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Quandle 2-cocycles define invariants of classical and virtual knots, and extensions of quandles. We show that the quandle 2-cocycle invariant with respect to a non-trivial [Formula: see text]-cocycle is constant, or takes some other restricted form, for classical knots when the corresponding extensions satisfy certain algebraic conditions. In particular, if an abelian extension is a conjugation quandle, then the corresponding cocycle invariant is constant. Specific examples are presented from the list of connected quandles of order less than 48. Relations among various quandle epimorphisms involved are also examined.

Quandle homology theories have been developed and cocycles have been used to construct invariants in state-sum form for knots using colorings of knot diagrams by quandles. Quandle 2-cocycles can be also used to define extensions as in the case of groups. There are relations among algebraic properties of quandles, their homology theories, and cocycle invariants; certain algebraic properties of quandles affect the values of the cocycle invariants, and identities satisfied by quandles induce subcomplexes of homology theory. Recent developments in these matters, as well as computational aspects of the invariant, are reviewed. Problems and conjectures pertinent to the subject are also listed.

Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror can be distinguished by quandle invariants of composite knots. We investigate this and related phenomena. Quandle cocycle invariants are studied in relation to the connected sum, and formulas are given for computing the cocycle invariant from the number of colorings of composite knots. Relations to corresponding abelian extensions of quandles are studied, and extensions are examined for the table of small connected quandles, called Rig quandles. Computer calculations are presented, and summaries of outputs are discussed., 37 pages

The knot coloring polynomial defined by Eisermann for a finite pointed group is generalized to an infinite pointed group as the longitudinal mapping invariant of a knot. In turn, this can be thought of as a generalization of the quandle 2-cocycle invariant for finite quandles. If the group is a topological group, then this invariant can be thought of as a topological generalization of the 2-cocycle invariant. The longitudinal mapping invariant is based on a meridian–longitude pair in the knot group. We also give an interpretation of the invariant in terms of quandle colorings of a 1-tangle for generalized Alexander quandles without use of a meridian–longitude pair in the knot group. The invariant values are concretely evaluated for the torus knots [Formula: see text], their mirror images, and the figure eight knot for the group SU(2).

For a permutation f of an n-dimensional vector space V over a finite field of order q we let k-affinity(f) denote the number of k-flats X of V such that f(X) is also a k-flat. By k-spectrum(n,q) we mean the set of integers k-affinity(f) where f runs through all permutations of V. The problem of the complete determination of k-spectrum(n,q) seems very difficult except for small or special values of the parameters. However, we are able to establish that k-spectrum(n,q) contains 0 in the following cases: (i) q>2 and 02. The maximum of k-affinity(f) is, of course, obtained when f is any semi-affine mapping. We conjecture that the next to largest value of k-affinity(f) is when f is a transposition and we are able to prove this when q=2, k=2, n>2 and when q>2, k=1, n>1., Comment: 25 pages

We present a set of 26 finite quandles that distinguish (up to reversal and mirror image) by number of colorings, all of the 2977 prime oriented knots with up to 12 crossings. We also show that 1058 of these knots can be distinguished from their mirror images by the number of colorings by quandles from a certain set of 23 finite quandles. We study the colorings of these 2977 knots by all of the 431 connected quandles of order at most 35 found by L. Vendramin. Among other things, we collect information about quandles that have the same number of colorings for all of the 2977 knots. For example, we prove that if $Q$ is a simple quandle of prime power order then $Q$ and the dual quandle $Q^*$ of $Q$ have the same number of colorings for all knots and conjecture that this holds for all Alexander quandles $Q$. We study a knot invariant based on a quandle homomorphism $f:Q_1\to Q_0$. We also apply the quandle colorings we have computed to obtain some new results for the bridge index, the Nakanishi index, the tunnel number, and the unknotting number. In an appendix we discuss various properties of the quandles in Vendramin's list. Links to the data computed and various programs in C, GAP and Maple are provided., A newer version of arXiv:1312.3307, Typos fixed

We prove that certain functions involving the gamma and q-gamma function are monotone. We also prove that (xmψ(x))(m+1) is completely monotonic. We conjecture that -(xmψ(m)(x))(m) is completely monotonic for m ≥ 2; and we prove it, with help from Maple, for 2 ≤ m ≤ 16. We give a very useful Maple procedure to verify this for higher values of m. A stronger result is also formulated where we conjecture that the power series coefficients of a certain function are all positive.

We explore a knot invariant derived from colorings of corresponding [Formula: see text]-tangles with arbitrary connected quandles. When the quandle is an abelian extension of a certain type the invariant is equivalent to the quandle [Formula: see text]-cocycle invariant. We construct many such abelian extensions using generalized Alexander quandles without explicitly finding [Formula: see text]-cocycles. This permits the construction of many [Formula: see text]-cocycle invariants without exhibiting explicit [Formula: see text]-cocycles. We show that for connected generalized Alexander quandles the invariant is equivalent to Eisermann’s knot coloring polynomial. Computations using this technique show that the [Formula: see text]-cocycle invariant distinguishes all of the oriented prime knots up to 11 crossings and most oriented prime knots with 12 crossings including classification by symmetry: mirror images, reversals, and reversed mirrors.

Let V be an n-dimensional vector space over a field F. We attempt to determine the least positive integer γ=(k, n, F) for which there exists a family U1 U2 ,⋯,Uγ of k-dimensional subspaces of V such that for every (n−k)-dimensional subspace W of V there is an ie{1,2,⋯,γ} satisfying Ui ⨷W=V. We find upper and lower bounds for γ{k, n, F). In a few special cases we find exact values.

A quandle is a self-distributive algebraic structure that appears in quasi-group and knot theories. For each abelian group A and c \in A we define a quandle G(A, c) on \Z_3 \times A. These quandles are generalizations of a class of non-medial Latin quandles defined by V. M. Galkin so we call them Galkin quandles. Each G(A, c) is connected but not Latin unless A has odd order. G(A, c) is non-medial unless 3A = 0. We classify their isomorphism classes in terms of pointed abelian groups, and study their various properties. A family of symmetric connected quandles is constructed from Galkin quandles, and some aspects of knot colorings by Galkin quandles are also discussed.

The k × n grid graph is the product Pk × Pn of a path of length k − 1 and a path of length n − 1. We prove here formulas found by E. O. Hare for the domination numbers of P5 × Pn and P6 × Pn. © 1993 John Wiley & Sons, Inc.

A subset S of an abelian group G is said to be sum-free if whenever a, b ∈ S, then a + b ∉ S. A maximal sum-free (msf) set S in G is a sum-free set which is not properly contained in another sum-free subset of G. We consider only the case where G is the vector space (V(n) of dimension n over GF(2). We are concerned with the problem of determining all msf sets in V(n). It is well known that if S is a msf set then |S| ⩽ 2n − 1. We prove that there are no msf sets S in V(n) with 5 × 2n − 4 < |S| < 2n − 1. (This bound is sharp at both ends.) Further, we construct msf sets S in V(n), n ⩾ 4, with |S| = 2n − s + 2s + t − 3 × 2t for 0 ⩽ t ⩽ n − 4 and 2 ⩽ s ⩽ [(n − t)2]. These methods suffice to construct msf sets of all possible cardinalities for n ⩽ 6. We also present some of the results of our computer searches for msf sets in V(n). Up to equivalence we found all msf-sets for n ⩽ 6. For n > 6 our searches used random sampling and, in this case, we find many more msf sets than our present methods of construction can account for.

A collection Q of linearly independent w-suhicfs of the n-dimensional vector space V(n) over GF(2) is a w-quilt if whenever X and Y are distinct elements of Q, then X is disjoint from the linear span of Y. The main problem is to determine the maximum possibility cardinality of a w-quilt in V(n) for fixed w and n. Here a graph T(Q) is associated with each quilt Q. The connected components of T(Q) are shown to be complete graphs and the structure of the subquilts corresponding to these components is completely determined. By use of Ramsey type arguments these results are shown to lead to new upper bounds on the cardinality of a w-quilt in V(n) where n = w + 2, a case of particular interest.

A combinatorial view of deadlock (as in Dijkstra's self-stabilizing systems) is presented which leads to precise lower bounds on the complexity of programs. Specifically, we consider a directed ring of k individual processors, each having n states, identical programs, and asynchronous activity. Our main theorem establishes the minimum size (i.e., complexity) of a program for which no global state is in deadlock.

A 1- blocking set in the projective space PG( m ,2), m ⩾2, is a set B of points such that any ( m −1)-flat meets B and no 1-flat is contained in B . A binary linear code is said to be uneven if it contains at least one codeword of odd weight. If B is a 1-blocking set in PG( r −1,2) and dim〈 B 〉= r −1 any matrix H whose columns are the vectors in B is a parity check matrix for an uneven binary code of length n =| B |, redundancy r , and minimum distance at least 4; Conversely, if B is the set of columns of the parity check matrix of such a code then it is a 1-blocking set. Using this and results on uneven binary codes of minimum distance 4, the author shows that there exists a 1-blocking set of cardinality n if and only if 5⩽ n ⩽5⋯2 m −3 .

Let $\gamma (n,\delta)$ denote the largest possible domination number for a graph of order $n$ and minimum degree $\delta$. This paper is concerned with the behavior of the right side of the sequence $$\gamma (n,0) \ge \gamma (n,1) \ge \cdots \ge \gamma (n,n-1) = 1. $$ We set $ \delta _k(n) = \max \{ \delta \, \vert \, \gamma (n,\delta) \ge k \}$, $k \ge 1.$ Our main result is that for any fixed $k \ge 2$ there is a constant $c_k$ such that for sufficiently large $n$, $$ n-c_kn^{(k-1)/k} \le \delta _{k+1}(n) \le n - n^{(k-1)/k}. $$ The lower bound is obtained by use of circulant graphs. We also show that for $n$ sufficiently large relative to $k$, $\gamma (n,\delta _k(n)) = k$. The case $k=3$ is examined in further detail. The existence of circulant graphs with domination number greater than 2 is related to a kind of difference set in ${\bf Z}_n$.

Let $\gamma(G)$ denote the domination number of a graph $G$ and let $G\square H$ denote the Cartesian product of graphs $G$ and $H$. We prove that $\gamma(G)\gamma(H) \le 2 \gamma(G\square H)$ for all simple graphs $G$ and $H$.

Let $\gamma (n,\delta)$ denote the largest possible domination number for a graph of order $n$ and minimum degree $\delta$. This paper is concerned with the behavior of the right side of the sequence $$\gamma (n,0) \ge \gamma (n,1) \ge \cdots \ge \gamma (n,n-1) = 1. $$ We set $ \delta _k(n) = \max \{ \delta \, \vert \, \gamma (n,\delta) \ge k \}$, $k \ge 1.$ Our main result is that for any fixed $k \ge 2$ there is a constant $c_k$ such that for sufficiently large $n$, $$ n-c_kn^{(k-1)/k} \le \delta _{k+1}(n) \le n - n^{(k-1)/k}. $$ The lower bound is obtained by use of circulant graphs. We also show that for $n$ sufficiently large relative to $k$, $\gamma (n,\delta _k(n)) = k$. The case $k=3$ is examined in further detail. The existence of circulant graphs with domination number greater than 2 is related to a kind of difference set in ${\bf Z}_n$.

In this paper we discuss the weight distribution of prime cyclic arithmetic codes. This is equivalent to the following number-theoretic problem: A norm | | dependent on a positive integer r is defined on Z p = {0, 1, …, p − 1} as follows: Let 〈r〉 denote the subgroup of the group of non-zero elements of Z p generated by r. Let |x| be the number of elements of the coset 〈r〉 x which lie in the interval M (p,r)= p r+t + 1 p r+1 + 2, … rp r+1 where the bracket denotes the greatest integer function. In coding theory |x| is called the weight of x. We study the deviation Δ(p,r) of the weight of x in Z p from the average weight of the non-zero elements of Z p. Several bounds are found for Δ(p,r), and using elementary facts concerning quadratic residues some new conditions are found which imply that Δ(p,r) = 0.

A chain ring is an associative, commutative ring with an identity whose ideals form a chain. We associate with each finite chain ring five invariants (integers) and determine (in certain cases) the number of isomorphism classes of rings with given invariants. These results yield immediate corollaries for Pappian Hjelmslev planes which are coordinatized by such rings.

By asemigroup-theoretical property we mean a property of semigroups which is preserved by isomorphism. Such a property iscategorical if it can be expressed in the language of categories: roughly, without using elements. We show that this is always possible with the proviso that in the case of one-sided properties we cannot refer in categorical terms to a specific side. For example, the property of having aleft identity cannot be described categorically in the category of semigroups, since the functor ()op which takes a semigroup into its “opposite” semigroup is a category automorphism. We show that ()op is the only non-trivial automorphism of the category of semigroups (up to natural equivalence of functors). In other words, the “automorphism group” of the category of semigroups has order two.

Let ℒ V denote the algebra of all linear transformations on an n-dimensional vector space V over a field Φ. A subsemigroup S of the multiplicative semigroup of ℒ V will be said to be an affine semigroup over Φ if S is a linear variety, i.e., a translate of a linear subspace of ℒ V.This concept in a somewhat different form was introduced and studied by Haskell Cohen and H. S. Collins [1]. In an appendix we give their definition and outline a method of describing possibly infinite dimensional affine semigroups in terms of algebras and supplemented algebras.

This paper is motivated by the observation that the property of having a left identity cannot be described “categorically” in the category of rings, since the functor ( )OP which takes a ring into its opposite ring is a category automorphism but does not preserve this property. In general, a property of objects, morphisms, etc., in a category V can be characterized category- theoretically if and only if it is invariant under the automorphisms (self- equivalences) of ‘27. This makes it desirable to know whether ( )OP is the only nontrivial automorphism of the category of rings up to equivalence of func- tors. We shall show that the answer is yes, and that in the case of commutative rings, there are no nontrivial automorphisms. More generally, let

Let R be a ring. We denote by o the so-called circle composition on R, denned by a o b = a + b — ab for a, b ∊ R. It is well known that this composition is associative and that R is a radical ring in the sense of Jacobson (see 6) if and only if the semigroup (R, o) is a group. We shall say that R is a generalized radical ring if (R, o) is a union of groups. Such rings might equally appropriately be called generalized strongly regular rings, since every strongly regular ring satisfies this property (see Theorem A below). This definition was in fact partially motivated by the observation of Jiang Luh (7) that a ring is strongly regular if and only if its multiplicative semigroup is a union of groups.

Let A be a finite-dimensional (associative) algebra over an arbitrary field F. We shall say that a semi-group S is a translate of A if there exist an algebra B over F and an epimorphism ϕ: B → F such that A = 0→-1 and S = 1→-1. It is shown in (2) that any such semi-group S has a kernel (defined below) that is completely simple in the sense of Rees. Following Stefan Schwarz (4), we define the radical R(S) of S to be the union of all ideals I of S such that some power In of I lies in the kernel K of S. First we prove that the radical of a translate of A is a translate of the radical of A. It follows that A is nilpotent if and only if it has a translate S such that R (S) = S.

Here it is proved that a cyclic ( n, k ) code over GF( q ) is equidistant if and only if its parity check polynomial is irreducible and has exponent e = (q k − 1) a where a divides q − 1 and ( a , k ) = 1. The length n may be any multiple of e . The proof of this theorem also shows that if a cyclic ( n,k ) code over GF( q ) is not a repetition of a shorter code and the average weight of its nonzero code words is integral, then its parity check polynomial is irreducible over GF( q ) with exponent n = (q k − 1) a where a divides q − 1.