Companions of the Unknot and Width Additivity

Blair, Ryan; Tomova, Maggy

doi:10.1142/s0218216511009352

Cited by 4 publications

(14 citation statements)

References 11 publications

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“…As in Figure 14, there is a disk G in M R R that is vertical with respect to h R and illustrates a parallelism between a subarc of x i , i 2 f1; 2; 3g, and an arc in F K . As in the proof of Claim 1, we have conclusion (2).…”

Section: Essential Surfaces In High Distance Tanglessupporting

confidence: 54%

“…Let˛be an outermost arc of F \ H in H . If˛meets one of y 1 , y 2 and y 3 in more than one point, then as in Claim 1, we have conclusion (2). Hence, we can assume that˛meets each of y 1 , y 2 and y 3 in at most one point.…”

Section: Essential Surfaces In High Distance Tanglesmentioning

confidence: 66%

“…However, the authors could not verify the width of the second knot in any of their pairs. In [2], the current authors established that, for most of these pairs, the second knot is not in thin position. Therefore, most of these pairs do not provide the desired counterexample to the conjectured equality w.K # K 0 / D w.K/ C w.K 0 / 2.…”

Section: Theorem 12mentioning

confidence: 78%

“…If any component of F \ H is a closed curve not disjoint from K , then, by appealing to the argument in Claim 1, we have conclusion (2). Hence, every component of F K \ H is an arc.…”

Section: Essential Surfaces In High Distance Tanglesmentioning

confidence: 89%

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Width is not additive

Blair

Tomova

2013

Geom. Topol.

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We develop the construction suggested by Scharlemann and Thompson in [14] to obtain an infinite family of pairs of knots Kα and K α so that w(Kα#K α ) = max{w(Kα), w(K α )}. This is the first known example of a pair of knots such that w(K#K ) < w(K) + w(K ) − 2 and it establishes that the lower bound w(K#K ) ≥ max{w(K), w(K )} obtained in [12] is best possible. Furthermore, the knots Kα provide an example of knots where the number of critical points for the knot in thin position is greater than the number of critical points for the knot in bridge position.

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Section: Essential Surfaces In High Distance Tanglessupporting

confidence: 54%

Section: Essential Surfaces In High Distance Tanglesmentioning

confidence: 66%

Section: Theorem 12mentioning

confidence: 78%

“…If any component of F \ H is a closed curve not disjoint from K , then, by appealing to the argument in Claim 1, we have conclusion (2). Hence, every component of F K \ H is an arc.…”

Section: Essential Surfaces In High Distance Tanglesmentioning

confidence: 89%

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Width is not additive

Blair

Tomova

2013

Geom. Topol.

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“…We refer the reader who is interested in the proof to Lemma 2.3 of [4]. Proof of Theorem 1.1 Let B ↑ and B ↓ be the two balls bounded by in S 3 .…”

Section: Lemma 32 Let T Be a Tangle Properly Embedded In A Cylindermentioning

confidence: 99%

Distance two links

et al. 2015

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In this paper, we characterize all links in S 3 with bridge number at least three that have a bridge sphere of distance two. We show that if a link L has a bridge sphere of distance at most two then it falls into at least one of three categories:• The exterior of L contains an essential meridional sphere.• L can be decomposed as a tangle product of a Montesinos tangle with an essential tangle in a way that respects the bridge surface and either the Montesinos tangle is rational or the essential tangle contains an incompressible, boundary-incompressible annulus. • L is obtained by banding from another link L that has a bridge sphere of the same Euler characteristic as the bridge sphere for L but of distance 0 or 1.This paper was written as part of an AIM SQuaREs project.

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Natural properties of the trunk of a knot

Davies

Zupan

2017

J. Knot Theory Ramifications

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ABSTRACT. The trunk of a knot in S 3 , defined by Makoto Ozawa, is a measure of geometric complexity similar to the bridge number or width of a knot. We prove that for any two knots K 1 and K 2 , we have tr(K 1 #K 2 ) = max{tr(K 1 ),tr(K 2 )}, confirming a conjecture of Ozawa. Another conjecture of Ozawa asserts that any width-minimizing embedding of a knot K also minimizes the trunk of K. We produce several families of probable counterexamples to this conjecture.

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Companions of the Unknot and Width Additivity

Cited by 4 publications

References 11 publications

Width is not additive

Width is not additive

Distance two links

Natural properties of the trunk of a knot

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