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3/19/2013 16:57:05Scott TaylorColby CollegeGraphs, Surfaces, and 3-manifoldsNote to organizers: My talk naturally splits into two pieces. I'd be happy to give two separate 30 min. talks or to just give one of the talks.

Part 1: A theta-graph in \R^3 is "almost unknotted" if it is non-planar but if the removal of any edge produces the unknot. I will provide new (conjectural) examples of almost unknotted theta graphs and discuss how recent results from 3-manifold theory shed light on their topological structure.

Part 2: One of the best ways of studying knots in 3-manifolds is to put them into "thin position". Until recently, it has been unclear what the best way to put graphs into thin position. I will describe a new method for doing this and explain what we hope to learn.
60 minutes
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3/22/2013 17:15:09Akio KawauchiOsaka City University Advanced Mathematical InstituteKnot theory for spatial graphs with the degree one vertices on a closed surfaceA notion of an "unknotted spatial graph" for every graph without degree one vertices
was introduced in [1, 2] so that any spatial graph can be transformed into an unknotted
spatial graph by a finite number of crossing changes on the edges. In this talk, this
notion is generalized to a notion for spatial graphs with the degree one vertices on a
fixed closed surface. The unknotting number of a spatial graph with the degree one vertices on a fixed closed surface is studied.

References

 A. Kawauchi, On a complexity of a spatial graph. in: Knots and soft-matter physics, Topology of polymers and related topics in physics, mathematics and biology, Bussei Kenkyu 92-1 (2009-4), 16-19.

 A. Kawauchi, On transforming a spatial graph into a plane graph,in: Statistical Physics and Topology of Polymers with Ramifications to Structure and Function of DNA and Proteins, Progress of Theoretical Physics Supplement, No. 191(2011), 235-244.
60 minutes
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3/28/2013 1:06:46Hwa Jeong LeeKAISTExactly fourteen intrinsically knotted graphs have $21$ edgesJohnson, Kidwell, and Michael showed that intrinsically knotted graphs have at least $21$ edges. Also it is known that $K_7$ and the thirteen graphs obtained from $K_7$ by $\nabla Y$ moves are intrinsically knotted graphs with $21$ edges. In this paper, we proved that only these $14$ graphs are intrinsically knotted graph with $21$ edges.30 minutes
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4/8/2013 7:03:38Hugh HowardsWake Forest UniversityForming the Borromean Rings from Random CurvesThe Borromean Rings in in their most common representation appear to be made out of circles, but a result of Freedman and Skora shows that this is an optical illusion. Although it cannot be built out of circles, the Borromean Rings can be built out of convex curves, for example, one can form it from two circles and an ellipse. On the other hand it is the only Brunnian link of 3 components which can be formed from convex components. We look at what other types of curves can be used to form the Borromean rings.30 minutes
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4/18/2013 20:35:10Allison MillerUniversity of Texas, AustinTangles and Generalized RavelsA generalized ravel is an embedded graph that is not planar but contains no nontrivial knots or links. More generally, embedded graphs can be formed from tangle diagrams by replacing some crossings with vertices and taking the vertex closure. In this talk I will characterize when a standard form diagram of a Montesinos tangle can be transformed in the above way to a generalized ravel.
This is joint work with Erica Flapan.
30 minutes
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4/23/2013 22:14:49Ryo NikkuniTokyo Woman's Christian UniversityConway-Gordon type theorem for the complete four-partite graph $K_{3,3,1,1}$We give a Conway-Gordon type formula for invariants of knots and links in a spatial complete four-partite graph $K_{3,3,1,1}$ in terms of the square of the linking number and the second coefficient of the Conway polynomial. As an application, we show that every rectilinear spatial $K_{3,3,1,1}$ contains a nontrivial Hamiltonian knot. This is a joint work with Hiroka Hashimoto. 60 minutes
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4/25/2013 22:33:33Seojung ParkKAISTQuadrisecants of linear embeddings of K6We will talk about the number of quadrisecants of linear embeddings of complete graphs, especially K6.30 minutes
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4/28/2013 21:50:38Reiko ShinjoKokushikan UniversityUniversal sequences of spatial graphsAn increasing sequence of integers is said to be universal
for links if every link has a projection to the sphere such that
the number of edges of each complementary face of the projection
comes from the given sequence.
In this talk, we define universal sequences for spatial graphs
and prove that the sequence (0,1,2,3,4,5) is universal for spatial
graphs without vertices of odd degree.
30 minutes
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4/29/2013 10:22:18Danielle O'DonnolImperial College LondonLegendrian theta-graphsWe investigate Legendrian $\theta$-graphs in $\mathbb{R}^3$ with the standard contact structure. We define the invariants Thurston-Bennequin number, tb, and rotation number, rot, for Legendrian graphs. We determine which Thurston-Bennequin number and rotation number can be realized for a Legendrian $\theta$-graph that is topologically planar. We investigate whether these invariants determine the graph up to Legendrian isotopy. Joint work with Elena Pavelescu. 60 minutes
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4/29/2013 19:48:15Youngsik HuhHanyang UniversityLinearly embedded graphs with free exteriorIn this talk we say that an embedding $f$ of a graph into $\mathbb{R}^3$ is {\em linear}, if the image $f(e)$ is a line segment for every edge $e$. The objects in our interests are graphs such that for any linear embedding its exterior is of free fundamental group. Those graphs include complete graphs and complete bipartite graphs. In addition we give some small graphs with such property.30 minutes
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4/29/2013 20:50:54Natsumi OyamaguchiOchanomizu UniversityEnumeration of prime 2-bouquet graphs with 6 crossings or less up to flat vertex isotopy
We provide an enumeration of a certain kind of spatial graphs, what we call prime 2-bouquet graphs, up to flat vertex isotopy. In order to do that, we construct such a graph from a prime 2-string tangle by connecting each end of one flat 4-valent vertex to that of the tangle, and we distinguish all the constructed 2-bouquet graphs up to six crossings by computing their Yamada polynomials. We then prove that there exist exactly 51 flat vertex isotopy classes of prime 2-bouquet graphs with six crossings or less.

30 minutes
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4/30/2013 7:32:47Joel FoisySUNY PotsdamThe conflict graphTutte showed that a graph is planar if and only if the conflict graph of each cycle is bipartite. We discuss an attempt to form an analogy of this result for graphs with a flat embedding. We conjecture that a graph has a flat embedding if and only if the (signed) conflict graph of every maximal planar graph is balanced. We will give the definition of conflict graph, discuss some evidence for this conjecture, and discuss some difficulties in proving it.30 minutes
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5/1/2013 1:09:22Ayaka ShimizuGunma National College of TechnologyRegion crossing change on spatial-graph diagramsA region crossing change is a local transformation on a knot or link diagram. The speaker showed that the region crossing change on a knot diagram is an unknotting operation, that is, we can deform any knot diagram into a diagram of the trivial knot by a finite number of region crossing changes. Z. Cheng and H. Gao gave the necessary and sufficiant condition for the region crossing change on link diagrams to be an unknotting operation. In this talk, we will discuss the region crossing change on spatial-graph diagrams, and show that the region crossing change on "inseparable" spatial-theta-curve and handcuff-graph diagram is an unknotting operation. This is a joint work with K. Hayano. 30 minutes
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5/2/2013 6:49:25Dorothy BuckImperial College LondonTopological and Graph Theoretic Freatures in BiologyWe will discuss a variety of topological and graph theoretic structures that arise in molecular biology, and the resulting biological implications. We will focus on knots and links within biopolymers, and theta graphs in DNA. (No previous biological knowledge necessary.)30 minutes
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5/3/2013 8:10:20Dana RowlandMerrimack CollegeBook Representations of Complete Graphs Book representations of graphs are good candidates for minimizing the number of and the complexity of knotted cycles and links found within the graph. In a book representation of a graph, the vertices lie on a circle and the edges are chords on internally disjoint disks sharing the circle as their boundary. The sheet number is the minimum possible number of disks required, and a knotted cycle in a book representation gives an arc presentation of a knot. We describe conditions which guarantee that two book representations are ambient isotopic, and we classify book representations for complete graphs with less than 5 sheets. This is joint work with students Amanda DeMarco, Joseph Mantoni, Stephen Francis, and Andrew O'Hara.30 minutes
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5/6/2013 15:00:12Thomas MattmanCSU, ChicoGraphs on 21 edges that are not 2-apex(joint with J. Barsotti) A graph of 20 or fewer edges is 2-apex, meaning there is a way to delete two or fewer vertices and arrive at a planar graph. The 20 graphs in the Heawood family are not 2-apex, and having 21 edges, are MMN2A or minor minimal with this property. We present evidence in support of a conjecture that the Heawood graphs are precisely the MMN2A graphs on 21 edges.30 minutes
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5/7/2013 5:14:20Hyuntae KimKAISTLinks in Linear Embeddings of K8A linear embedding of K8 which contains no figure-eight knot as its cycle is known before; here, we present another embedding which does not contain any (2,4)-torus link. Nevertheless, we conjecture that every linear embedding of K8 must contain either a figure-eight knot or a (2,4)-torus link. In an attempt to prove this we utilize the theory of oriented matroids, and discuss how it is used to describe the appearance of (2,4)-torus link in a linear embedding of K8.30 minutes
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5/8/2013 19:33:59Ted StanfordNew Mexico State UniversityLinking numbers for spatial graphsFor a fixed, finite, oriented, labeled graph Gamma, we give several descriptions of the linking numbers of embeddings of Gamma in in R3. These generalize the usual linking number of two disjoint cycles in a spatial graph, but in some cases they exist even when Gamma has no pair of disjoint cycles. We describe some of the properties of these linking numbers.30 minutes
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5/10/2013 8:18:35Lew LudwigDenison UniversityKnot MosaicsIn 2008, Lomonaco and Kauffman introduced knot mosaics, which have been shown to be equivalent to tame knot theory. This relatively new topic is easily accessible to undergraduate students as it is very hands-on. We explore several areas of this new approach to knot theory including the mosaic number (the smallest integer $n$ required to fit a given knot on an $n\times n$ mosaic board). We also consider the mosaic number for different families of knots and answer an open question posed by Colin Adams. Several open questions will be posed.60 minutes
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5/14/2013 8:14:32Dwayne ChambersPomona CollegeTopological Symmetry Groups of Small Complete GraphsFor each n < 7, we characterize all the groups that can occur as the orientation preserving topological symmetry group or the topological symmetry group of some embedding of the complete graph $K_n$ in $S^3$.30 minutes
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5/23/2013 11:44:22Erica FlapanPomona CollegeReduced Wu and Generalized Simon InvariantsWe introduce invariants of spatial graphs related to the Wu invariant and the Simon invariant, and apply them to prove that certain graphs are intrinsically chiral, and to obtain lower bounds for the minimal crossing number of embedded graphs.30 minutes
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5/28/2013 11:27:13Blake MellorLoyola Marymount UniversityColoring Spatial GraphsWe study colorings of spatial graphs, first introduced by Ishii and Yasuhara. We define a determinant for spatial graphs, analogous to the knot determinant, and explore its relationship to graph coloring. We use similar ideas to extend the Alexander polynomial to spatial graphs. This is a preliminary report of joint work with Terry Kong, Alec Lewald and Vadim Pigrish.30 minutes
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