Unknotting Tunnels for P(-2,3,7)

Heath, Daniel J.; Song, Hyun-Jong

doi:10.1142/s0218216505004238

Cited by 11 publications

(4 citation statements)

References 12 publications

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“…Tunnels of 2-bridge knots are fully classified due to work of several authors, and they satisfy Case III. D. Heath and H. Song [12] proved that the (−2, 3, 7)-pretzel knot satisfies Case III, and there are expected to be other examples.…”

Section: Case IIImentioning

confidence: 99%

Iterated splitting and the classification of knot tunnels

Cho

McCullough

2013

J. Math. Soc. Japan

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For a genus-1 1-bridge knot in S 3 , that is, a (1, 1)-knot, a middle tunnel is a tunnel that is not an upper or lower tunnel for some (1, 1)-position. Most torus knots have a middle tunnel, and nontorus-knot examples were obtained by Goda, Hayashi, and Ishihara. In a previous paper, we generalized their construction and calculated the slope invariants for the resulting examples. We give an iterated version of the construction that produces many more examples, and calculate their slope invariants. If one starts with the trivial knot, the iterated constructions produce all the 2-bridge knots, giving a new calculation of the slope invariants of their tunnels. In the final section we compile a list of the known possibilities for the set of tunnels of a given tunnel number 1 knot.

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Section: Case IIImentioning

confidence: 99%

Iterated splitting and the classification of knot tunnels

Cho

McCullough

2013

J. Math. Soc. Japan

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“…The tunnels of the (−2, 3, 7)-pretzel knot, denoted by K, were determined by D. Heath and H. Song [5]. There are exactly four mutually non-isotopic unknotting tunnels for K. Figure 15 shows K along with its four tunnels.…”

Section: Figure 11mentioning

confidence: 99%

Braid group and leveling of a knot

Cho¹,

Koda

Seo³

2018

Preprint

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Any knot K in genus-1 1-bridge position can be moved by isotopy to lie in a union of n parallel tori tubed by n − 1 tubes so that K intersects each tube in two spanning arcs, which we call a leveling of the position. The minimal n for which this is possible is an invariant of the position, called the level number. In this work, we describe the leveling by the braid group on two points in the torus, which yields a numerical invariant of the position, called the (1, 1)-length. We show that the (1, 1)-length equals the level number. We then find braid descriptions for (1, 1)-positions of all 2-bridge knots providing upper bounds for their level numbers, and also show that the (−2, 3, 7)-pretzel knot has level number two.

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“…D. Heath and H.-J. Song prove in [4] that the pretzel knot P (−2, 3, 7) has four non-isotopic tunnels. It is well known that it is μ-primitive.…”

Section: Proof Suppose Now That S 3 − N(l(7 17)) Contains An Incompmentioning

confidence: 99%

Heegaard splittings of twisted torus knots

Moriah

Sedgwick

2009

Topology and its Applications

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Little is known on the classification of Heegaard splittings for hyperbolic 3-manifolds. Although Kobayashi gave a complete classification of Heegaard splittings for the exteriors of 2-bridge knots, our knowledge of other classes is extremely limited. In particular, there are very few hyperbolic manifolds that are known to have a unique minimal genus splitting.Here we demonstrate that an infinite class of hyperbolic knot exteriors, namely exteriors of certain "twisted torus knots" originally studied by Morimoto, Sakuma and Yokota, have a unique minimal genus Heegaard splitting of genus two. We also conjecture that these manifolds possess irreducible yet weakly reducible splittings of genus three. There are no known examples of such Heegaard splittings.

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Unknotting Tunnels for P(-2,3,7)

Abstract: There are exactly four mutually non-isotopic unknotting tunnels τi, i = 1,2,3,4 for the pretzel knot P(-2,3,7). Moreover, there are at most 3 non-stabilized genus 3 Heegaard splittings.

Cited by 11 publications

References 12 publications

Iterated splitting and the classification of knot tunnels

Iterated splitting and the classification of knot tunnels

Braid group and leveling of a knot

Heegaard splittings of twisted torus knots

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