arXiv subjects

Charles Livingston

Publications and source records attributed to Charles Livingston.

At least 19 recordsLinked to original sources

arXiv2025

The computation of higher order Alexander invariants

In 1928, Alexander defined a sequence of knot polynomials, D_i(K). The first, D_1(K), is the classical Alexander polynomial. These are easily defined in terms of the homology of the infinite cyclic cover of the knot. In theory they can be computed by putting an associated Alexander matrix in Smith normal form. However, standard algorithms for computing the Smith form become impractically slow, even for some 16 crossing knots. Here, methods are developed that can effectively compute the Alexander polynomials of knots with up to 100 crossings.

math.GT↗

arXiv2025

Knot primality: knot Floer homology, metacyclic representations and twisted homology

We develop purely algebraic methods for proving that a knot is prime. Our approach uses the Heegaard Floer polynomial in conjunction with classical knot-theoretic methods: cyclic, dihedral, and metacyclic covering spaces. The theory of twisted homology allows us to view these approaches from a unified perspective. Collectively, the primality tests developed here have proved primality for over 99.67% of knots in a large family of prime knots that includes all prime knots with 15 or fewer crossings. There are additional ways in which our approach highlights the power of Heegaard Floer methods. For one, a single computation can prove the primality of an infinite family of knots. We also illustrate the application of our approach to the setting of general three-manifolds by proving the primality of a knot in a nontrivial homology sphere.

math.GT↗

arXiv2023

Using knot Floer invariants to detect prime knots

We present knot primality tests that are built from knot Floer homology. The most basic of these is a simply stated and elementary consequence of Heegaard Floer theory: if the two-variable knot Floer polynomial of a knot K is irreducible, then K is prime. Improvements in this test yield a primality condition that has been over 90 percent effective in identifying prime knots of up to 30 crossings. As another illustration of the strength of these tools, there are 1,315 non-hyperbolic prime knots with crossing number 20 or less; the tests we develop prove the primality of over 96 percent of them. The filtered chain homotopy class of the knot Floer hat complex of a knot K has a unique minimal-dimension representative that is the direct sum of a one-dimensional complex and two-dimensional complexes, each of which can be assigned a parity. Let delta(K), b_e(K), and b_o(K) denote the dimension of this minimal representative and the number of even and odd two-dimensional summands, respectively. For a composite knot K, we observe that there is a non-trivial factorization delta(K) = mn satisfying (m-1)(n-1) \le 4 min(b_e(K), b_o(K)). This yields another knot primality test. One corollary is a simple proof of Krcatovich's result that L-space knots are prime.

math.GT↗

arXiv2022

Rank-expanding satellite operators on the topological knot concordance group

Given a fixed knot P in a solid torus and any knot K in S^3, one can form the satellite of K with pattern P. This operation induces a self-map of the concordance group of knots in S^3. It has been proved by Dai, Hedden, Mallick, and Stoffregen that in the smooth category there exist P for which this function is rank-expanding; that is, for some K, the set {P(nK)} generates an infinite rank subgroup. Here we demonstrate that similar examples exist in the case of the topological locally flat concordance group. Such examples cannot exist in the algebraic concordance group.

math.GT↗

arXiv2022

An Upsilon torsion function for knot Floer homology

Heegaard Floer theory produces chain complexes associated to knots. Viewed as modules over polynomial rings, such complexes yield torsion invariants that offer constraints on cobordisms between knots. For instance, Juhasz, Miller and Zemke used torsion invariants to bound the number of local maxima and minima in cobordisms between pairs of knots. Gong and Marengon defined a related torsion invariant and used it to study nonorientable knot cobordisms. In this paper we define a one parameter family of Heegaard Floer torsion invariants that yields a piecewise linear function defined on the interval [0,2]. We call this the Upsilon torsion function; it is closely related to the Heegaard Floer Upsilon function defined by Ozsvath, Stipsicz and Szabo. In a natural way, this Upsilon torsion function interpolates between the Juhasz-Miller-Zemke invariant and the Gong-Marengon invariant. In addition to bounding the number of local maxima and minima in knot cobordisms, the Upsilon torsion function provides new obstructions related to the Gordian distance between knots.

math.GT↗

arXiv2022

Signed clasp numbers and four-genus bounds

There exist knots having positive and negative four-dimensional clasp numbers zero but having four-genus, and hence clasp number, arbitrarily large. Such examples were first constructed by Allison Miller, answering a question of Juhasz-Zemke. Further examples are constructed here, complementing those of Miller in that they include examples that are of infinite order in concordance.

math.GT↗

arXiv2022

Branched covers and rational homology balls

The concordance group of knots in the three-sphere contains an infinite subgroup generated by elements of order two, each one of which is represented by a knot K with the property that for every n > 0, the n-fold cyclic cover of S^3 branched over K bounds a rational homology ball. This implies that the kernel of the canonical homomorphism from the knot concordance group to the infinite direct sum of rational homology cobordism groups (defined via prime-power branched covers) contains an infinitely generated two-torsion subgroup.

math.GT↗

arXiv2021

Intrinsic symmetry groups of links

The set of isotopy classes of ordered n-component links in the 3-sphere is acted on by the symmetric group via permutation of the components. The intrinsic symmetry group of the link, S(L), is defined to be the set of elements in the symmetric group that preserve the ordered isotopy type of L as an unoriented link. The study of these groups was initiated in 1969, but the question of whether or not every subgroup of the symmetric group arises as an intrinisic symmetry group of some link has remained open. We provide counterexamples; in particular, if n > 5, then there does not exist an n-component link L for which S(L) is the alternating group.

math.GT↗

arXiv2021

Cobordism distance on the projective space of the knot concordance group

The cobordism distance on the knot concordance group is used to define a measure of how close two knots are to being linearly dependent. Roughly stated, d(K,J) is defined by minimizing the cobordism distance between pairs of knots in cyclic subgroups containing K and J. When made precise, this leads to the definition of the projective space of the knot concordance group. We explore basic properties of this projective space and its integer-valued metric by considering torus knots. Twist knots are used to show that the associated simplicial complex contains an infinite set of simplices of arbitrarily large dimension.

math.GT↗

arXiv2021

Critical point counts in knot cobordisms: abelian and metacyclic invariants

For pairs of knots K and J in the three-sphere, we consider the set of four-tuples of integers (g,x,y,z) for which there is a cobordism from K to J of genus g having x, y, and z, critical points of index 0, 1, and 2, respectively. We describe basic properties that such sets must satisfy and then build obstructions to membership in the set. These obstructions are based on homological invariants arising from cyclic and metacyclic branched covering spaces. A series of examples is presented. A concluding example demonstrates that for each pair of integers g and n, there exists a ribbon knot K for which any genus g cobordism from K to its reverse K^r must have at least n critical points of each index.

math.GT↗

arXiv2021

Connected sums of codimension two locally flat submanifolds

Let X and Y be oriented topological manifolds of dimension n + 2, and let K and J be connected, locally-flat, oriented, n-dimensional submanifolds of X and Y. We show that up to orientation preserving homeomorphism there is a well-defined connected sum K # J in X # Y. For n = 1, the proof is classical, relying on results of Rado and Moise. For dimensions n = 3 and n > 5, results of Edwards-Kirby, Kirby, and Kirby-Siebenmann concerning higher dimensional topological manifolds are required. For n = 2, 4, and 5, Freedman and Quinn's work on topological four-manifolds is needed. The truth of the corresponding statement for higher codimension seems to be unknown.

math.GT↗

arXiv2020

The cobordism distance between a knot and its reverse

The cobordism distance between knots, d(K,J), equals the four-genus g_4(K # -J). We consider d(K,K^r), where K^r is the reverse of K. It is elementary that 0 \le d(K,K^r) \le 2g_4(K) and it is known that there are knots K for which d(K,K^r) is arbitrarily large. Here it is shown that for any knot for which g_4(K) = g_3(K) (such as non-slice knots with g_3(K) = 1 or strongly quasi-positive knots), one has that d(K,K^r) is strictly less that twice g_4(K). It is shown that for arbitrary positive g, there exist knots for which d(K,K^r) = g = g_4(K). There are no known examples for which d(K,K^r) > g_4(K).

math.GT↗

arXiv2020

Notes on the knot concordance invariant Upsilon

The knot concordance invariant Upsilon, recently defined by Ozsvath, Stipsicz, and Szabo, takes values in the group of piecewise linear functions on the closed interval [0,2]. This paper presents a description of one approach to defining Upsilon and of proving its basic properties related to the knot 3-genus, 4-genus, and concordance genus.

math.GT↗

arXiv2020

The concordance classification of low crossing number knots

We present the complete classification of the subgroup of the classical knot concordance group generated by knots with eight or fewer crossings. Proofs are presented in summary. We also describe extensions of this work to the case of nine crossing knots.

math.GT↗

arXiv2020

Unknotting with a single twist

Given a knot in the three-sphere, is it possible to unknot it by performing a single twist, and if so, what are the possible linking numbers of such a twist? We develop obstructions to unknotting using a twist of a specified linking number. The obstructions we describe are built using classical knot invariants, Casson-Gordon invariants, and Heegaard Floer theory.

math.GT↗

arXiv2020

Knot reversal acts non-trivially on the concordance group of topologically slice knots

We construct an infinite family of topologically slice knots that are not smoothly concordant to their reverses. More precisely, if T denotes the concordance group of topologically slice knots and R is the involution of T induced by string reversal, then T/Fix(R) contains an infinitely generated free subgroup. The result remains true modulo the subgroup of T generated by knots with trivial Alexander polynomial.

math.GT↗

arXiv2019

Primary decompositions of knot concordance

For all n > 0 there is a homomorphism from the smooth concordance group of knots in dimension 2n + 1 to an algebraically defined group called the rational algebraic concordance group. This algebraic concordance group splits as a direct sum of groups indexed by polynomials. For n > 1 the homomorphism is injective. This leads to what is called a primary decomposition theorem for knot concordance. In the classical dimension, the kernel of this homomorphism includes the smooth concordance group of topologically slice knots, and Jae Choon Cha has begun studying possible primary decompositions of this subgroup. Here we will show that primary decompositions of a strong type cannot exist. In more detail, it is shown that there exists a topologically slice knot K for which there is a factorization of its Alexander polynomial as f(t)g(t), where f(t) and g(t) are relatively prime and each is the Alexander polynomial of a topologically slice knot, but K is not smoothly concordant to any connected sum of a pair of knots with Alexander polynomials f(t) and g(t).

math.GT↗

arXiv2019

Null-homologous unknottings

Every knot can be unknotted with two generalized twists; this was first proved by Ohyama. Here we prove that any knot of genus g can be unknotted with 2g null-homologous twists and that there exist genus g knots that cannot be unknotted with fewer than 2g null-homologous twists.

math.GT↗