Knot concordances and alternating knots

Friedl, Stefan; Livingston, Charles; Zentner, Raphael

doi:10.1307/mmj/1491465685

Cited by 14 publications

(19 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Furthermore, Friedl, Livingston, and Zentner recently showed the following. Theorem There is an infinitely generated free subgroup

H \subset {scriptC}_{T S}

such that if

K

represents a nontrivial class in

scriptH

, then

K

is not concordant to any alternating knot.…”

Section: Introductionmentioning

confidence: 91%

“…Additionally, Ozsváth, Stipsicz, and Szabó [32] gave another proof that C T S has a Z ∞ summand using the knot concordance invariant Υ. Furthermore, Friedl, Livingston, and Zentner [10] recently showed the following.…”

Section: Applicationsmentioning

confidence: 97%

“…The sphere S1,0 is homotopic to a copy of S 2δ 1 −1 ⊆ S2,0 in F (00) S1,0 S2,0 S 2(δ 1 +δ 2 )−1 . Furthermore, S 2δ 1 −1 ⊆ S2,0 is homotopic to S1,1 in F (10) . Thus, we have found a homotopy n−1 i=1 Si,0 → j( n−1 i=1 Si,0) for n = 2.…”

Section: Smash Productsmentioning

confidence: 98%

“…Theorem 1.10 [10]. There is an infinitely generated free subgroup H ⊂ C T S such that if K represents a nontrivial class in H, then K is not concordant to any alternating knot.…”

Section: Applicationsmentioning

confidence: 99%

See 3 more Smart Citations

Manolescu invariants of connected sums

Stoffregen

2017

Proc. London Math. Soc.

View full text Add to dashboard Cite

Abstract. We give inequalities for the Manolescu invariants α, β, γ under the connected sum operation. We compute the Manolescu invariants of connected sums of some Seifert fiber spaces. Using these same invariants, we provide a proof of Furuta's Theorem, the existence of a Z 8 subgroup of the homology cobordism group. To our knowledge, this is the first proof of Furuta's Theorem using monopoles. We also provide information about Manolescu invariants of the connected sum of n copies of a three-manifold Y , for large n.

show abstract

“…Furthermore, Friedl, Livingston, and Zentner recently showed the following. Theorem There is an infinitely generated free subgroup

H \subset {scriptC}_{T S}

such that if

K

represents a nontrivial class in

scriptH

, then

K

is not concordant to any alternating knot.…”

Section: Introductionmentioning

confidence: 91%

Section: Applicationsmentioning

confidence: 97%

Section: Smash Productsmentioning

confidence: 98%

“…Theorem 1.10 [10]. There is an infinitely generated free subgroup H ⊂ C T S such that if K represents a nontrivial class in H, then K is not concordant to any alternating knot.…”

Section: Applicationsmentioning

confidence: 99%

See 2 more Smart Citations

Manolescu invariants of connected sums

Stoffregen

2017

Proc. London Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Heegaard Floer theory has provided powerful new tools to investigate this idea. For instance, Friedl-Livingston-Zentner [7] show that alternating knots generate a subgroup with infinitely generated quotient in the concordance group. Aceto-Alfieri [1] have studied the question (given in [7]) of which sums of torus knots are concordant to alternating knots.…”

Section: Introductionmentioning

confidence: 99%

Concordances from differences of torus knots to $L$-space knots

Allen

2019

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

It is known that connected sums of positive torus knots are not concordant to L-space knots. Here we consider differences of torus knots. The main result states that the subgroup of the concordance group generated by two positive torus knots contains no nontrivial L-space knots other than the torus knots themselves. Generalizations to subgroups generated by more than two torus knots are also considered.

show abstract

On the Upsilon invariant and satellite knots

2018

View full text Add to dashboard Cite

We study the effect of satellite operations on the Upsilon invariant of Ozsváth-Stipsicz-Szabó. We obtain results concerning when a knot and its satellites are independent; for example, we show that the setis a basis for an infinite rank summand of the group of smooth concordance classes of topologically slice knots, for D the positive clasped untwisted Whitehead double of any knot with positive τ-invariant, e.g. the right-handed trefoil. We also prove that the image of the Mazur satellite operator on the smooth knot concordance group contains an infinite rank subgroup of topologically slice knots.

show abstract

Knot concordances and alternating knots

Cited by 14 publications

References 32 publications

Manolescu invariants of connected sums

Manolescu invariants of connected sums

Concordances from differences of torus knots to $L$-space knots

On the Upsilon invariant and satellite knots

Contact Info

Product

Resources

About