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13
Gauge theory for embedded surfaces
 I, Topology
, 1993
"... (i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly ..."
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Cited by 71 (6 self)
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(i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly
Knot Floer Homology and the fourball genus
 Geom. Topol
"... Abstract. We use the knot filtration on the Heegaard Floer complex ĈF to define an integer invariant τ(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to Z. As such, it gives lower bounds for the slice genus (and hence also the unknotti ..."
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Cited by 67 (8 self)
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Abstract. We use the knot filtration on the Heegaard Floer complex ĈF to define an integer invariant τ(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to Z. As such, it gives lower bounds for the slice genus (and hence also the unknotting number) of a knot; but unlike the signature, τ gives sharp bounds on the fourball genera of torus knots. As another illustration, we use calculate the invariant for several tencrossing knots. 1.
SLICE AND GORDIAN NUMBERS OF TRACK KNOTS
, 2005
"... Abstract. We present a class of knots associated with labelled generic immersions of intervals into the plane and compute their Gordian numbers and 4dimensional invariants. At least 10 % of the knots in Rolfsen’s table belong to this class of knots. We call them track knots. They are contained in t ..."
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Cited by 5 (3 self)
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Abstract. We present a class of knots associated with labelled generic immersions of intervals into the plane and compute their Gordian numbers and 4dimensional invariants. At least 10 % of the knots in Rolfsen’s table belong to this class of knots. We call them track knots. They are contained in the class of quasipositive knots. In this connection, we classify quasipositive knots and strongly quasipositive knots up to 10 crossings. 1.
ON THE UNKNOTTING NUMBER OF MINIMAL DIAGRAMS
, 2003
"... Answering negatively a question of Bleiler, we give examples of knots where the difference between minimal and maximal unknotting number of minimal crossing number diagrams grows beyond any extent. ..."
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Cited by 2 (0 self)
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Answering negatively a question of Bleiler, we give examples of knots where the difference between minimal and maximal unknotting number of minimal crossing number diagrams grows beyond any extent.
On Polynomials of Variously Positive Links
, 2002
"... It is known that the minimal degree of the Jones polynomial of a positive knot is equal to its genus, and the minimal coefficient is 1. We extend this result to almost positive links and partly identify the 3 following coefficients for special types of positive links. We also give counterexamples to ..."
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It is known that the minimal degree of the Jones polynomial of a positive knot is equal to its genus, and the minimal coefficient is 1. We extend this result to almost positive links and partly identify the 3 following coefficients for special types of positive links. We also give counterexamples to the Jones polynomialribbon genus conjectures for a quasipositive knot. Then we show that the Alexander polynomial completely detects the minimal genus and fiber property of canonical Seifert surfaces associated to almost positive (and almost alternating) link diagrams.
The Granny And The Square Tangle And The Unknotting Number
, 1999
"... . We show that a knot with a diagram with n granny and square tangles has unknotting number at least n, bridge number ? n, and braid index ? n. As an application, we construct exponentially many in the crossing number slice knots with arbitrarily high unknotting number. Keywords: tangle, Jones pol ..."
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. We show that a knot with a diagram with n granny and square tangles has unknotting number at least n, bridge number ? n, and braid index ? n. As an application, we construct exponentially many in the crossing number slice knots with arbitrarily high unknotting number. Keywords: tangle, Jones polynomial, unknotting number, braid index, bridge number, slice knot. AMS subject classification: 57M25 (primary), 57M15, 57N70 (secondary). 1. Introduction In a recent beautiful paper [Kr], David Krebes introduced an invariant of 4tangles with values in Z \Theta Z=(p;q) (\Gamma p; \Gammaq) (a set one can think of as rational numbers with possibly zero denominators and not allowed to be reduced), that generalizes the classical iterated fraction of rational tangles, see [GK]. Up to a sign, p and q are the evaluations of the determinant D (\Gamma1) =V (\Gamma1) [J2, x12] on both closures of the tangle. Here D as the Alexander polynomial [Al] and V the Jones polynomial [J]. Krebes's invarian...
Genus generators and the positivity of the signature
, 2006
"... It is a conjecture that the signature of a positive link is bounded below by an increasing function of its negated Euler characteristic. In relation to this conjecture, we apply the generator description for canonical genus to show that the boundedness of the genera of positive knots with given sig ..."
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It is a conjecture that the signature of a positive link is bounded below by an increasing function of its negated Euler characteristic. In relation to this conjecture, we apply the generator description for canonical genus to show that the boundedness of the genera of positive knots with given signature can be algorithmically partially decided. We relate this to the result that the set of knots of canonical genus ≥ n is dominated by a finite subset of itself in the sense of Taniyama’s partial order.
UNKNOTTING SEQUENCES FOR TORUS KNOTS
, 809
"... Abstract. The unknotting number of a knot is bounded from below by its slice genus. It is a wellknown fact that the genera and unknotting numbers of torus knots coincide. In this note we characterize quasipositive knots for which the genus bound is sharp: the slice genus of a quasipositive knot equ ..."
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Abstract. The unknotting number of a knot is bounded from below by its slice genus. It is a wellknown fact that the genera and unknotting numbers of torus knots coincide. In this note we characterize quasipositive knots for which the genus bound is sharp: the slice genus of a quasipositive knot equals its unknotting number, if and only if the given knot appears in an unknotting sequence of a torus knot. 1.
BENNEQUIN’S INEQUALITY AND THE POSITIVITY OF THE SIGNATURE
"... Abstract. We use an algorithm for special diagrams to prove a Bennequin type inequality for the signature of an arbitrary link diagram, related to its Murasugi sum decomposition. We apply this inequality to show that the signature of a nontrivial positive 3braid knot is greater than its genus, and ..."
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Abstract. We use an algorithm for special diagrams to prove a Bennequin type inequality for the signature of an arbitrary link diagram, related to its Murasugi sum decomposition. We apply this inequality to show that the signature of a nontrivial positive 3braid knot is greater than its genus, and that the signature of a positive braid link is minorated by an increasing function of its negated Euler characteristic. The latter property is conjectured to extend to positive links. 1. Introduction and
ON THE KINKINESS OF CLOSED BRAIDS
, 2001
"... Abstract. In this note, we prove a lower bound for the positive kinkiness of a closed braid which we then use to derive an estimate for the positive kinkiness of a link in terms of its Seifert system. As an application, we show that certain pretzel knots cannot be unknotted using only positive cross ..."
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Abstract. In this note, we prove a lower bound for the positive kinkiness of a closed braid which we then use to derive an estimate for the positive kinkiness of a link in terms of its Seifert system. As an application, we show that certain pretzel knots cannot be unknotted using only positive crossing changes. We also describe a subgroup of infinite rank in the smooth knot concordance group of which no element has a strongly quasipositive representative. In 1993, L. Rudolph proved a lower bound for the slice genus of a knot in terms of a presentation as the closure of a braid. It is clear that the same estimate holds for the number of double points of any properly immersed disk in the 4–ball spanning the knot, for such a disk which has r self–intersection points can be turned into an embedded surface of genus r by replacing all the self–intersection points by handles. In this paper, we show that there is a similar bound for the minimal number of positive self–intersection points of such an immersion, a knot invariant introduced by R. Gompf which is called the positive kinkiness. First let us recall the definition of the unknotting number and the positive respectively