Amphichiral knots with large 4-genus

Miller, Allison N.

doi:10.48550/arxiv.2011.09346

Cited by 1 publication

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“…This raises the question of what can be the difference between the T -genus and the balanced 4-dimensional clasp number. In [Mil20], Miller proves the existence of knots with arbitrarily large slice genus and trivial positive and negative 4-dimensional clasp number, which implies that the difference T s − c ± 4 can be arbitrarily large. Question 2.4.…”

Section: Definitions and Main Statementsmentioning

confidence: 99%

Slice genus, $T$-genus and $4$-dimensional clasp number

Moussard

2021

Preprint

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The T -genus of a knot is the minimal number of borromean-type triple points on a normal singular disk with no clasp bounded by the knot; it is an upper bound for the slice genus. Kawauchi, Shibuya and Suzuki characterized the slice knots by the vanishing of their T -genus. We generalize this to provide a 3-dimensional characterization of the slice genus. Further, we prove that the T -genus majors the 4-dimensional positive clasp number and we deduce that the difference between the T -genus and the slice genus can be arbitrarily large. We introduce the ribbon counterpart of the T -genus and prove that it is an upper bound for the ribbon genus. Interpreting the T -genera in terms of ∆-distance, we show that the T -genus and the ribbon T -genus coincide for all knots if and only if all slice knots are ribbon. We work in the more general setting of algebraically split links and we also discuss the case of colored links. Finally, we express Milnor's triple linking number of an algebraically split 3-component link as the algebraic intersection number of three immersed disks bounded by the three components.

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Section: Definitions and Main Statementsmentioning

confidence: 99%

Slice genus, $T$-genus and $4$-dimensional clasp number

Moussard

2021

Preprint

View full text Add to dashboard Cite

show abstract

Amphichiral knots with large 4-genus

Cited by 1 publication

References 19 publications

Slice genus, $T$-genus and $4$-dimensional clasp number

Slice genus, $T$-genus and $4$-dimensional clasp number

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