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The present paper is an introduction to a combinatorial theory arising as a natural generalization of classical and virtual knot theory. There is a way to encode links by a class of "realizable" graphs. When passing to generic graphs with the same equivalence relations we get "graph-links". On one hand graph-links generalize the notion of virtual link, on the other hand they do not detect link mutations. We define the Jones polynomial for graph-links and prove its invariance. We also prove some a generalization of the Kauffman–Murasugi–Thistlethwaite theorem on "minimal diagrams" for graph-links. [ABSTRACT FROM AUTHOR]

This paper records some observations concerning geodesic growth functions. If a nilpotent group is not virtually cyclic then it has exponential geodesic growth with respect to all finite generating sets. On the other hand, if a finitely generated group G has an element whose normal closure is abelian and of finite index, then G has a finite generating set with respect to which the geodesic growth is polynomial (this includes all virtually cyclic groups). [ABSTRACT FROM AUTHOR]

For a ring R, we denote by R[C] the free R-module spanned by the isotopy classes of singular links in S3. Given two invertible elements x,t ∊ R, the HOMFLY-PT skein module of singular links in S3 (relative to the triple (R,t,x)) is the quotient of R[C] by local relations, called skein relations, that involve t and x. We compute the HOMFLY-PT skein module of singular links for any R such that (f-1 --t + x ) and (t-1 -t-x) are invertible. In particular, we deduce the Conway skein module of singular links. [ABSTRACT FROM AUTHOR]

We introduce the odd Jones-Kauffman polynomial and odd Miyazawa polynomials of virtual link diagrams by using the parity of virtual link diagrams given in [Y. H. Im and K. I. Park, A parity and a multi-variable polynomial invariant for virtual links, J. Knot Theory Ramifications 22(13) (2013), Article ID: 1350073, 18pp.], which are different from the original Jones-Kauffman and Miyazawa polynomials. Also, we give a family of parities and odd polynomials for virtual knots so that many virtual knots can be distinguished. [ABSTRACT FROM AUTHOR]

Given a finite (associative, unital) ring R, let K(R) denote the set of polynomials in R[x] that send each element of R to 0 under evaluation. We study K(R) and its elements. We conjecture that K(R) is a two-sided ideal of R[x] for any finite ring R, and prove the conjecture for several classes of finite rings (including commutative rings, semisimple rings, local rings, and all finite rings of odd order). We also examine a connection to sets of integer-valued polynomials. [ABSTRACT FROM AUTHOR]

We study rack polynomials and the link invariants they define. We show that constant action racks are classified by their generalized rack polynomials and show that nsata-quandles are not classified by their generalized quandle polynomials. We use subrack polynomials to define enhanced rack counting invariants, generalizing the quandle polynomial invariants. [ABSTRACT FROM AUTHOR]

Let sq(n) denote the sum of the digits in the q-ary expansion of an integer n. In 2005, Melfi examined the structure of n such that s2(n) = s2(n2). We extend this study to the more general case of generic q and polynomials p(n), and obtain, in particular, a refinement of Melfi's result. We also give a more detailed analysis of the special case p(n) = n2, looking at the subsets of n where sq(n) = sq(n2) = k for fixed k. [ABSTRACT FROM AUTHOR]

Let K be a number field of degree n over ℚ, Â be the set of integers of K that are primitive over ℚ and let I(K) be its index. The prime factors of I(K) are called common factors of indices or inessential discriminant divisors. We show that these primes divide another index i(K) previously defined by Gunji and McQuillan as i(K) = lcmθ∈Âi(θ), where i(θ) = gcdx∈ℤFθ(x) and Fθ(x) is the characteristic polynomial of θ over ℚ. It is shown that there exists θ ∈ Â such that i(K) = i(θ) and an algorithm is given for the computation of such an integer. For any prime p|i(K), an integer ρK(p) defined as the number of $\bar{\theta} \in A{/}pA$ such that p|i(θ) is investigated. It is shown that this integer determines in some cases the splitting type of p in K. Some open questions related to I(K), i(K) and ρK(p) are stated. [ABSTRACT FROM AUTHOR]

We establish the volume conjecture for (m, 2)-cables of the figure 8 knot, when m is odd. For (m, 2)-cables of general knots where m is even, we show that the limit in the volume conjecture depends on the parity of the color (of the Kashaev invariant). There are many cases when the volume conjecture for cables of the figure 8 knot is false if one considers all the colors, but holds true if one restricts the colors to a subset of the set of positive integers. [ABSTRACT FROM AUTHOR]