- Recent Developments in Spatial Graph Theory with Erica Flapan, Thomas Mattman, Ramin Naimi and Ryo Nikkuni (in
*Knots, Links, Spatial Graphs and Algebraic Invariants*, ed. E. Flapan et. al.,*Contemporary Mathematics*, vol. 689, 2017, pp. 81-102)

**Abstract:**This article presents a survey of some recent results in the theory of spatial graphs. In particular, we highlight results related to intrinsic knotting and linking and results about symmetries of spatial graphs. In both cases we consider spatial graphs in $S^3$ as well as in other 3-manifolds. - The Alexander polynomial for virtual twist knots with Isaac Benioff (
*Journal of Knot Theory and its Ramifications*, vol. 26, no. 1, 2017)

**Abstract:**We define a family of virtual knots generalizing the classical twist knots. We develop a recursive formula for the Alexander polynomial $\Delta_0$ (as defined by Silver and Williams \cite{sw}) of these virtual twist knots. These results are applied to provide evidence for a conjecture that the odd writhe of a virtual knot can be obtained from $\Delta_0$. - Alexander and writhe polynomials for virtual knots (
*Journal of Knot Theory and its Ramifications*, vol. 25, no. 8, 2016)

**Abstract:**We give a new interpretation of the Alexander polynomial $\Delta_0$ for virtual knots due to Sawollek and Silver and Williams, and use it to show that, for any virtual knot, $\Delta_0$ determines the writhe polynomial of Cheng and Gao (equivalently, Kauffman's affine index polynomial). We also use it to define a second-order writhe polynomial, and give some applications. - Colorings, determinants and Alexander polynomials for spatial graphs with Terry Kong, Alec Lewald and Vadim Pigrish (
*Journal of Knot Theory and its Ramifications*, vol. 25, no. 4, 2016)

**Abstract:**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. - Topological symmetry groups of complete bipartite graphs with Kathleen Hake and Matthew Pittluck (
*Tokyo Journal of Mathematics*, vol. 39, 2016, pp. 133-156)

**Abstract:**The symmetries of complex molecular structures can be modeled by the*topological symmetry group*of the underlying embedded graph. It is therefore important to understand which topological symmetry groups can be realized by particular abstract graphs. This question has been answered for complete graphs; it is natural next to consider complete bipartite graphs. In previous work we classified the complete bipartite graphs that can realize topological symmetry groups isomorphic to $A_4$, $S_4$ or $A_5$; in this paper we determine which complete bipartite graphs have an embedding in $S^3$ whose topological symmetry group is isomorphic to $\mathbb{Z}_m$, $D_m$, $\mathbb{Z}_r \times \mathbb{Z}_s$ or $(\mathbb{Z}_r \times \mathbb{Z}_s) \ltimes \mathbb{Z}_2$. - The forbidden number of a knot with Alissa Crans and Sandy Ganzell (
*Kyungpook Mathematical Journal*, vol. 55, 2015, pp. 485-506)

**Abstract:**Every classical or virtual knot is equivalent to the unknot via a sequence of extended Reidemeister moves and the so-called forbidden moves. The minimum number of forbidden moves necessary to unknot a given knot is a new invariant we call the*forbidden number*. We relate the forbidden number to several known invariants, and calculate bounds for some classes of virtual knots. - "The Mathematics of Symmetry and Attitudes towards Mathematics," in Doing the Scholarship of Teaching and Learning in Mathematics, ed. C. Bennett and J. Dewar, MAA Notes no. 83, 2015, pp. 157-169

- Complete bipartite graphs whose topological symmetry groups are polyhedral (
*Tokyo Journal of Mathematics*, vol. 37, 2014, pp. 135-158)

**Abstract:**We determine for which $n$, the complete graph $K_{n,n}$ has an embedding in $S^3$ whose topological symmetry group is isomorphic to one of the polyhedral groups: $A_4$, $A_5$, or $S_4$. - Symmetries of embedded complete bipartite graphs with Erica Flapan, Nicole Lehle, Matt Pittluck and Xan Vongsathorn (
*Fundamenta Mathematicae*, vol. 226, 2014, pp. 1-16)

**Abstract:**We characterize which automorphisms of an arbitrary complete bipartite graph $K_{n,m}$ can be induced by a homeomorphism of some embedding of the graph in $S^3$. - Classification of topological symmetry groups of $K_n$ with Erica Flapan, Ramin Naimi and Michael Yoshizawa (
*Topology Proceedings*, vol. 43, 2014, pp. 209-233)

**Abstract:**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. - Counting Links and Knots in Complete Graphs
with Loren Abrams and Lowell Trott (
*Tokyo Journal of Mathematics*, vol. 36, no. 2, 2013, pp. 429-458)

**Abstract:**We investigate the minimal number of links and knots in complete partite graphs. We provide exact values or bounds on the minimal number of links for all complete partite graphs with all but 4 vertices in one partition, or with 9 vertices in total. In particular, we find that the minimal number of links for $K_{4,4,1}$ is 74. We also provide exact values or bounds on the minimal number of knots for all complete partite graphs with 8 vertices.*Gordian*, which is available on my*Gordian*page. - Using tiling theory to generate angle weaves with beads
with Gwen Fisher (
*Journal of Mathematics and the Arts*, vol. 6, no. 4, 2012, pp. 141-158; available online at http://journalsonline.tandf.co.uk/)

**Abstract:**Tilings of the plane, especially periodic tilings, can be used as the basis for flat bead weaving patterns called angle weaves. We describe specific ways to create intricate and beautiful angle weaves from periodic tilings, by placing beads on or near the vertices or edges of a tiling and weaving them together with thread. We also introduce the notion of star tilings and their associated angle weaves. We organize the angle weaves that we create into several classes, and explore some of the relationships among them. We then use the results to design graphic illustrations of many layered patterns. Finally, we prove that every normal tiling induces an angle weave, providing many opportunities for further exploration. - Spatial Graphs with Local Knots
with Erica Flapan and Ramin Naimi (
*Revista Matemática Complutense*, vol. 25, no. 2, 2012, pp. 493-510, DOI: 10.1007/s13163-011-0072-9)

**Abstract:**It is shown that for any locally knotted edge of a 3-connected graph in $S^3$, there is a ball that contains all of the local knots of that edge and is unique up to an isotopy setwise fixing the graph. This result is applied to the study of topological symmetry groups of graphs embedded in $S^3$. - "Student Surveys: What Do They Think?" with H. Zullo, K. Cline, et. al., in
*Teaching Mathematics with Classroom Voting: With and Without Clickers*, ed. K. Cline and H. Zullo, Mathematical Association of America, 2011

- Complete graphs whose topological symmetry groups are polyhedral
with Erica Flapan and Ramin Naimi (
*Algebraic and Geometric Topology*, vol. 11, 2011, pp. 1405-1433)

**Abstract:**We determine for which $m$, the complete graph $K_m$ has an embedding in $S^3$ whose topological symmetry group is isomorphic to one of the polyhedral groups: $A_4$, $A_5$, or $S_4$. - Drawing a triangle on the Thurston Model of Hyperbolic Space
with Curtis Bennett and Patrick Shanahan (
*Mathematics Magazine*, vol. 83, 2010, pp. 83-99)

**Abstract:**Experiments with a common physical model of the hyperbolic plane presented the authors with surprising difficulties in drawing a large triangle. Understanding these difficulties led to an intriguing exploration of the geometry of the Thurston model of the hyperbolic plane. In this exploration we encountered topics ranging from combinatorics and Pick’s Theorem to differential geometry and the Gauss-Bonnet Theorem. - Counting Links in Complete Graphs
with Thomas Fleming (
*Osaka Journal of Mathematics*, vol. 46, 2009, pp. 1-29)

**Abstract:**We find the minimal number of links in an embedding of any complete*k*-partite graph on 7 vertices (including $K_7$, which has at least 21 links). We give either exact values or upper and lower bounds for the minimal number of links for all complete*k*-partite graphs on 8 vertices. We also look at larger complete bipartite graphs, and state a conjecture relating minimal linking embeddings with minimal book embeddings. - Intrinsic linking and knotting are arbitrarily complex
with Erica Flapan and Ramin Naimi (
*Fundamenta Mathematicae*, vol. 201, no. 2, 2008, pp. 131-148)

**Abstract:**We show that, given any $n$ and $\alpha$, every embedding of any sufficiently large complete graph in $\mathbb{R}^3$ contains an oriented link with components $Q_1$, ..., $Q_n$ such that for every $i\not =j$, $|\lk(Q_i,Q_j)|\geq\alpha$ and $|a_2(Q_i)|\geq\alpha$, where $a_{2}(Q_i)$ denotes the second coefficient of the Conway polynomial of $Q_i$. - Tree Diagrams for String Links II: Determining Chord Diagrams (
*Journal of Knot Theory and its Ramifications*, vol. 17, no. 6, 2008, pp. 649-664)

**Abstract:**In previous work, we defined the intersection graph of a chord diagram associated with a string link (as in the theory of finite type invariants). In this paper, we look at the case when this graph is a tree, and we show that in many cases these trees determine the chord diagram (modulo the usual 1-term and 4-term relations). - Weight Systems for Milnor Invariants
(
*Journal of Knot Theory and its Ramifications*, vol. 17, no. 2, 2008, pp. 213-230)

**Abstract:**We use Polyak's skein relation to give a new proof that Milnor's string link homotopy invariants are finite type invariants, and to develop a recursive relation for their associated weight systems. We show that the obstruction to the triviality of these weight systems is the presence of a certain kind of spanning tree in the intersection graph of a chord diagram. - Virtual Spatial Graphs
with Thomas Fleming (
*Kobe Journal of Mathematics*, vol. 24, no. 2, 2007, pp. 67-85)

**Abstract:**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. - Three dimensional finite point groups and the symmetries of beaded beads
with Gwen Fisher (
*Journal of Mathematics and the Arts*, vol. 1, no. 2, 2007, pp. 85-96; available online at http://journalsonline.tandf.co.uk/)

**Abstract:**Beaded beads are clusters of beads woven together (usually around one or more large holes). Their groups of symmetries are classified by the three-dimensional finite point groups, i.e. the finite subgroups of the orthogonal group of degree three, O(3). The question we answer is whether every finite subgroup of O(3) can be realized as the group of symmetries of a beaded bead. We show that this is possible, and we describe general weaving techniques we used to accomplish this feat, as well as examples of a beaded bead realizing each finite subgroup of O(3) or, in the case of the seven infinite classes of finite subgroups, at least one representative beaded bead for each class. - An Introduction to Virtual Spatial Graph Theory
*(Survey Paper)*with Thomas Fleming (in*Proceedings of the International Workshop on Knot Theory for Scientific Objects*, OCAMI Studies, Vol. 1 (A. Kawauchi, editor), Osaka Municipal Universities Press, 2007)

**Abstract:**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. We state the definitions, provide some examples, and survey the known results. We hope that this paper will help lead to rapid development of the area. - Intrinsic Linking and Knotting in Virtual Spatial Graphs
with Thomas Fleming (
*Algebraic and Geometric Topology*, vol. 7, 2007, pp. 583-601)

**Abstract:**We introduce a notion of intrinsic linking and knotting for virtual spatial graphs. Our theory gives two filtrations of the set of all graphs, allowing us to measure, in a sense, how intrinsically linked or knotted a graph is; we show that these filtrations are descending and non-terminating. We also provide several examples of intrinsically virtually linked and knotted graphs. As a byproduct, we introduce the {\it virtual unknotting number} of a knot, and show that any knot with non-trivial Jones polynomial has virtual unknotting number at least 2. - Tree Diagrams for String Links
(
*Journal of Knot Theory and its Ramifications*, vol. 15, no. 10, 2006, pp. 1303-1318)

**Abstract:**In previous work, the author defined the intersection graph of a chord diagram associated with a string link (as in the theory of finite type invariants). In this paper, we classify the trees which can be obtained as intersection graphs of string link diagrams. - Intrinsic linking and knotting of graphs in arbitrary 3-manifolds
with Erica Flapan, Hugh Howards and Don Lawrence (
*Algebraic and Geometric Topology*, vol. 6, 2006, pp. 1025-1035)

**Abstract:**We prove that a graph is intrinsically linked in an arbitrary 3-manifold*M*if and only if it is intrinsically linked in*S*^{3}. Also, assuming the Poincare Conjecture, a graph is intrinsically knotted in*M*if and only if it is intrinsically knotted in*S*^{3}. - Intersection Graphs for String Links
(
*Journal of Knot Theory and its Ramifications*, vol. 15, no. 1, 2006, pp. 53-72)

**Abstract:**We extend the notion of intersection graphs for knots in the theory of finite type invariants to string links. We use our definition to develop weight systems for string links via the adjacency matrix of the intersection graphs, and show that these weight systems are related to the weight systems induced by the Conway and Homfly polynomials. - On the Topology of Celtic Knot Designs with Gwen Fisher (
*Proceedings of the 7th Annual International Conference of Bridges*, 2004, published in*Visual Mathematics*, vol. 7, no. 1, 2005)

**Abstract:**We derive formulas for counting the number of strands in a variety of knotwork designs inspired by traditional Celtic designs, including rectangular panels, circular borders, rectangular borders, and half frames. - A Geometric Interpretation of Milnor's triple invariants
with Paul Melvin (
*Algebraic and Geometric Topology*, vol. 3, 2003, pp. 557-568)

**Abstract:**We give a geometric interpretation of Milnor's invariants $\bar{\mu}(ijk)$ in terms of triple intersection points of Seifert surfaces for the three link components. This generalizes ideas of Cochran to links which are not algebraically split. - A few weight systems arising from intersection graphs
(
*Michigan Math. J.*, vol. 51, no. 3, 2003, pp. 509-536)

**Abstract:**We show that the adjacency matrices of the intersection graphs of chord diagrams satisfy the 2-term relations of Bar-Natan and Garoufalides, and hence give rise to weight systems. Among these weight systems are those associated with the Conway and HOMFLYPT polynomials. We extend these ideas to looking at a space of*marked*chord diagrams modulo an extended set of 2-term relations, define a set of generators for this space, and again derive weight systems from the adjacency matrices of the (marked) intersection graphs. Among these weight systems are those associated with the Kauffman polynomial. - On the existence of finite type link homotopy invariants
with Dylan Thurston (
*Journal of Knot Theory and Its Ramifications*, vol. 10, no. 7, 2001, pp. 1025-1040)

**Abstract:**We show that for links with at most 5 components, the only finite type homotopy invariants are products of the linking numbers. In contrast, we show that for links with at least 9 components, there must exist finite type homotopy invariants which are not products of the linking numbers. This corrects previous errors of the first author. - Finite Type Link Homotopy Invariants II: Milnor's invariants
(
*Journal of Knot Theory and Its Ramifications*, vol. 9, no. 6, 2000, pp. 735-758)

**Abstract:**We define a notion of finite type invariants for links with a fixed linking matrix. We show that Milnor's triple link homotopy invariant is a finite type invariant, of type 1, in this sense. We also generalize the approach to Milnor's higher order homotopy invariants and show that they are also, in a sense, of finite type. Finally, we compare our approach to another approach for defining finite type invariants within linking classes. - Finite Type Link Concordance Invariants
(
*Journal of Knot Theory and Its Ramifications*, vol. 9, no. 3, 2000, pp. 367-385)

**Abstract:**This paper is a generalization of the author's previous work on link homotopy to link concordance. We show that the only real-valued finite type link concordance invariants are the linking numbers of the components. (Note: This main result later turned out to be false - see "On the existence of finite type link homotopy invariants") - The Intersection Graph Conjecture for Loop Diagrams
(
*Journal of Knot Theory and Its Ramifications*, vol. 9, no. 2, 2000, pp. 187-211)

**Abstract:**Vassiliev invariants can be studied by studying the spaces of chord diagrams associated with singular knots. To these chord diagrams are associated the intersection graphs of the chords. We extend results of Chmutov, Duzhin and Lando to show that these graphs determine the chord diagram if the graph has at most one loop. We also compute the size of the subalgebra generated by these "loop diagrams." - Finite Type Link Homotopy Invariants
(
*Journal of Knot Theory and Its Ramifications*, vol. 8, no. 6, 1999, pp. 773-787)

**Abstract:**Bar-Natan used Chinese characters to show that finite type invariants classify string links up to homotopy. In this paper, I construct the correct spaces of chord diagrams and Chinese characters for links up to homotopy. I use these spaces to show that the only rational finite type invariants of link homotopy are the pairwise linking numbers of the components. (Note: This main result later turned out to be false - see "On the existence of finite type link homotopy invariants")

*Gordian*, with Loren Abrams and Lowell Trott, is a program to count the linked and knotted cycles in a spatial graph.

- Review of
*Viewpoints: Mathematical Perspective and Fractal Geometry in Art*, by Marc Frantz and Annalisa Crannell (*Journal of Mathematics and the Arts*, vol. 5, no. 4, 2011, pp. 221-222)

- The Exhibition of Mathematical Art at the 2008 Joint Mathematics Meetings
(
*Journal of Mathematics and the Arts*, vol. 2, no. 1, 2008, pp. 39-45; available online at http://journalsonline.tandf.co.uk/)

- Chord Diagrams and Gauss Codes for Graphs
with Thomas Fleming (preprint, August 2005)

**Abstract:**Chord diagrams on circles and their intersection graphs (also known as circle graphs) have been intensively studied, and have many applications to the study of knots and knot invariants, among others. However, chord diagrams on more general graphs have not been studied, and are potentially equally valuable in the study of spatial graphs. We will define chord diagrams for planar embeddings of planar graphs and their intersection graphs, and prove some basic results. Then, as an application, we will introduce Gauss codes for immersions of graphs in the plane and give algorithms to determine whether a particular crossing sequence is realizable as the Gauss code of an immersed graph. - Spin^c Manifolds
*(Postscript file)*(unpublished exposition, September 1995)

For more papers on finite type invariants, check out the bibliography maintained by Sergei Duzhin.

Most of my papers are posted on the mathematics arXiv. A list of my papers on the arXiv is here.

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