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Register a new userInvariants of Spatial Graphs
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Authors
Blake Mellor
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This is a short review article on invariants of spatial graphs, written for A Concise Encyclopedia of Knot Theory (ed. Adams et. al.). The emphasis is on combinatorial and polynomial invariants of spatial graphs, including the Alexander polynomial, the fundamental quandle of a graph, and the Yamada polynomial.
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We say that a set of pairs of disjoint cycles $Lambda(G)$ of a graph $G$ is linked if for any spatial embedding $f$ of $G$ there exists an element $lambda$ of $Lambda(G)$ such that the $2$-component link $f(lambda)$ is nonsplittable, and also say minimally linked if none of its proper subsets are linked. In this paper, we show that: (1) the set of all pairs of disjoint cycles of $G$ is minimally linked if and only if $G$ is essentially same as a graph in the Petersen family, and (2) for any two integers $p,qge 3$, we exhibit a minimally linked set of Hamiltonian $(p,q)$-pairs of cycles of the complete graph $K_{p+q}$ with at most eighteen elements.
A {em balanced} spatial graph has an integer weight on each edge, so that the directed sum of the weights at each vertex is zero. We describe the Alexander module and polynomial for balanced spatial graphs (originally due to Kinoshita cite{ki}), and examine their behavior under some common operations on the graph. We use the Alexander module to define the determinant and $p$-colorings of a balanced spatial graph, and provide examples. We show that the determinant of a spatial graph determines for which $p$ the graph is $p$-colorable, and that a $p$-coloring of a graph corresponds to a representation of the fundamental group of its complement into a metacyclic group $Gamma(p,m,k)$. We finish by proving some properties of the Alexander polynomial.
A neighborhood homotopy is an equivalence relation on spatial graphs which is generated by crossing changes on the same component and neighborhood equivalence. We give a complete classification of all 2-component spatial graphs up to neighborhood homotopy by the elementary divisor of a linking matrix with respect to the first homology group of each of the connected components. This also leads a kind of homotopy classification of 2-component handlebody-links.
We study finite type invariants of nullhomologous knots in a closed 3-manifold $M$ defined in terms of certain descending filtration ${mathscr{K}_n(M)}_{ngeq 0}$ of the vector space $mathscr{K}(M)$ spanned by isotopy classes of nullhomologous knots in $M$. The filtration ${mathscr{K}_n(M)}_{n geq 0}$ is defined by surgeries on special kinds of claspers in $M$ having one special leaf. More precisely, when $M$ is fibered over $S^1$ and $H_1(M)=mathbb{Z}$, we study how far the natural surgery map from the space of $mathbb{Q}[t^{pm 1}]$-colored Jacobi diagrams on $S^1$ of degree $n$ to the graded quotient $mathscr{K}_n(M)/mathscr{K}_{n+1}(M)$ can be injective for $nleq 2$. To do this, we construct a finite type invariant of nullhomologous knots in $M$ up to degree 2 that is an analogue of the invariant given in our previous paper arXiv:1503.08735, which is based on Lescops construction of $mathbb{Z}$-equivariant perturbative invariant of 3-manifolds.
C. Giller proposed an invariant of ribbon 2-knots in S^4 based on a type of skein relation for a projection to R^3. In certain cases, this invariant is equal to the Alexander polynomial for the 2-knot. Gillers invariant is, however, a symmetric polynomial -- which the Alexander polynomial of a 2-knot need not be. After modifying a 2-knot into a Montesinos twin in a natural way, we show that Gillers invariant is related to the Seiberg-Witten invariant of the exterior of the twin, glued to the complement of a fiber in E(2).
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