We present a model of how DNA knots and links are formed as a result of a single recombination ev... more We present a model of how DNA knots and links are formed as a result of a single recombination event, or multiple rounds of (processive) recombination events, starting with an unknotted, unlinked, or a (2,m)-torus knot or link substrate. Given these substrates, according to our model all DNA products of a single recombination event or processive recombination fall into a
Site-specific recombination on supercoiled circular DNA yields a variety of knotted or catenated ... more Site-specific recombination on supercoiled circular DNA yields a variety of knotted or catenated products. Here, we present a topological model of this process and characterize all possible products of the most common substrates: unknots, unlinks, and torus knots and catenanes. This model tightly prescribes the knot or catenane type of previously uncharacterized data. We also discuss how the model helps
We present the concept of the topological symmetry group as a way to analyze the symmetries of no... more We present the concept of the topological symmetry group as a way to analyze the symmetries of non-rigid molecules. Then we characterize all of the groups which can occur as the topological symmetry group of an embedding of the complete graph K_{4r+3} in S^3.
Proceedings of the American Mathematical Society, Jun 11, 2009
In contrast with knots, whose properties depend only on their extrinsic topology in $S^3$, there ... more In contrast with knots, whose properties depend only on their extrinsic topology in $S^3$, there is a rich interplay between the intrinsic structure of a graph and the extrinsic topology of all embeddings of the graph in $S^3$ . For example, it was shown in [2] that every embedding of the complete graph $K_7$ in $S^3$ contains a non-trivial knot. Later in it was shown that for every $m \in N$, there is a complete graph $K_n$ such that every embedding of $K_n$ in $S_3$ contains a knot $Q$ whose minimal crossing number is at least $m$. Thus there are arbitrarily complicated knots in every embedding of a sufficiently large complete graph in $S^3$. We prove here the contrasting result that every graph has an embedding in $S^3$ such that every non-trivial knot in that embedding is hyperbolic. Our theorem implies that every graph has an embedding in $S^3$ which contains no composite or satellite knots.
We develop a topological model of site-specific recombination that applies to substrates which ar... more We develop a topological model of site-specific recombination that applies to substrates which are the connected sum of two torus links of the form $T(2,n)\#T(2,m)$. Then we use our model to prove that all knots and links that can be produced by site-specific recombination on such substrates are contained in one of two families, which we illustrate.
We show that the edges of every 3-connected planar graph except $K_4$ can be colored with two col... more We show that the edges of every 3-connected planar graph except $K_4$ can be colored with two colors so that every embedding of the graph in $S^3$ is asymmetric, and we characterize all planar graphs whose edges can be 2-colored so that every embedding of the graph in $S^2$ is asymmetric.
The main result of this paper is that for every closed, connected, orientable, irreducible 3-mani... more The main result of this paper is that for every closed, connected, orientable, irreducible 3-manifold $M$, there is an integer $ n_M$ such that any abstract graph with no automorphism of order 2 which has a 3-connected minor whose genus is more than $n_M$ has no achiral embedding in $M$. By contrast, the paper also proves that for every graph $\gamma$, there are infinitely many closed, connected, orientable, irreducible 3-manifolds $M$ such that some embedding of $\gamma$ in $M$ is pointwise fixed by an orientation reversing involution of $M$.
In this paper we complete the classification of topological symmetry groups for complete graphs $... more In this paper we complete the classification of topological symmetry groups for complete graphs $K_n$ by characterizing which $K_n$ can have a cyclic group, a dihedral group, or a subgroup of $D_m \times D_m$ where $m$ is odd, as its topological symmetry group.
We present a model of how DNA knots and links are formed as a result of a single recombination ev... more We present a model of how DNA knots and links are formed as a result of a single recombination event, or multiple rounds of (processive) recombination events, starting with an unknotted, unlinked, or a (2,m)-torus knot or link substrate. Given these substrates, according to our model all DNA products of a single recombination event or processive recombination fall into a
Site-specific recombination on supercoiled circular DNA yields a variety of knotted or catenated ... more Site-specific recombination on supercoiled circular DNA yields a variety of knotted or catenated products. Here, we present a topological model of this process and characterize all possible products of the most common substrates: unknots, unlinks, and torus knots and catenanes. This model tightly prescribes the knot or catenane type of previously uncharacterized data. We also discuss how the model helps
We present the concept of the topological symmetry group as a way to analyze the symmetries of no... more We present the concept of the topological symmetry group as a way to analyze the symmetries of non-rigid molecules. Then we characterize all of the groups which can occur as the topological symmetry group of an embedding of the complete graph K_{4r+3} in S^3.
Proceedings of the American Mathematical Society, Jun 11, 2009
In contrast with knots, whose properties depend only on their extrinsic topology in $S^3$, there ... more In contrast with knots, whose properties depend only on their extrinsic topology in $S^3$, there is a rich interplay between the intrinsic structure of a graph and the extrinsic topology of all embeddings of the graph in $S^3$ . For example, it was shown in [2] that every embedding of the complete graph $K_7$ in $S^3$ contains a non-trivial knot. Later in it was shown that for every $m \in N$, there is a complete graph $K_n$ such that every embedding of $K_n$ in $S_3$ contains a knot $Q$ whose minimal crossing number is at least $m$. Thus there are arbitrarily complicated knots in every embedding of a sufficiently large complete graph in $S^3$. We prove here the contrasting result that every graph has an embedding in $S^3$ such that every non-trivial knot in that embedding is hyperbolic. Our theorem implies that every graph has an embedding in $S^3$ which contains no composite or satellite knots.
We develop a topological model of site-specific recombination that applies to substrates which ar... more We develop a topological model of site-specific recombination that applies to substrates which are the connected sum of two torus links of the form $T(2,n)\#T(2,m)$. Then we use our model to prove that all knots and links that can be produced by site-specific recombination on such substrates are contained in one of two families, which we illustrate.
We show that the edges of every 3-connected planar graph except $K_4$ can be colored with two col... more We show that the edges of every 3-connected planar graph except $K_4$ can be colored with two colors so that every embedding of the graph in $S^3$ is asymmetric, and we characterize all planar graphs whose edges can be 2-colored so that every embedding of the graph in $S^2$ is asymmetric.
The main result of this paper is that for every closed, connected, orientable, irreducible 3-mani... more The main result of this paper is that for every closed, connected, orientable, irreducible 3-manifold $M$, there is an integer $ n_M$ such that any abstract graph with no automorphism of order 2 which has a 3-connected minor whose genus is more than $n_M$ has no achiral embedding in $M$. By contrast, the paper also proves that for every graph $\gamma$, there are infinitely many closed, connected, orientable, irreducible 3-manifolds $M$ such that some embedding of $\gamma$ in $M$ is pointwise fixed by an orientation reversing involution of $M$.
In this paper we complete the classification of topological symmetry groups for complete graphs $... more In this paper we complete the classification of topological symmetry groups for complete graphs $K_n$ by characterizing which $K_n$ can have a cyclic group, a dihedral group, or a subgroup of $D_m \times D_m$ where $m$ is odd, as its topological symmetry group.
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Papers by Erica Flapan