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Charles Livingston

Publications and source records attributed to Charles Livingston.

69 records · Page 4Linked to original sources
arXiv2004

The concordance genus of knots

In knot concordance three genera arise naturally, g(K), g_4(K), and g_c(K): these are the classical genus, the 4-ball genus, and the concordance genus, defined to be the minimum genus among all knots concordant to K. Clearly 0 <= g_4(K) <= g_c(K) <= g(K). Casson and Nakanishi gave examples to show that g_4(K) need not equal g_c(K). We begin by reviewing and extending their results. For knots representing elements in A, the concordance group of algebraically slice knots, the relationships between these genera are less clear. Casson and Gordon's result that A is nontrivial implies that g_4(K) can be nonzero for knots in A. Gilmer proved that g_4(K) can be arbitrarily large for knots in A. We will prove that there are knots K in A with g_4(K) = 1 and g_c(K) arbitrarily large. Finally, we tabulate g_c for all prime knots with 10 crossings and, with two exceptions, all prime knots with fewer than 10 crossings. This requires the description of previously unnoticed concordances.

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arXiv2003

Four-manifolds of large negative deficiency

For every N > 0 there exists a group of deficiency less than -N that arises as the fundamental group of a smooth homology 4-sphere and also as the fundamental group of the complement of a compact contractible submanifold of the 4-sphere. A group is the fundamental group of the complement of a contractible submanifold of the n-sphere, n > 4, if and only if it is the fundamental group of a homology n-sphere. There exist fundamental groups of homology n-spheres, n > 4, that cannot arise as the fundamental group of the complement of a contractible submanifold of the 4-sphere.

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arXiv2003

Knot mutation: 4-genus of knots and algebraic concordance

Kearton observed that mutation can change the concordance class of a knot. A close examination of his example reveals that it is of 4-genus 1 and has a mutant of 4-genus 0. The first goal of this paper is to construct examples to show that for any pair of nonnegative integers m and n there is a knot of 4-genus m with a mutant of 4-genus n. A second result of this paper is a crossing change formula for the algebraic concordance class of a knot, which is then applied to prove the invariance of the algebraic concordance class under mutation. The paper concludes with an application of crossing change formulas to give a short new proof of Long's theorem that strongly positive amphicheiral knots are algebraically slice.

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arXiv2003

The Hausmann-Weinberger 4-manifold invariant of abelian groups

The Hausmann-Weinberger invariant of a group G is the minimal Euler characteristic of a closed orientable 4-manifold M with fundamental group G. We compute this invariant for finitely generated free abelian groups and estimate the invariant for all finitely generated abelian groups.

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arXiv2003

Knot signature functions are independent

To each unit complex number with positive imaginary part there is defined a Tristram-Levine knot signature function. The set of all such signature functions is linearly independent as a set of functions defined on the set of all knots. The set of averaged signature functions forms a linearly independent set of homomorophisms on the knot concordance group. However, for each unit root of an Alexander polynomial, there is a slice knot with nonvanishing signature at that root and its conjugate, and nowhere else. These results hold for knots in all odd dimension.

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arXiv2002

The slicing number of a knot

An open question asks if every knot of 4-genus g_s can be changed into a slice knot by g_s crossing changes. A counterexample is given.

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arXiv2002

Seifert forms and concordance

If a knot K has Seifert matrix V_K and has a prime power cyclic branched cover that is not a homology sphere, then there is an infinite family of non-concordant knots having Seifert matrix V_K.

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arXiv2001

Observations on Lickorish knotting of contractible 4-manifolds

Lickorish has constructed large families of contractible 4--manifolds that have knotted embeddings in the 4--sphere and has also shown that every finitely presented perfect group with balanced presentation occurs as the fundamental group of the complement of a knotted contractible manifold. Here we make a few observations regarding Lickorish's construction, showing how to extend it to construct contractible 4--manifolds which have an infinite number of knotted embeddings and also to construct knotted embeddings of the Mazur manifold for which the complement has trivial fundamental group.

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arXiv2001

Infinite Order Amphicheiral Knots

In answer to a question of Long, Flapan constructed an example of a prime strongly positive amphicheiral knot that is not slice. Long had proved that all such knots are algebraically slice. Here we show that the concordance group of algebraically slice knots contains an infinitely generated free subgroup that is generated by prime strongly positive amphicheiral knots. A simple corollary of this result is the existence of positive amphicheiral knots that are not of order two in concordance.

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arXiv2000

Pseudo-slice knots

For n >1, if the Seifert form of a knotted 2n-1 sphere K in S^{2n+1} has a metabolizer, then the knot is slice. Casson and Gordon proved that this is false in dimension three (n = 1). However, in the three dimensional case it is true that if the metabolizer has a basis represented by a strongly slice link then K is slice. The question has been asked as to whether it is sufficient that each basis element is represented by a slice knot to assure that K is slice. For genus one knots this is of course true; here we present a genus two counterexample.

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arXiv2000

Maps of surface groups to finite groups with no simple loops in the kernel

Let $F_g$ denote the closed orientable surface of genus $g$. What is the least order finite group, $G_g$, for which there is a homomorphism $ψ$ from $π_1(F_g)$ to $G_g$ so that no nontrivial simple closed curve on $F_g$ represents an element in Ker($ψ$)? For the torus it is easily seen that $G_1 = Z_2 \times Z_2$ suffices. We prove here that $G_2$ is a group of order 32 and that an upper bound for the order of $G_g$ is given by $g^{2g +1}$. The previously known upper bound was greater than $2^{g{2^{2g}}}$.

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arXiv1999

Knot Concordance and Torsion

Let K be a knot in the 3-sphere with 2-fold branched covering space M. If for some prime p congruent to 3 mod 4 the p-torsion in the first homology of M is cyclic with odd exponent, then K is of infinite order in the knot concordance group. As one application, recall that the n-twisted double of an arbitrary knot has order 4 in Levine's algebraic concordance group if and only if n is positive and some prime congruent to 3 mod 4 has odd exponent in 4n+1; we show that all such knots are of infinite order in the knot concordance group. As a second application, the 2-bridge knot K(r,s) has infinite order in the knot concordance group if some prime congruent to 3 mod 4 has odd exponent in r.

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arXiv1999

Order 2 Algebraically Slice Knots

The concordance group of algebraically slice knots is the subgroup of the classical knot concordance group formed by algebraically slice knots. Results of Casson and Gordon and of Jiang showed that this group contains in infinitely generated free (abelian) subgroup. Here it is shown that the concordance group of algebraically slice knots also contain elements of finite order; in fact it contains an infinite subgroup generated by elements of order 2.

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arXiv1998

Obstructing 4-torsion in the classical knot concordance group

We prove that if the order of the first homology of the 2-fold branched cover of a knot K in the 3-sphere is given by pm where p is a prime congruent to 3 mod 4 and gcd(p,m) =1, then K is of infinite order in the knot concordance group. This provides an obstruction to classical knots being of order 4. In particular, there are 11 prime knots with 10 or fewer crossings that are of order 4 in the algebraic concordance group; all are infinite order in concordance. Another corollary states that any knot with Alexander polynomial 5t^2 - 11t + 5 is of infinite order in concordance; Levine proved that in higher dimensions all such knots are of order 4.

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