J. S. Birman :
“Book reviews: Gerhard Burde and Heiner Zieschang, ‘Knots’, Louis H. Kauffman, ‘On knots’ Bull. Am. Math. Soc. (N.S.)
19 : 2
(1988 ),
pp. 550–558 .
MR
1567720 article
People
BibTeX
@article {key1567720m,
AUTHOR = {Birman, Joan S.},
TITLE = {Book reviews: Gerhard Burde and Heiner
Zieschang, ``{K}nots'', Louis H. Kauffman,
``{O}n knots''},
JOURNAL = {Bull. Am. Math. Soc. (N.S.)},
FJOURNAL = {Bulletin of the American Mathematical
Society. New Series},
VOLUME = {19},
NUMBER = {2},
YEAR = {1988},
PAGES = {550--558},
DOI = {10.1090/S0273-0979-1988-15740-0},
NOTE = {MR:1567720.},
ISSN = {0273-0979},
CODEN = {BAMOAD},
}
J. S. Birman and W. W. Menasco :
“On Markov’s theorem 295–310
in
Knots 2000 Korea (Volume 1)
(Yongpyong, Korea, 31 July–5 August 2000 ),
published as J. Knot Theor. Ramif.
11 : 3 .
Issue edited by J. S. Birman, C. M. Gordon, G. T. Jin, L. H. Kauffman, A. Kawauchi, K. H. Ko, J. P. Levine, and Y. Matsumoto .
World Scientific (Singapore ),
2002 .
MR
1905686 Zbl
1059.57002 incollection
Abstract
People
BibTeX
Let \( \chi \) be an oriented link type in the oriented 3-sphere \( S^3 \) or
\[ \mathbb{R}^3 = S^3 - \{\infty\} .\]
A representative \( X \in \chi \) is said to be a closed braid if there is an unknotted curve
\[ \mathbf{A} \subset S^3 - X \]
(the axis ) and a choice of fibration \( \mathscr{H} \) of the open solid torus \( S^3 - \mathbf{A} \) by meridian discs
\[ \{H_{\theta}: \theta\in [0, 2\pi]\} ,\]
such that whenever \( X \) meets a fiber \( H_{\theta} \) the intersection is transverse.
Closed braid representations of \( \chi \) are not unique, and Markov’s well-known theorem asserts that any two are related by a finite sequence of elementary moves. The main result in this paper is to give a new proof of Markov’s theorem. We hope that our new proof will be of interest because it gives new insight into the geometry.
@article {key1905686m,
AUTHOR = {Birman, Joan S. and Menasco, William
W.},
TITLE = {On {M}arkov's theorem},
JOURNAL = {J. Knot Theor. Ramif.},
FJOURNAL = {Journal of Knot Theory and its Ramifications},
VOLUME = {11},
NUMBER = {3},
YEAR = {2002},
PAGES = {295--310},
DOI = {10.1142/S0218216502001627},
NOTE = {\textit{Knots 2000 Korea (Volume 1)}
(Yongpyong, Korea, 31 July--5 August
2000). Issue edited by J. S. Birman,
C. M. Gordon, G. T. Jin,
L. H. Kauffman, A. Kawauchi,
K. H. Ko, J. P. Levine,
and Y. Matsumoto. MR:1905686.
Zbl:1059.57002.},
ISSN = {0218-2165},
}
J. S. Birman, M. Rampichini, P. Boldi, and S. Vigna :
“Towards an implementation of the B–H algorithm for recognizing the unknot 601–645
in
Knots 2000 Korea (Volume 2)
(Yongpyong, Korea, 31 July–5 August 2000 ),
published as J. Knot Theor. Ramif.
11 : 4 .
Issue edited by J. S. Birman, C. M. Gordon, G. T. Jin, L. H. Kauffman, A. Kawauchi, K. H. Ko, J. P. Levine, and Y. Matsumoto .
World Scientific (Hackensack, NJ ),
2002 .
MR
1915499 Zbl
1007.57004 incollection
Abstract
People
BibTeX
In the manuscript [Birman and Hirsch 1998] the first author and Michael Hirsch presented a then-new algorithm for recognizing the unknot. The first part of the algorithm required the systematic enumeration of all discs which support a ‘braid foliation’ and are embeddable in 3-space. The boundaries of these ‘foliated embeddable discs’ (FEDs) are the collection of all closed braid representatives of the unknot, up to conjugacy, and the second part of the algorithm produces a word in the generators of the braid group which represents the boundary of the previously listed FEDs. The third part tests whether a given closed braid is conjugate to the boundary of a FED on the list.
In this paper we describe implementations of the first and second parts of the algorithm. We also give some of the data which we obtained. The data suggests that FEDs have unexplored and interesting structure. Open question are interspersed throughout the manuscript.
The third part of the algorithm was studied in [Birman, Ko and Lee 1998] and [Birman, Ko and Lee 2001], and implemented by S. J. Lee [preprint]. At this writing his algorithm is polynomial for \( n\leq 4 \) and exponential for \( n\geq 5 \) .
@article {key1915499m,
AUTHOR = {Birman, Joan S. and Rampichini, Marta
and Boldi, Paolo and Vigna, Sebastiano},
TITLE = {Towards an implementation of the {B}--{H}
algorithm for recognizing the unknot},
JOURNAL = {J. Knot Theor. Ramif.},
FJOURNAL = {Journal of Knot Theory and its Ramifications},
VOLUME = {11},
NUMBER = {4},
YEAR = {2002},
PAGES = {601--645},
DOI = {10.1142/S0218216502001858},
NOTE = {\textit{Knots 2000 Korea (Volume 2)}
(Yongpyong, Korea, 31 July--5 August
2000). Issue edited by J. S. Birman,
C. M. Gordon, G. T. Jin,
L. H. Kauffman, A. Kawauchi,
K. H. Ko, J. P. Levine,
and Y. Matsumoto. MR:1915499.
Zbl:1007.57004.},
ISSN = {0218-2165},
}
Knots 2000 Korea (Volume 1) Yongpyong, South Korea, 31 July–5 August 2000 ),
published as J. Knot Theor. Ramif.
11 : 3 .
Issue edited by J. S. Birman, C. M. Gordon, G. T. Jin, L. H. Kauffman, A. Kawauchi, K. H. Ko, J. P. Levine, and Y. Matsumoto .
World Scientific (Singapore ),
2002 .
Zbl
0995.00510 book
People
BibTeX
@book {key0995.00510z,
TITLE = {Knots 2000 Korea (Volume 1)},
EDITOR = {Birman, J. S. and Gordon, C. M. and
Jin, G. T. and Kauffman, L. H. and Kawauchi,
A. and Ko, K. H. and Levine, J. P. and
Matsumoto, Y.},
PUBLISHER = {World Scientific},
ADDRESS = {Singapore},
YEAR = {2002},
PAGES = {283--473},
URL = {http://www.worldscientific.com/toc/jktr/11/03},
NOTE = {(Yongpyong, South Korea, 31 July--5
August 2000). Published as \textit{J.
Knot Theor. Ramif.} \textbf{11}:3. Zbl:0995.00510.},
ISSN = {0218-2165},
}
Knots 2000 Korea (Volume 3) Yongpyong, South Korea, 31 July–5 August 2000 ),
published as J. Knot Theor. Ramif.
11 : 6 .
Issue edited by J. S. Birman, C. M. Gordon, G. T. Jin, L. H. Kauffman, A. Kawauchi, K. H. Ko, J. P. Levine, and Y. Matsumoto .
World Scientific (Singapore ),
September 2002 .
Zbl
1018.00508 book
People
BibTeX
@book {key1018.00508z,
TITLE = {Knots 2000 Korea (Volume 3)},
EDITOR = {Birman, J. S. and Gordon, C. M. and
Jin, G. T. and Kauffman, L. H. and Kawauchi,
A. and Ko, K. H. and Levine, J. P. and
Matsumoto, Y.},
PUBLISHER = {World Scientific},
ADDRESS = {Singapore},
MONTH = {September},
YEAR = {2002},
PAGES = {833--1016},
URL = {http://www.worldscientific.com/toc/jktr/11/06},
NOTE = {(Yongpyong, South Korea, 31 July--5
August 2000). Published as \textit{J.
Knot Theor. Ramif.} \textbf{11}:6. Zbl:1018.00508.},
ISSN = {0218-2165},
}
