Results 1  10
of
47
The number of Reidemeister Moves Needed for Unknotting
, 2008
"... There is a positive constant c1 such that for any diagram D representing the unknot, there is a sequence of at most 2 c1n Reidemeister moves that will convert it to a trivial knot diagram, where n is the number of crossings in D. A similar result holds for elementary moves on a polygonal knot K embe ..."
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Cited by 44 (11 self)
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There is a positive constant c1 such that for any diagram D representing the unknot, there is a sequence of at most 2 c1n Reidemeister moves that will convert it to a trivial knot diagram, where n is the number of crossings in D. A similar result holds for elementary moves on a polygonal knot K embedded in the 1skeleton of the interior of a compact, orientable, triangulated PL 3manifold M. There is a positive constant c2 such that for each t ≥ 1, if M consists of t tetrahedra, and K is unknotted, then there is a sequence of at most 2 c2t elementary moves in M which transforms K to a triangle contained inside one tetrahedron of M. We obtain explicit values for c1 and c2.
Interactive Topological Drawing
, 1998
"... The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in threedimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues ..."
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Cited by 31 (3 self)
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The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in threedimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues. Because the domain of application is mathematics, topological drawing is also concerned with the correct representation and display of these objects on a computer. By correctness we mean that the essential topological features of objects are maintained during interaction. We have chosen to limit the scope of topological drawing to knot theory, a domain that consists essentially of one class of object (embedded circles in threedimensional space) yet is rich enough to contain a wide variety of difficult problems of research interest. In knot theory, two embedded circles (knots) are considered equivalent if one may be smoothly deformed into the other without any cuts or selfintersections. This notion of equivalence may be thought of as the heart of knot theory. We present methods for the computer construction and interactive manipulation of a
On the LinksGould invariant of links
 J. KNOT THEORY RAM
, 2998
"... We introduce and study in detail an invariant of (1,1) tangles. This invariant, derived from a family of four dimensional representations of the quantum superalgebra Uq[gl(21)], will be referred to as the Links–Gould invariant. We find that our invariant is distinct from the Jones, HOMFLY and Kauff ..."
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Cited by 19 (14 self)
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We introduce and study in detail an invariant of (1,1) tangles. This invariant, derived from a family of four dimensional representations of the quantum superalgebra Uq[gl(21)], will be referred to as the Links–Gould invariant. We find that our invariant is distinct from the Jones, HOMFLY and Kauffman polynomials (detecting chirality of some links where these invariants fail), and that it does not distinguish mutants or inverses. The method of evaluation is based on an abstract tensor state model for the invariant that is quite useful for computation as well as theoretical exploration.
and Akira Yasuhara. Crosscap number of a knot
 Pacific J. Math
, 1995
"... B. E. Clark defined the crosscap number of a knot to be the minimum number of the first Betti numbers of nonorientable surfaces bounding it. In this paper, we investigate the crosscap numbers of knots. We show that the crosscap number of 74 is equal to 3. This gives an affirmative answer to a ques ..."
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Cited by 15 (0 self)
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B. E. Clark defined the crosscap number of a knot to be the minimum number of the first Betti numbers of nonorientable surfaces bounding it. In this paper, we investigate the crosscap numbers of knots. We show that the crosscap number of 74 is equal to 3. This gives an affirmative answer to a question given by Clark. In general, the crosscap number is not additive under the connected sum. We give a necessary and sufficient condition for the crosscap number to be additive under the connected sum. 0. Introduction. We study knots in the 3sphere S3. The genus g{K) of a knot K is the minimum number of the genera of Seifert surfaces for it [11]. Here a Seifert surface means a connected, orientable surface with boundary K. In 1978, B. E. Clark [3] defined the crosscap number C(K) oϊK to be the minimum num
Unknot diagrams requiring a quadratic number of Reidemeister moves to untangle
, 2007
"... We present a sequence of diagrams of the unknot for which the minimum number of Reidemeister moves required to pass to the trivial diagram is quadratic with respect to the number of crossings. These bounds apply both in S 2 and in R 2. 1 ..."
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Cited by 10 (1 self)
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We present a sequence of diagrams of the unknot for which the minimum number of Reidemeister moves required to pass to the trivial diagram is quadratic with respect to the number of crossings. These bounds apply both in S 2 and in R 2. 1
Tectonic evolution of
 the Coral Sea Basin, J. Geophys. Res
, 1979
"... of unknotting strategies for knots and braids ..."
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Quandles knot invariants and the Nfold branched cover
, 1984
"... This paper focuses on the class of algebraic objects called involutory quandles and, in particular, on the connection between involutory quandles and topological knots and links. The paper is intended to be a selfcontained treatment of the topic and should be accessible to readers with various back ..."
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Cited by 9 (0 self)
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This paper focuses on the class of algebraic objects called involutory quandles and, in particular, on the connection between involutory quandles and topological knots and links. The paper is intended to be a selfcontained treatment of the topic and should be accessible to readers with various backgrounds.
An algebraic approach to symmetry with applications to knot theory
, 1979
"... This edition of my dissertation differs little from the original 1979 version. This edition is typeset in LATEX, whereas the original was typed on a typewriter and the figures were hand drawn. The page numbers and figure numbers are changed, the table of contents is expanded to include sections, a l ..."
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Cited by 8 (0 self)
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This edition of my dissertation differs little from the original 1979 version. This edition is typeset in LATEX, whereas the original was typed on a typewriter and the figures were hand drawn. The page numbers and figure numbers are changed, the table of contents is expanded to include sections, a list of figures is included, and the index appears at the end instead of the front. I’ve corrected a few typos (and probably added others), and I added figure 4.5 that was missing from the original.
Invariants of Knot Diagrams
 MATHEMATISCHE ANNALEN
, 2008
"... We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams. ..."
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Cited by 8 (2 self)
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We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams.