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67
New points of view in knot theory
 Bull. Am. Math. Soc., New Ser
, 1993
"... In this article we shall give an account of certain developments in knot theory which followed upon the discovery of the Jones polynomial [Jo3] in 1984. The focus of our account will be recent glimmerings of understanding of the topological meaning of the new invariants. A second theme will be the c ..."
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Cited by 91 (0 self)
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In this article we shall give an account of certain developments in knot theory which followed upon the discovery of the Jones polynomial [Jo3] in 1984. The focus of our account will be recent glimmerings of understanding of the topological meaning of the new invariants. A second theme will be the central role that braid
Heegaard Floer homology and alternating knots
, 2002
"... In [23] we introduced a knot invariant for a nullhomologous knot K in an oriented threemanifold Y, which is closely related to the Heegaard Floer homology of Y (c.f. [21]). In this paper we investigate some properties of these knot homology groups for knots in the threesphere. We give a combinato ..."
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Cited by 60 (16 self)
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In [23] we introduced a knot invariant for a nullhomologous knot K in an oriented threemanifold Y, which is closely related to the Heegaard Floer homology of Y (c.f. [21]). In this paper we investigate some properties of these knot homology groups for knots in the threesphere. We give a combinatorial description for the generators of the chain complex and their gradings. With the help of this description, we determine the knot homology for alternating knots, showing that in this special case, it depends only on the signature and the Alexander polynomial of the knot (compare [24]). Applications include new restrictions on the Alexander polynomial of alternating knots.
The computational Complexity of Knot and Link Problems
 J. ACM
, 1999
"... We consider the problem of deciding whether a polygonal knot in 3dimensional Euclidean space is unknotted, capable of being continuously deformed without selfintersection so that it lies in a plane. We show that this problem, unknotting problem is in NP. We also consider the problem, unknotting pr ..."
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Cited by 55 (6 self)
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We consider the problem of deciding whether a polygonal knot in 3dimensional Euclidean space is unknotted, capable of being continuously deformed without selfintersection so that it lies in a plane. We show that this problem, unknotting problem is in NP. We also consider the problem, unknotting problem of determining whether two or more such polygons can be split, or continuously deformed without selfintersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in PSPACE. We also give exponential worstcase running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.
The Alexander polynomial of a 3manifold and the Thurston norm on cohomology
 Ann. Sci. École Norm. Sup
, 2001
"... Let M be a connected, compact, orientable 3manifold with b1 (M) > 1, whose boundary (if any) is a union of tori. Our main result is the inequality kkA kkT between the Alexander norm on H 1 (M;Z), dened in terms of the Alexander polynomial, and the Thurston norm, dened in terms of the Euler ch ..."
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Cited by 53 (3 self)
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Let M be a connected, compact, orientable 3manifold with b1 (M) > 1, whose boundary (if any) is a union of tori. Our main result is the inequality kkA kkT between the Alexander norm on H 1 (M;Z), dened in terms of the Alexander polynomial, and the Thurston norm, dened in terms of the Euler characteristic of embedded surfaces. (A similar result holds when b1 (M) = 1.) Using this inequality we determine the Thurston norm for most links with 9 or fewer crossings. Contents 1
Fundamentals of Kauffman bracket skein modules
 Kobe J. Math
, 1999
"... Skein modules are the main objects of an algebraic topology based on knots (or position). In the same spirit as Leibniz we would call our approach algebra situs. When looking at the panorama of skein modules 1, we see, past the rolling hills of homologies and homotopies, distant mountains the Kauff ..."
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Cited by 19 (6 self)
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Skein modules are the main objects of an algebraic topology based on knots (or position). In the same spirit as Leibniz we would call our approach algebra situs. When looking at the panorama of skein modules 1, we see, past the rolling hills of homologies and homotopies, distant mountains the Kauffman bracket skein module, and farther off in the distance skein modules based on other quantum invariants. We concentrate here on the basic properties of the Kauffman bracket skein module; properties fundamental in further development of the theory. In particular we consider the relative Kauffman bracket skein module, and we analyze skein modules of I bundles over surfaces. History of skein modules from my personal perspective I would like to use this opportunity, of informal presentation, 2 to give my personal history of algebraic topology based on knots (a more formal account was given in [Pr7]). In July 1986 I left Poland invited by Dale Rolfsen for a visiting position at UBC. In January of 1987, Jim Hoste gave a talk at the first Cascade Mountains Conference (in Vancouver) and described his work on multivariable generalization of the JonesConway ([HOMFLY][PT]) polynomial of links in S 3. He was convinced that his construction works for 2 colors when the first color is represented only by a trivial component. He had already succeeded in the case of 2component 2bridge links. His method, following Nakanishi, was to analyze link diagrams in an annulus (the trivial component being z axis). We immediately noticed (with Jim) that the analogous construction for the Kauffman bracket polynomial has an easy solution [HP1]. In March
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 18 (1 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 18 (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.
Automatic evaluation of the Links–Gould invariant for all prime knots of up to 10 crossings
 Journal of Knot Theory and its Ramifications
, 2000
"... This paper describes a method for the automatic evaluation of the Links– Gould twovariable polynomial link invariant (LG) for any link, given only a braid presentation. This method is currently feasible for the evaluation of LG for links for which we have a braid presentation of string index at mos ..."
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Cited by 8 (8 self)
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This paper describes a method for the automatic evaluation of the Links– Gould twovariable polynomial link invariant (LG) for any link, given only a braid presentation. This method is currently feasible for the evaluation of LG for links for which we have a braid presentation of string index at most 5. Data are presented for the invariant, for all prime knots of up to 10 crossings and various other links. LG distinguishes between these links, and also detects the chirality of those that are chiral. In this sense, it is more sensitive than the wellknown twovariable HOMFLY and Kauffman polynomials. When applied to examples which defeat the HOMFLY invariant, interestingly, LG ‘almost’ fails. The automatic method is in fact applicable to the evaluation of any such state sum invariant for which an appropriate R matrix and cap and cup matrices have been determined.