Links with non-trivial Alexander polynomial which are topologically concordant to the Hopf link

Kim, Min Hoon; Krcatovich, David; Park, JungHwan

doi:10.1090/tran/7389

Cited by 6 publications

(6 citation statements)

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“…Similar procedures show that CFK ∞ (T (7,9)) is not stably equivalent to CFK ∞ (T (2, 7) # T (7, 8)), and, in fact, the following general theorem holds. Theorem 1.2.…”

Section: Introductionmentioning

confidence: 77%

“…We have that ( 4 5 ), we compute how far the line of slope − 3 2 needs to be moved so that the elements represented by (1,8) and (3,5) are homologous in F 4 5 ,r . In the diagram we see that we need F 4 5 ,r to contain the elements represented by (3,7) and (2,8). The minimal r which accomplishes this is r = 23 5 , as shown in ( 4 5 ) < − 8 5 .…”

Section: Resultsmentioning

confidence: 99%

“…For example, with a change of basis one can see that the knot complexes CFK ∞ (T (2, 3) # T (2, 3)) and CFK ∞ (T (2, 5)) are stably equivalent. In a recent paper, Kim, Krcatovich, and Park [7] gave a condition for the knot complex of the connected sum of two L-space knots to be stably equivalent to a staircase complex. Using this result, we can see that CFK ∞ (T (p − 1, p) # T (p, p + 1)) and CFK ∞ (T (p, 2p − 1)) are stably equivalent.…”

mentioning

confidence: 99%

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Using secondary Upsilon invariants to rule out stable equivalence of knot complexes

Allen

2020

Algebr. Geom. Topol.

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Two Heegaard Floer knot complexes are called stably equivalent if an acyclic complex can be added to each complex to make them filtered chain homotopy equivalent. Hom showed that if two knots are concordant, then their knot complexes are stably equivalent. Invariants of stable equivalence include the concordance invariants τ , ε, and Υ. Feller and Krcatovich gave a relationship between the Upsilon invariants of torus knots. We use secondary Upsilon invariants defined by Kim and Livingston to show that these relations do not extend to stable equivalence.

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“…Similar procedures show that CFK ∞ (T (7,9)) is not stably equivalent to CFK ∞ (T (2, 7) # T (7, 8)), and, in fact, the following general theorem holds. Theorem 1.2.…”

Section: Introductionmentioning

confidence: 77%

Section: Resultsmentioning

confidence: 99%

mentioning

confidence: 99%

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Using secondary Upsilon invariants to rule out stable equivalence of knot complexes

Allen

2020

Algebr. Geom. Topol.

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“…In fact, a careful check of the Alexander and the algebraic filtrations shows that

C F K^{\infty} false(T_{3 q + 2, 3} false) = {Span}_{{double-struckZ}_{2} false[U, U^{- 1} false]} false⟨ z_{0}^{'}, \dots, z_{2 q + 2}^{'} false⟩ .

Summing up we get that

C F K^{\infty} false(T_{3 q + 1, 3} false) \otimes C F K^{\infty} false(T_{3, 2} false) = C F K^{\infty} false(T_{3 q + 2, 3} false) \oplus A_{*}

with

A_{*}

acyclic. Note that this can also be seen as a consequence of [, Lemma 3.18]. To prove the corresponding statement for

C F K I^{\infty}

we need to check that (1)the involution of

C F K I^{\infty} false(T_{3 q + 1, 3} false) \tilde{\otimes} C F K I^{\infty} false(T_{3, 2} false)

restricts to

ι_{T_{3 q + 2, 3}}

on the sub complex spanned by

z_{0}^{'}, \dots, z_{2 q + 2}^{'}

, (2)

ι_{T_{3 q + 1, 3}} \times ι_{T_{3, 2}}

leaves

A_{*}

invariant. Here

C F K I^{\infty} false(T_{3 q + 1, 3} false) \tilde{\otimes} C F K I^{\infty} false(T<...>

…”

Section: Obstructions From the Kim‐livingston Secondary Invariantmentioning

confidence: 97%

On sums of torus knots concordant to alternating knots

Aceto

Alfieri

2018

Bull. London Math. Soc.

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We consider the question, asked by Friedl, Livingston and Zentner, of which sums of torus knots are concordant to alternating knots. After a brief analysis of the problem in its full generality, we focus on sums of two torus knots. We describe some effective obstructions based on Heegaard Floer homology.be a sum of torus knots. Suppose that p i > q i , and that there is no q i coefficient appearing with repetitions in the list of coefficients ). Denote by Υ a,b (t) the upsilon function of the torus knot T a,b with a > b, thenConsequently, if q i and r i denote, respectively, the quotients and the remainders occurring in the Euclidean algorithm for a and b (so that r 0 = a, r −1 = b, and r i−1 = q i r i + r i+1 ), we have that Υ a,b (t) = n i=0 q i · Υ ri+1,ri (t).

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“…For knots K and J, if for some s ∈ (0, 2) and m ∈ R we have Remark 4.11. As noted in Allen's paper [1], in [6] Kim, Krcatovich and Park gave a condition for the knot complex of the connected sum of two L-space knots to be stably equivalent to a staircase complex. In particular, Lemma 3.18 in [6] implies that CFK ∞ (T (p, 2p − 1)) is stably equivalent to CFK ∞ (T (p − 1, p)#T (p, p + 1)).…”

Section: Calculations Of Secondary Upsilon Invariantmentioning

confidence: 99%

On the secondary Upsilon invariant

2018

Preprint

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Links with non-trivial Alexander polynomial which are topologically concordant to the Hopf link

Abstract: We give infinitely many 2-component links with unknotted components which are topologically concordant to the Hopf link, but not smoothly concordant to any 2-component link with trivial Alexander polynomial. Our examples are pairwise non-concordant.

Cited by 6 publications

References 40 publications

Using secondary Upsilon invariants to rule out stable equivalence of knot complexes

Using secondary Upsilon invariants to rule out stable equivalence of knot complexes

On sums of torus knots concordant to alternating knots

On the secondary Upsilon invariant

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