****[T]**M. B. Thistlethwaite,*A spanning tree expansion of the Jones polynomial*, Topology (to appear). MR**899051 (88h:57007)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
57M25

Retrieve articles in all journals with MSC: 57M25

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