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Charles Livingston

Publications and source records attributed to Charles Livingston.

At least 37 records · Page 2Linked to original sources
arXiv2018

Chiral smoothings of knots

Given knots K and J, one can ask whether a single smoothing of a crossing in a diagram for K can convert it into a diagram for J. As an interesting example, Zekovic discovered that the torus knot T(2,5) can be converted into T(2,-5) with a single smoothing. On the other hand, Moore and Vasquez have shown that among torus knots of the form T(2,m), T(2,5) is the only one which can be converted into its mirror image with a single smoothing, assuming that m is square free. In this paper, Casson-Gordon theory is applied to extend the Moore-Vasquez result to all torus knots T(2,m), with the exception of T(2,9), which remains an open case. Applications of similar techniques to the general problem are also described, as are tools arising from Heegaard Floer theory.

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arXiv2018

Unknown values in the table of knots

This paper, to be regularly updated, lists those prime knots with the fewest possible number of crossings for which values of basic knot invariants, such as the unknotting number or the smooth 4-genus, are unknown. This list is being developed in conjunction with "KnotInfo" (www.indiana.edu/~knotinfo), a web-based table of knot invariants.

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arXiv2018

Signature invariants related to the unknotting number

New lower bounds on the unknotting number of a knot are constructed from the classical knot signature function. These bounds can be twice as strong as previously known signature bounds. They can also be stronger than known bounds arising from Heegaard Floer and Khovanov homology. Results include new bounds on the Gordian distance between knots and information about four-dimensional knot invariants. By considering a related non-balanced signature function, bounds on the unknotting number of slice knots are constructed; these are related to the property of double-sliceness.

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arXiv2017

An involutive upsilon knot invariant

Using the theory of involutive Heegaard Floer knot theory developed by Hendricks-Manolescu, we define two involutive analogs of the Upsilon knot concordance invariant of Ozsvath-Stipsicz-Szabo. These involutive invariants are piecewise linear functions defined on the interval [0,2]. Each is a concordance invariant and provides bounds on the three-genus of a knot.

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arXiv2017

Signature functions of knots

The signature function of a knot is an integer-valued step function on the unit circle in the complex plane. Necessary and sufficient conditions for a function to be the signature function of a knot are presented.

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arXiv2016

Secondary Upsilon invariants of knots

The knot invariant Upsilon, defined by Ozsvath, Stipsicz, and Szabo, induces a homomorphism from the smooth knot concordance group to the group of piecewise linear functions on the interval [0,2]. Here we define a set of related secondary invariants, each of which assigns to a knot a piecewise linear function on [0,2]. These secondary invariants provide bounds on the genus and concordance genus of knots. Examples of knots for which Upsilon vanishes but which are detected by these secondary invariants are presented.

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arXiv2016

Splitting numbers of links and the four-genus

The splitting number of a link is the minimum number of crossing changes between distinct components that is required to convert the link into a split link. We provide a bound on the splitting number in terms of the four-genus of related knots.

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arXiv2015

Knot concordances and alternating knots

There is an infinitely generated free subgroup of the smooth knot concordance group with the property that no nontrivial element in this subgroup can be represented by an alternating knot. This subgroup has the further property that every element is represented by a topologically slice knot.

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arXiv2015

The four-genus of connected sums of torus knots

We study the four-genus of linear combinations of torus knots: aT(p,q) # -bT(p',q'). Fixing positive p, q, p', and q', our focus is on the behavior of the four-genus as a function of positive a and b. Three types of examples are presented: in the first, for all a and b the four-genus is completely determined by the Tristram-Levine signature function; for the second, the recently defined Upsilon function of Ozsvath-Stipsicz-Szabo determines the four-genus for all a and b; for the third, a surprising interplay between signatures and Upsilon appears.

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arXiv2015

Doubly slice knots with low crossing number

A knot in the three-sphere is doubly slice if it is the cross-section of an unknotted two-sphere in the four-sphere. For low-crossing knots, the most complete work to date gives a classification of doubly slice knots through 9 crossings. We extend that work through 12 crossings, resolving all but four cases among the 2,977 prime knots in that range. The techniques involved in this analysis include considerations of the Alexander module and signature functions as well as calculations of the twisted Alexander polynomials for higher-order branched covers. We give explicit illustrations of the double slicing for each of the 20 knots shown to be smoothly doubly slice. We place the study of doubly slice knots in a larger context by introducing the double slice genus of a knot.

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arXiv2014

Plane algebraic curves of arbitrary genus via Heegaard Floer homology

Suppose C is a singular curve in CP^2 and it is topologically an embedded surface of genus g; such curves are called cuspidal. The singularities of C are cones on knots K_i. We apply Heegaard Floer theory to find new constraints on the sets of knots {K_i} that can arise as the links of singularities of cuspidal curves. We combine algebro-geometric constraints with ours to solve the existence problem for curves with genus one, d>33, that possess exactly one singularity which has exactly one Puiseux pair (p;q). The realized triples (p,d,q) are expressed as successive even terms in the Fibonacci sequence.

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arXiv2014

Semigroups, d-invariants and deformations of cuspidal singular points of plane curves

We apply Heegaard Floer homology to study deformations of singularities of plane algebraic curves. Our main result provides an obstruction to the existence of a deformation between two singularities. Generalizations include the case of multiple singularities. The obstruction is formulated in terms of a semicontinuity property for semigroups associated to the singularities.

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arXiv2014

Topologically slice knots of smooth concordance order two

The existence of topologically slice knots that are of infinite order in the knot concordance group followed from Freedman's work on topological surgery and Donaldson's gauge theoretic approach to 4-manifolds. Here, as an application of Ozsvath and Szabo's Heegaard-Floer theory, we show the existence of an infinite subgroup of the smooth concordance group generated by topologically slice knots of concordance order two. In addition, no nontrivial element in this subgroup can be represented by a knot with Alexander polynomial one.

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arXiv2013

Heegaard Floer homology and rational cuspidal curves

We apply the methods of Heegaard Floer homology to identify topological properties of complex curves in the complex projective plane. As one application, we resolve an open conjecture that constrains the Alexander polynomial of the link of the singular point of the curve in the case that there is exactly one singular point, having connected link, and the curve is of genus 0. Generalizations apply in the case of multiple singular points.

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arXiv2012

On surgery curves for genus one slice knots

If a knot K bounds a genus one Seifert surface F in the 3-sphere and F contains an essential simple closed curve alpha that has induced framing 0 and is smoothly slice, then K is smoothly slice. Conjecturally, the converse holds. It is known that if K is slice, then there are strong constraints on the algebraic concordance class of such alpha, and it was thought that these constraints might imply that alpha is at least algebraically slice. We present a counterexample; in the process we answer negatively a question of Cooper and relate the result to a problem of Kauffman. Results of this paper depend on the interplay between the Casson-Gordon invariants of K and algebraic invariants of alpha.

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arXiv2012

Non-splittability of the rational homology cobordism group of 3-manifolds

The smooth rational homology cobordism group of rational homology three spheres, T, contains subgroups T_p generated by 3-manifolds with first homology p-torsion, where p is a prime. Rochlin's theorem and gauge theoretic methods show that the inclusion of the direct sum of the T_p into T has infinitely generated kernel. We use Heegaard-Floer methods to show that the cokernel is infinitely generated. On the level of Witt groups of linking forms and conjecturally in the topological category, the inclusion is an isomorphism. One application is the demonstration of the failure in dimension three of an algebraic and higher dimensional theorem of Stoltzfus regarding primary splittings in the knot concordance group. We also build on work of the second author with Hedden and Ruberman to prove that the homology cobordism group of rational homology three spheres that bound topological rational homology 4-balls remains infinitely generated when modded out by the homology cobordism group of integral homology 3-spheres.

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