arXiv subjects

Greg Kuperberg

Publications and source records attributed to Greg Kuperberg.

71 records · Page 4Linked to original sources

arXiv1997

Detecting knot invertibility

We discuss the consequences of the possibility that Vassiliev invariants do not detect knot invertibility as well as the fact that quantum Lie group invariants are known not to do so. On the other hand, finite group invariants, such as the set of homomorphisms from the knot group to M_11, can detect knot invertibility. For many natural classes of knot invariants, including Vassiliev invariants and quantum Lie group invariants, we can conclude that the invariants either distinguish all oriented knots, or there exist prime, unoriented knots which they do not distinguish.

q-alg↗

arXiv1997

Another homogeneous, non-bihomogeneous Peano continuum

K. Kuperberg found a locally connected, finite-dimensional continuum which is homogeneous but not bihomogeneous. We give a similar but simpler example. Like previous constructions, the example is locally a Cartesian product of Menger spaces. The new idea is to choose a fundamental group in which not every element is conjugate to its inverse.

math.GN↗

arXiv1997

Another proof of the alternating sign matrix conjecture

Robbins conjectured, and Zeilberger recently proved, that there are 1!4!7!...(3n-2)!/n!/(n+1)!/.../(2n-1)! alternating sign matrices of order n. We give a new proof of this result using an analysis of the six-vertex state model (also called square ice) based on the Yang-Baxter equation.

math.CO↗

arXiv1997

Spiders for rank 2 Lie algebras

A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or group-like object. We define certain combinatorial spiders by generators and relations that are isomorphic to the representation theories of the three rank two simple Lie algebras, namely A2, B2, and G2. They generalize the widely-used Temperley-Lieb spider for A1. Among other things, they yield bases for invariant spaces which are probably related to Lusztig's canonical bases, and they are useful for computing quantities such as generalized 6j-symbols and quantum link invariants.

q-alg↗

arXiv1996

Jaeger's Higman-Sims state model and the B_2 spider

Jaeger [Geom. Dedicata 44 (1992), 23-52] discovered a remarkable checkerboard state model based on the Higman-Sims graph that yields a value of the Kauffman polynomial, which is a quantum invariant of links. We present a simple argument that the state model has the desired properties using the combinatorial $B_2$ spider [Comm. Math. Phys. 180 (1996), 109-151].

math.QA↗

arXiv1995

Highly saturated packings and reduced coverings

We introduce and study certain notions which might serve as substitutes for maximum density packings and minimum density coverings. A body is a compact connected set which is the closure of its interior. A packing $\cal P$ with congruent replicas of a body $K$ is $n$-saturated if no $n-1$ members of it can be replaced with $n$ replicas of $K$, and it is completely saturated if it is $n$-saturated for each $n\ge 1$. Similarly, a covering $\cal C$ with congruent replicas of a body $K$ is $n$-reduced if no $n$ members of it can be replaced by $n-1$ replicas of $K$ without uncovering a portion of the space, and it is completely reduced if it is $n$-reduced for each $n\ge 1$. We prove that every body $K$ in $d$-dimensional Euclidean or hyperbolic space admits both an $n$-saturated packing and an $n$-reduced covering with replicas of $K$. Under some assumptions on $K\subset \mathbb{E}^d$ (somewhat weaker than convexity), we prove the existence of completely saturated packings and completely reduced coverings, but in general, the problem of existence of completely saturated packings and completely reduced coverings remains unsolved. Also, we investigate some problems related to the the densities of $n$-saturated packings and $n$-reduced coverings. Among other things, we prove that there exists an upper bound for the density of a $d+2$-reduced covering of $\mathbb{E}^d$ with congruent balls, and we produce some density bounds for the $n$-saturated packings and $n$-reduced coverings of the plane with congruent circles.

math.MG↗

arXiv1995

Asymptotically optimal covering designs

A (v,k,t) covering design, or covering, is a family of k-subsets, called blocks, chosen from a v-set, such that each t-subset is contained in at least one of the blocks. The number of blocks is the covering's size}, and the minimum size of such a covering is denoted by C(v,k,t). It is easy to see that a covering must contain at least (v choose t)/(k choose t) blocks, and in 1985 Rödl [European J. Combin. 5 (1985), 69-78] proved a long-standing conjecture of Erdős and Hanani [Publ. Math. Debrecen 10 (1963), 10-13] that for fixed k and t, coverings of size (v choose t)/(k choose t) (1+o(1)) exist (as v \to \infty). An earlier paper by the first three authors [J. Combin. Des. 3 (1995), 269-284] gave new methods for constructing good coverings, and gave tables of upper bounds on C(v,k,t) for small v, k, and t. The present paper shows that two of those constructions are asymptotically optimal: For fixed k and t, the size of the coverings constructed matches Rödl's bound. The paper also makes the o(1) error bound explicit, and gives some evidence for a much stronger bound.

math.CO↗

arXiv1995

Four symmetry classes of plane partitions under one roof

In previous paper, the author applied the permanent-determinant method of Kasteleyn and its non-bipartite generalization, the Hafnian-Pfaffian method, to obtain a determinant or a Pfaffian that enumerates each of the ten symmetry classes of plane partitions. After a cosmetic generalization of the Kasteleyn method, we identify the matrices in the four determinantal cases (plain plane partitions, cyclically symmetric plane partitions, transpose-complement plane partitions, and the intersection of the last two types) in the representation theory of sl(2,C). The result is a unified proof of the four enumerations.

math.CO↗

arXiv1995

A volume-preserving counterexample to the Seifert conjecture

We prove that every 3-manifold possesses a $C^1$, volume-preserving flow with no fixed points and no closed trajectories. The main construction is a volume-preserving version of the Schweitzer plug. We also prove that every 3-manifold possesses a volume-preserving, $C^\infty$ flow with discrete closed trajectories and no fixed points (as well as a PL flow with the same geometry), which is needed for the first result. The proof uses a Dehn-twisted Wilson-type plug which also preserves volume.

math.DS↗

arXiv1995

New constructions for covering designs

A $(v,k,t)$ {\em covering design}, or {\em covering}, is a family of $k$-subsets, called blocks, chosen from a $v$-set, such that each $t$-subset is contained in at least one of the blocks. The number of blocks is the covering's {\em size}, and the minimum size of such a covering is denoted by $C(v,k,t)$. This paper gives three new methods for constructing good coverings: a greedy algorithm similar to Conway and Sloane's algorithm for lexicographic codes~\cite{lex}, and two methods that synthesize new coverings from preexisting ones. Using these new methods, together with results in the literature, we build tables of upper bounds on $C(v,k,t)$ for $v \leq 32$, $k \leq 16$, and $t \leq 8$.%

math.CO↗

arXiv1994

Self-complementary plane partitions by Proctor's minuscule method

A method of Proctor [European J. Combin. 5 (1984), no. 4, 331-350] realizes the set of arbitrary plane partitions in a box and the set of symmetric plane partitions as bases of linear representations of Lie groups. We extend this method by realizing transposition and complementation of plane partitions as natural linear transformations of the representations, thereby enumerating symmetric plane partitions, self-complementary plane partitions, and transpose-complement plane partitions in a new way.

math.CO↗

arXiv1994

Symmetries of plane partitions and the permanent-determinant method

In the paper [J. Combin. Theory Ser. A 43 (1986), 103--113], Stanley gives formulas for the number of plane partitions in each of ten symmetry classes. This paper together with results by Andrews [J. Combin. Theory Ser. A 66 (1994), 28-39] and Stembridge [Adv. Math 111 (1995), 227-243] completes the project of proving all ten formulas. We enumerate cyclically symmetric, self-complementary plane partitions. We first convert plane partitions to tilings of a hexagon in the plane by rhombuses, or equivalently to matchings in a certain planar graph. We can then use the permanent-determinant method or a variant, the Hafnian-Pfaffian method, to obtain the answer as the determinant or Pfaffian of a matrix in each of the ten cases. We row-reduce the resulting matrix in the case under consideration to prove the formula. A similar row-reduction process can be carried out in many of the other cases, and we analyze three other symmetry classes of plane partitions for comparison.

math.CO↗

arXiv1994

Average kissing numbers for non-congruent sphere packings

The Koebe circle packing theorem states that every finite planar graph can be realized as the nerve of a packing of (non-congruent) circles in R^3. We investigate the average kissing number of finite packings of non-congruent spheres in R^3 as a first restriction on the possible nerves of such packings. We show that the supremum k of the average kissing number for all packings satisfies 12.566 ~ 666/53 <= k < 8 + 4*sqrt(3) ~ 14.928 We obtain the upper bound by a resource exhaustion argument and the upper bound by a construction involving packings of spherical caps in S^3. Our result contradicts two naive conjectures about the average kissing number: That it is unbounded, or that it is supremized by an infinite packing of congruent spheres.

math.MG↗

arXiv1992

A low-technology estimate in convex geometry

Let $K$ be an $n$-dimensional symmetric convex body with $n \ge 4$ and let $K\dual$ be its polar body. We present an elementary proof of the fact that $$(\Vol K)(\Vol K\dual)\ge \frac{b_n^2}{(\log_2 n)^n},$$ where $b_n$ is the volume of the Euclidean ball of radius 1. The inequality is asymptotically weaker than the estimate of Bourgain and Milman, which replaces the $\log_2 n$ by a constant. However, there is no known elementary proof of the Bourgain-Milman theorem.

math.MG↗

arXiv1991

The quantum G_2 link invariant

We derive an inductive, combinatorial definition of a polynomial-valued regular isotopy invariant of links and tangled graphs. We show that the invariant equals the Reshetikhin-Turaev invariant corresponding to the exceptional simple Lie algebra G_2. It is therefore related to G_2 in the same way that the HOMFLY polynomial is related to A_n and the Kauffman polynomial is related to B_n, C_n, and D_n. We give parallel constructions for the other rank 2 Lie algebras and present some combinatorial conjectures motivated by the new inductive definitions.

math.QA↗

arXiv1991

Alternating sign matrices and domino tilings

We introduce a family of planar regions, called Aztec diamonds, and study the ways in which these regions can be tiled by dominoes. Our main result is a generating function that not only gives the number of domino tilings of the Aztec diamond of order $n$ but also provides information about the orientation of the dominoes (vertical versus horizontal) and the accessibility of one tiling from another by means of local modifications. Several proofs of the formula are given. The problem turns out to have connections with the alternating sign matrices of Mills, Robbins, and Rumsey, as well as the square ice model studied by Lieb.

math.CO↗

arXiv1990

Involutory Hopf algebras and 3-manifold invariants

We establish a 3-manifold invariant for each finite-dimensional, involutory Hopf algebra. If the Hopf algebra is the group algebra of a group $G$, the invariant counts homomorphisms from the fundamental group of the manifold to $G$. The invariant can be viewed as a state model on a Heegaard diagram or a triangulation of the manifold. The computation of the invariant involves tensor products and contractions of the structure tensors of the algebra. We show that every formal expression involving these tensors corresponds to a unique 3-manifold modulo a well-understood equivalence. This raises the possibility of an algorithm which can determine whether two given 3-manifolds are homeomorphic.

math.QA↗