Related papers: Invariants of Spatial Graphs
The fundamental quandle is a powerful invariant of knots, links and spatial graphs, but it is often difficult to determine whether two quandles are isomorphic. One approach is to look at quotients of the quandle, such as the $n$-quandle…
We investigate the property of a spatial graph of having a leveled embedding and characterize the abstract graphs with this property. We show that all leveled embeddings are free and we compare leveled and paneled (also known as flat)…
Link homotopy has been an active area of research for knot theorists since its introduction by Milnor in the 1950s. We introduce a new equivalence relation on spatial graphs called component homotopy, which reduces to link homotopy in the…
This is a survey article for the forthcoming `A Concise Encyclopedia of Knot Theory.' We focus on the topology of spatial graphs with few vertices and edges, paying particular attention to Brunnian $\theta$-graphs.
This paper is a self-contained development of an invariant of graphs embedded in three-dimensional Euclidean space using the Jones polynomial and skein theory. Some examples of the invariant are computed. An unlinked embedded graph is one…
We raise some questions about graph polynomials, highlighting concepts and phenomena that may merit consideration in the development of a general theory. Our questions are mainly of three types: When do graph polynomials have reduction…
The leading coefficient of the Alexander polynomial of a knot is the most informative element in this invariant, and the growth of orders of the first homology of cyclic branched covering spaces is also a familiar subject. Accordingly,…
We introduce an Alexander polynomial for MOY graphs. For a framed trivalent MOY graph $\mathbb{G}$, we refine the construction and obtain a framed ambient isotopy invariant $\Delta_{(\mathbb{G},c)}(t)$. The invariant $\Delta_{(\mathbb{G},…
We describe a multivariable polynomial invariant for certain class of non isolated hypersurface singularities generalizing the characteristic polynomial on monodromy. The starting point is an extension of a theorem due to Le Dung Trang and…
The Alexander polynomial of a knot has been generalized in three different ways to give twisted invariants. The resulting invariants are usually referred to as twisted Alexander polynomials, higher-order Alexander polynomials and…
Two natural generalizations of knot theory are the study of spatially embedded graphs, and Kauffman's theory of virtual knots. In this paper we combine these approaches to begin the study of virtual spatial graphs.
Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs which are generalizations of Milnor's link-homotopy. We introduce some edge (resp. vertex)-homotopy invariants of spatial graphs by applying the Sato-Levine…
We give a new construction of the one-variable Alexander polynomial of an oriented knot or link, and show that it generalizes to a vector valued invariant of oriented tangles.
In the present paper, we introduce the concept of universal graph series. We then present four invariants of graphs and discuss some of their properties. In particular, one of these invariants is a generalization of the chromatic symmetric…
Two natural generalizations of knot theory are the study of spatial graphs and virtual knots. Our goal is to unify these two approaches into the study of virtual spatial graphs. This paper is a survey, and does not contain any new results.…
The notion of a pseudoknot is defined as an equivalence class of knot diagrams that may be missing some crossing information. We provide here a topological invariant schema for pseudoknots and their relatives, 4-valent rigid vertex spatial…
We generalize the notion of the quandle polynomial to the case of singquandles. We show that the singquandle polynomial is an invariant of finite singquandles. We also construct a singular link invariant from the singquandle polynomial and…
The normalized Yamada polynomial is a polynomial invariant in variable A for theta-curves. In this work, we show that the coefficients of the power series obtained from this polynomial by the substitution A=e^x=1+x+x^2/2+x^3/6+... are…
Polynomials on stranded graphs are higher dimensional generalization of Tutte and Bollob\'as-Riordan polynomials [Math. Ann. 323 (2002), 81-96]. Here, we deepen the analysis of the polynomial invariant defined on rank 3 weakly-colored…
We strengthen and put in a broader perspective previous results of the first two authors on colliding permutations. The key to the present approach is a new non-asymptotic invariant for graphs.