Computations of the Ozsv ath{Szab o knot concordance invariant

By Charles Livingston. 2004

121 citations recorded by the source corpus.

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Citations in the corpus

The Superpolynomial for Knot Homologies (2005, methodology): “…with the previous inequality to obtain the following estimate for S: w(D) − c(D) + 1 ≤ S(K) ≤ w(D) + c(D) − 1, where D is any planar diagram of K. Zoltán Szabó pointed out to us that using the work of Livingston [Livingston 04], it is not difficult to see that s and τ satisfy similar estimates.”
A concordance invariant from the Floer homology of double branched covers (2005, background): “ m,n ≥1, m 6= n. Then the connected sum Wh(K 1)#(−Wh(K 2)) has σ′ = τ = s′ = 0 but δ 6= 0 . Let us denote by C ts the smooth concordance group of topologically slice knots. It was shown by Livingston [14] that τ : C ts →Z is a surjective homomorphism, and therefore C ts has a Z summand. Using the pa”
The knot Floer complex and the smooth concordance group (2011, background): “concordance group Cto the integers: ˝: C!Z; which gives a new proof of the Milnor conjecture [OS03b] and is strong enough to obstruct topologically slice knots from being smoothly slice (for example, [Liv04]). Often, we would like to be able to show that a collection of nknots is linearly independen”
Bordered Heegaard Floer homology and the tau‐invariant of cable knots (2012, background): “Studying τ has yielded many nice results, such as a new proof of the Milnor conjecture [12], and examples of Alexander polynomial one knots which are not smoothly slice (for example, [4, 10]).”
Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation (2008, unknown): “A few more recent articles include [4,5,9,32,34,35].”
Khovanov homology and the slice genus (2004, background): “Combining the results of [4], [11], [13], [14], and [15] with some unpublished computations of the author, it appears that the value of τ has been determined for all but two knots of 10 crossings and fewer.”
Topologically slice knots of smooth concordance order two (2012, methodology): “Techniques derived from Heegaard Floer theory and Khovanov homology (specifically the Rasmussen invariant [39]) were used to show that CTS contains a summand isomorphic to Z(3) [23, 24, 25].”
The slice-ribbon conjecture for 3-stranded pretzel knots (2007, result): “ollary 1.4) only relied on Donaldson’s diagonalization theorem. It is not hard, however, to come up with a purely Heegaard Floer theoretic proof of their result: it follows from work of C. Livingston [13] and Ozsv´ath-Szab´o [21] that the concordance invariant τ(cf. [17]) for three stranded pretzel ”
Notes on the knot concordance invariant Upsilon (2014, background): “The argument is essentially the same as used in [2] to prove the corresponding fact about τ(K).”
The Ozsváth-Szabó and Rasmussen Concordance Invariants are not Equal (2005, background): “Next, recall that τ (and s) satisfy the following inequality under the operation of changing a crossing in a given knot diagram (see [11] or [18] for a proof):”
An infinite-rank summand of topologically slice knots (2013, methodology): “[HLR12] Matthew Hedden, Charles Livingston, and Daniel Ruberman, Topologically slice knots with nontrivial Alexander polynomial, Adv.”