Narrow Results

Levine introduced clover links to investigate the indeterminacy of Milnor invariants of links. He proved that for a clover link, Milnor numbers of length up to $2k+1$ are well-defined if those of length $\le k$ vanish, and that Milnor numbers of length at least $2k+2$ are not well-defined if those of length $k+1$ survive. For a clover link $c$ with vanishing Milnor numbers of length $\le k$, we show that the Milnor number ${\mu }_c(I)$ for a sequence $I$ is well-defined by taking modulo the greatest common divisor of the ${\mu }_c{(J)}^{\prime }s$, where $J$ is any proper subsequence of $I$ obtained by removing at least $k+1$ indices. Moreover, if $I$ is a non-repeated sequence of length $2k+2$, the possible range of ${\mu }_c(I)$ is given explicitly. As an application, we give an edge-homotopy classification of $4$-clover links.

We introduce Niebrzydowski algebras, algebraic structures with a ternary operation and a partially defined multiplication, with axioms motivated by the Reidemeister moves for $Y$-oriented trivalent spatial graphs and handlebody-links. As part of this definition, we identify generating sets of $Y$-oriented Reidemeister moves. We give some examples to demonstrate that the counting invariant can distinguish some $Y$-oriented trivalent spatial graphs and handlebody-links.

What is the smallest positive integer m=m(L) such that every linear spatial representation of the complete graph with n vertices, n≥m contain cycles isotopic to link L? In this paper, we show that . The proof uses the well-known cyclic polytope and its combinatorial description in terms of oriented matroids.

For every natural number n, we exhibit a graph with the property that every embedding of it in ℝ³ contains a non-split n-component link. Furthermore, we prove that our graph is minor minimal in the sense that every minor of it has an embedding in ℝ³ that contains no non-split n-component link.

A generic map from a finite graph to the 2-space is called identifiable if any two embeddings of the graph into the 3-space obtained by lifting the map with respect to the natural projection from the 3-space to the 2-space are ambient isotopic in the 3-space. We show that only planar graphs have identifiable maps. We characterize the identifiable maps for some planar graphs.

We describe a family of graphs that contain a knot or a non-split 3-component link in every spatial embedding. We exhibit a graph in this family that has a knotless embedding, and a 3-component linkless embedding.

We prove that every embedding of K_2n+1,2n+1 into ℝ³ contains a non-split link of n components. Further, given an embedding of K_2n+1,2n+1 in ℝ³, every edge of K_2n+1,2n+1 is contained in a non-split n-component link in K_2n+1,2n+1.

It is well-known that the "all right" metric structure on the Dehn complex of a knot is non-positively curved, or CAT(0), if and only if the knot projection is prime and alternating. We extend these ideas to spatial graphs and introduce the idea of ψ-alternating graphs as the appropriate analog of alternating knots.

In this paper, colorings by symmetric quandles for spatial graphs and handlebody-links are introduced. We also introduce colorings by LH-quandles for LH-links. LH-links are handlebody-links, some of whose circle components are specified, which are related to Heegaard splittings of link exteriors. We also discuss quandle (co)homology groups and cocycle invariants.

Motivated by proposed entangled molecular structures known as ravels, we introduce a method for constructing such entanglements from 2-string tangles. We then show that for most (but not all) arborescent tangles this construction yields either a planar θ₄ graph or contains a knot.

In this paper, we compute the graph skein algebra of the punctured disk with two holes. Then, we apply the graph skein techniques developed here to establish necessary conditions for a spatial graph to have a symmetry of order p, where p is a prime. The obstruction criteria introduced here extend some results obtained earlier for symmetric spatial graphs.

We show that deleting an edge of a 3-cycle in an intrinsically knotted graph gives an intrinsically linked graph.

A 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 S. Kinoshita, Alexander polynomials as isotopy invariants I, Osaka Math. J.10 (1958) 263–271.), 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 $\mathrm{\Gamma }(p,m,k)$. We finish by proving some properties of the Alexander polynomial.

This paper introduces a number of new intrinsically 3-linked graphs through five new constructions. We prove that intrinsic 3-linkedness is not preserved by $\text{Y}\nabla$ moves. We will see that the graph $M$, which is obtained through a $\text{Y}\nabla$ move on ${(PG)}_{\ast }^{\ast }(PG)$, is not intrinsically 3-linked.

We extend the concepts of trivializing and knotting numbers for knots to spatial graphs and 2-bouquet graphs, in particular. Furthermore, we calculate the trivializing and knotting numbers for projections and pseudodiagrams of 2-bouquet spatial graphs based on the number of precrossings and the placement of the precrossings in the pseudodiagram of the spatial graph.

A book representation of a graph is a particular way of embedding a graph in three-dimensional space so that the vertices lie on a circle and the edges are chords on disjoint topological disks. We describe a set of operations on book representations that preserves ambient isotopy, and apply these operations to $K_6$, the complete graph with six vertices. We prove there are exactly 59 distinct book representations for $K_6$, and we identify the number and type of knotted and linked cycles in each representation. We show that book representations of $K_6$ contain between one and seven links, and up to nine knotted cycles. Furthermore, all links and cycles in a book representation of $K_6$ have crossing number at most four.

In order to apply quantum topology methods to nonplanar graphs, we define a planar diagram category that describes the local topology of embeddings of graphs into surfaces. These virtual graphs are a categorical interpretation of ribbon graphs. We describe an extension of the flow polynomial to virtual graphs, the $S$-polynomial, and formulate the $𝔰𝔩(N)$ Penrose polynomial for non-cubic graphs, giving contraction–deletion relations. The $S$-polynomial is used to define an extension of the Yamada polynomial to virtual spatial graphs, and with it we obtain a sufficient condition for non-classicality of virtual spatial graphs. We conjecture the existence of local relations for the $S$-polynomial at squares of integers.

In 2003, Ozsváth and Szabó defined the concordance invariant $\tau$ for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of $\tau$ for knots in $S^3$ and a combinatorial proof that $\tau$ gives a lower bound for the slice genus of a knot. Recently, Harvey and O’Donnol defined a relatively bigraded combinatorial Heegaard Floer homology theory for transverse spatial graphs in $S^3$, extending HFK for knots. We define a $\mathbb{Z}$-filtered chain complex for balanced spatial graphs whose associated graded chain complex has homology determined by Harvey and O’Donnol’s graph Floer homology. We use this to show that there is a well-defined $\tau$ invariant for balanced spatial graphs generalizing the $\tau$ knot concordance invariant. In particular, this defines a $\tau$ invariant for links in $S^3$. Using techniques similar to those of Sarkar, we show that our $\tau$ invariant is an obstruction to a link being slice.

We present four models for a random graph and show that, in each case, the probability that a graph is intrinsically knotted goes to one as the number of vertices increases. We also argue that, for $n\ge 18$, most graphs of order $n$ are intrinsically knotted and, for $n\ge 2m+9$, most of order $n$ are not $m$-apex. We observe that $p(n)=1/n$ is the threshold for intrinsic knotting and linking in Gilbert’s model.

We characterize all groups which can occur as the topological symmetry group or the orientation preserving topological symmetry group of some embedding of the Petersen graph in $S^3$.