Abstract
Basic concepts that appear throughout the monograph are summarized here. ‘W.l.o. g.’ will abbreviate ‘without loss of generality’; ‘r.h.s.’ will stand for ‘right hand-side’. The notations ‘: = ’ and ‘ =: ’ indicate definition, with the defined symbol standing on the hand-side of the colon.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
J.W. Alexander, A lemma on systems of knotted curves. Proc. Natl. Acad. Sci. USA 9, 93–95 (1923)
D. Bennequin, Entrelacements et équations de Pfaff. Soc. Math. de France, Astérisque 107–108, 87–161 (1983)
J.S. Birman, K. Ko, S.J. Lee, A new approach to the word and conjugacy problems in the braid groups. Adv. Math. 139(2), 322–353 (1998)
R.D. Brandt, W.B.R. Lickorish, K. Millett, A polynomial invariant for unoriented knots and links. Invent. Math. 74, 563–573 (1986)
J.S. Birman, W.W. Menasco, Studying knots via braids III: classifying knots which are closed 3 braids. Pac. J. Math. 161, 25–113 (1993)
J.S. Birman, W.W. Menasco, Studying links via closed braids II: On a theorem of Bennequin. Topol. Appl. 40(1), 71–82 (1991)
O. Dasbach, X.-S. Lin, On the head and the tail of the colored Jones polynomial. Compos. Math. 142(5), 1332–1342 (2006)
T. Fiedler, A small state sum for knots. Topology 32(2), 281–294 (1993)
T. Fiedler, Gauss Sum Invariants for Knots and Links. Mathematics and Its Applications, vol. 532 (Kluwer Academic Publishers, Boston, 2001)
J. Franks, R.F. Williams, Braids and the Jones-Conway polynomial. Trans. Am. Math. Soc. 303, 97–108 (1987)
P. Freyd, J. Hoste, W.B.R. Lickorish, K. Millett, A. Ocneanu, D. Yetter, A new polynomial invariant of knots and links. Bull. Am. Math. Soc. 12, 239–246 (1985)
J. Hoste, M. Thistlethwaite, KnotScape, a knot polynomial calculation and table access program. Available at http://www.math.utk.edu/~morwen
V.F.R. Jones, A polynomial invariant of knots and links via von Neumann algebras. Bull. Am. Math. Soc. 12, 103–111 (1985)
L.H. Kauffman, An invariant of regular isotopy. Trans. Am. Math. Soc. 318, 417–471 (1990)
L.H. Kauffman, State models and the Jones polynomial. Topology 26, 395–407 (1987)
W.B.R. Lickorish, K.C. Millett, A polynomial invariant for oriented links. Topology 26(1), 107–141 (1987)
W.B.R. Lickorish, M.B. Thistlethwaite, Some links with non-trivial polynomials and their crossing numbers. Comment. Math. Helv. 63, 527–539 (1988)
H.R. Morton, Seifert circles and knot polynomials. Proc. Camb. Philos. Soc. 99, 107–109 (1986)
K. Murasugi, J. Przytycki, The skein polynomial of a planar star product of two links. Math. Proc. Camb. Philos. Soc. 106(2), 273–276 (1989)
K. Murasugi, J. Przytycki, An index of a graph with applications to knot theory. Mem. Am. Math. Soc. 106(508) (1993)
W.W. Menasco, M.B. Thistlethwaite, The Tait flyping conjecture. Bull. Am. Math. Soc. 25(2), 403–412 (1991)
K. Murasugi, On the braid index of alternating links. Trans. Am. Math. Soc. 326(1), 237–260 (1991)
Y. Ohyama, On the minimal crossing number and the braid index of links. Canad. J. Math. 45(1), 117–131 (1993)
M. Polyak, O. Viro, Gauss diagram formulas for Vassiliev invariants. Int. Math. Res. Notes 11, 445–454 (1994)
M. Polyak, O. Viro, On the Casson knot invariant. J. Knot Theory Ramifications 10(5), 711–738 (2001). Knots in Hellas ’98, vol. 3 (Delphi)
D. Rolfsen, Knots and Links (Publish or Perish, New York, 1976)
L. Rudolph, Quasipositivity as an obstruction to sliceness. Bull. Am. Math. Soc. (N.S.) 29(1), 51–59 (1993)
A. Stoimenow, Coefficients and non-triviality of the Jones polynomial, J. Reine Angew. Math. 657 (2011), 1–55; see also math.GT/0606255
M. B. Thistlethwaite, On the Kauffman polynomial of an adequate link. Invent. Math. 93(2), 285–296 (1988)
M.B. Thistlethwaite, A spanning tree expansion for the Jones polynomial. Topology 26, 297–309 (1987)
P. Vogel, Representation of links by braids: a new algorithm. Comment. Math. Helv. 65, 104–113 (1990)
P. Xu, The genus of closed 3-braids. J. Knot Theory Ramifications 1(3), 303–326 (1992)
S. Yamada, The minimal number of Seifert circles equals the braid index. Invent. Math. 88, 347–356 (1987)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2017 The Author(s)
About this chapter
Cite this chapter
Stoimenow, A. (2017). Preliminaries, Basic Definitions, and Conventions. In: Properties of Closed 3-Braids and Braid Representations of Links . SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-68149-8_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-68149-8_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-68148-1
Online ISBN: 978-3-319-68149-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)