On tunnel numbers of a cable knot and its companion

Wang, Junhua; Zou, Yanqing

doi:10.1016/j.topol.2020.107319

Cited by 2 publications

(1 citation statement)

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“…Schirmer [34] showed that this is true when the wrapping number is 1; that is, for composite knots. Wang and Zou [48] showed that the tunnel number of a cable knot is at least the tunnel number of its companion. Li [24] has shown that t(T ) is at least that of its pattern; however, it seems to be an open problem whether or not it is at least that of the companion K. Our work points towards some of the difficulties of proving this and the existence of counter-examples to Schubert's inequality when g = 1 perhaps indicates some skepticism is in order.…”

Section: Comparisons and Consequencesmentioning

confidence: 99%

Thin position for knots, links, and graphs in 3–manifolds

Taylor

Tomova

2018

Algebr. Geom. Topol.

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We define a new notion of thin position for a graph in a 3-manifold which combines the ideas of thin position for manifolds first originated by Scharlemann and Thompson with the ideas of thin position for knots first originated by Gabai. This thin position has the property that connect summing annuli and pairs-of-pants show up as thin levels. In a forthcoming paper, this new thin position allows us to define two new families of invariants of knots, links, and graphs in 3-manifolds. The invariants in one family are similar to bridge number and the invariants in the other family are similar to Gabai's width for knots in the 3-sphere. The invariants in both families detect the unknot and are additive under connected sum and trivalent vertex sum.

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Section: Comparisons and Consequencesmentioning

confidence: 99%