Chern-Simons functional, singular instantons, and the four-dimensional clasp number

Daemi, Aliakbar; Scaduto, Christopher

doi:10.48550/arxiv.2007.13160

Cited by 8 publications

(12 citation statements)

References 23 publications

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“…This bound agrees with the bound in Lemma 2.5 for alternating knots and L-space knots, in particular for torus knots. In addition, Daemi and Scaduto [DS20] and Ballinger [Bal20] defined invariants h s and t in gauge theory and Khovanov homology, respectively, from which similar inequalities can be obtained. Again, for many knots, including alternating knots and torus knots, these bounds are no better than the one from Lemma 2.5.…”

Section: Background On the Nonorientable 4-ball Genusmentioning

confidence: 93%

On the nonorientable four-ball genus of torus knots

Binns¹,

Kang²,

Simone³

et al. 2021

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The nonorientable four-ball genus of a knot K in S 3 is the minimal first Betti number of nonorientable surfaces in B 4 bounded by K. By amalgamating ideas from involutive knot Floer homology and unoriented knot Floer homology, we give a new lower bound on the smooth nonorientable four-ball genus γ 4 of any knot. This bound is sharp for several families of torus knots, including T 4n,(2n±1) 2 for even n ≥ 2, a family Longo showed were counterexamples to Batson's conjecture. We also prove that, whenever p is an even positive integer and p 2 is not a perfect square, the torus knot Tp,q does not bound a locally flat Möbius band for almost all integers q relatively prime to p.

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Section: Background On the Nonorientable 4-ball Genusmentioning

confidence: 93%

On the nonorientable four-ball genus of torus knots

Binns¹,

Kang²,

Simone³

et al. 2021

Preprint

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“…For any algebraically split link L, c b 4 (L) ≤ T s (L). In [DS20], Daemi and Scaduto minored the difference c + 4 (K) − g s (K) for the connected sums of copies of the knot 7 4 .…”

Section: Definitions and Main Statementsmentioning

confidence: 99%

“…This raises the question of wether this difference can be greater than one; we prove that it can be arbitrarily large. For this, we show that the T -genus is an upper bound for the 4-dimensional positive clasp number and we use a recent result of Daemi and Scaduto [DS20] that states that the difference between the 4-dimensional positive clasp number and the slice genus can be arbitrarily large.…”

Section: Introductionmentioning

confidence: 99%

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

Moussard

2021

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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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“…In fact, most of the results listed below Question 1.1 were proved using instanton Floer theory. Although we only focus on the 3-dimensional case in this paper, we want to point out that instanton Floer theory has also been used to study SU(2)-representations of the fundamental groups of 4-manifolds (see [DLVVW19,Tan19,Dae20,DS20]).…”

Section: Introductionmentioning

confidence: 99%

On meridian-traceless SU(2)-representations of link groups

Zhang¹

2021

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Suppose L is a link in S 3 . We show that π 1 (S 3 − L) admits an irreducible meridian-traceless representation in SU(2) if and only if L is not the unknot, the Hopf link, or a connected sum of Hopf links. As a corollary, π 1 (S 3 − L) admits an irreducible representation in SU(2) if and only if L is neither the unknot nor the Hopf link. This result generalizes a theorem of Kronheimer and Mrowka [KM10b, Corollary 7.17] to the case of links.

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Chern-Simons functional, singular instantons, and the four-dimensional clasp number

Cited by 8 publications

References 23 publications

On the nonorientable four-ball genus of torus knots

On the nonorientable four-ball genus of torus knots

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

On meridian-traceless SU(2)-representations of link groups

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