Fall 2005 Schedule

special days, times or locations are in red

 Date Speaker Title and Abstract Tuesday Sept 6 Dave Bachman Pitzer College Title: Heegaard splittings and tori Abstract: I will present joint work with Ryan Derby-Talbot, where we almost completely characterize those Seifert Fibered Spaces that admit infinitely many non-isotopic Heegaard splittings of the same genus. Tuesday Sept 13 Sam Nelson UC Riverside Title: Recent results on knots and quandles Abstract: In this talk, we'll see some recent results on the relationship between knot diagrams and quandles as well as recent results on the structure of finite quandles. Tuesday Sept 20 Sam NelsonUCR Title: More recent results on knots and finite quandles Abstract: Results surveyed will include quandle matrices, the orbit decomposition of finite quandles, symbolic computation with finite quandles and an algorithm for finding Alexander presentations of finite quandles. Tuesday Sept 27 Erica FlapanPomona College Title: Intrinsic linking and knotting of graphs in arbitrary 3-manifolds (joint with: Hugh Howards, Don Lawrence, Blake Mellor) 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. Tuesday Oct 4 Erica Flapan continuation of last week Tuesday Oct 11 Tuesday Oct 18 Fall Break, no meeting Tuesday Oct 25 Danny C. CalegariCal Tech Title: Distortion in Transformation Groups (joint with M. Freedman) Abstract: An element g in a finitely generated group G is distorted if the word length of g^n grows sublinearly as a function of n. An element g in an arbitrary group G is distorted if it is distorted (as above) in some finitely generated subgroup H. We show that rigid rotations of spheres are distorted in groups of diffeomorphisms (C^1 or C^infty depending on the dynamics of the rotation). We also show that every homeomorphism of the sphere is arbitrarily badly distorted in the group of all homeomorphisms. A corollary, pointed out by Yves de Cornulier, is that the group of homeomorphisms of a sphere, thought of as a discrete group, has the *strongly bounded* property, introduced by G. Bergman. Tuesday Nov 1 Mike KrebsCal State LA Title: Toledo invariants on 2-orbifolds Abstract: To each connected component in the space of semisimple representations from the orbifold fundamental group of the base orbifold of a Seifert fibered homology 3-sphere into the Lie group U(2,1), we associate a real number called the orbifold Toledo invariant.'' Using the theory of Higgs bundles, we explicitly compute all values this invariant takes on. Tuesday Nov 8 Afra ZomorodianStanford University Title: Localized Homology Theory Abstract: In this talk, I introduce "localized homology", a theory for finding local geometric descriptions for topological attributes. Given a space and a cover of subspaces, we construct the blowup complex, a derived space that contains the local and global topologies at different scales. The persistent homology of the blowup complex localizes the topological attributes of the space. Our theory is general and applies in all dimensions. After giving an informal discussion, I formalize the approach for general spaces, adapt it for simplicial spaces, and develop a simple algorithm that works directly on the input and avoids the construction. I then show results using a complete implementation of the ideas presented. Tuesday Nov 15 Marty ScharlemannUCSB Title: Meditation on the Schoenflies Conjecture Abstract: The Schoenflies Conjecture remains unsolved only in one dimension and in that dimension only in the Diff/PL category. I'll give a brief overview both of the importance and history of the conjecture and discuss some recent developments and connections to problems in 3-manifolds. Tuesday Nov 22 Lisa HernandezUC Riverside Title: An application of TQFT: determining the girth of a knot. Abstract: A knot diagram can be divide by a circle into two parts, such that each part can be coded by a planar tree with integer weights on its edges. A half of the number of intersection points of this circle with the knot diagram is called the girth. The girth of a knot is then the minimal girth of all diagrams of this knot. The girth of a knot minus 1 is an upper bound of the Heegaard genus of the 2-fold branched covering of that knot. We will use TQFTs coming from the Kauffman bracket to determine the girth of some knots. Consequently, our method can be used to determine the Heegaard genus of 2-fold branched covering of some knots. Tuesday Nov 29 Blake MellorLoyola Marymount University Title: Intrinsic linking and knotting in Virtual Spatial Graphs. (Joint with Tom Fleming) Abstract: Kauffman's theory of virtual knots can be extended to a theory of virtual spatial graphs. We look at what it means for a graph to be virtually intrinsically linked. We show that the space of intrinsically linked graphs can be decomposed into an infinite sequence of nested spaces of graphs, which are virtually intrinsically linked of various degrees. We also look at several examples of particular graphs, such as K_6 and the Petersen graph. Time permitting, we will also consider virtually intrinsically knotted graphs. Tuesday Dec 6 Xiao-Song LinUC Riverside Title: Loop braids and the motion group of the unlink in the 3-space. Abstract: The motions of components of the unlink in the 3-space can be described by loop braids. In this talk, we will give a complete description of the relations among loop braids. We shall also touch upon the question of finding representations of the group of loop braids. (Notes: The concept of a group of motions of a submanifold N in a manifold M was introduced by David Dahm in his unpublished Ph.D. thesis (Princeton, 1962). In the mathematics literature, this concept was almost only further explored in the work of Deboral Goldsmith published in 1981. More physicists seem to be aware of the work of Dahm and Goldsmith than mathematicians. It is our hope to offer a remedy to this situation through this talk.)

Spring 2006 Schedule

special days, times or locations are in red

 Date Speaker Title and Abstract TuesdayJan 24 Peter MilleyUC Riverside Title: Constructing all 1-cusped hyperbolic manifolds with volume less than 2.7 (Joint with D. Gabai and R. Meyerhoff) Abstract: Every 1-cusped hyperbolic 3-manifold with volume less than 2.7 can be obtained by Dehn surgery on one of 21 2-cusped manifolds. This result, while simple in principle and easy to demonstrate, was surprisingly difficult to prove; the proof spans two papers currently in progress. In this talk I will sketch out the proof and explain the connection to the still-open problem of finding the minimum-volume closed hyperbolic manifold. I will also describe some of the difficulties we had nailing down this result, including the hopefully amusing tale of Lemma 8.8, case 7, subcase 2, sub-subcase 6, sub-sub-subcase 2. Finally, I will perform tricks with SnapPea. TuesdayJan 31 Ko HondaUniversity of Southern California Title: Right-veering diffeomorphisms of a compact surface with boundary Abstract: We initiate the study of the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary. The monoid strictly contains the monoid of products of positive Dehn twists. We explain the relationship to tight contact structures and open book decompositions. This is joint work with W. Kazez and G. Mati\'c. TuesdayFeb 7 Stefano Vidussi UC Riverside Taubes' Conjecture and Twisted Alexander Polynomials Abstract: It is well-known that the Seiberg-Witten invariants of a $4$--manifold provide obstructions to the existence of a symplectic structure. When the $4$--manifold is of the form $S^{1} \times N$, these obstructions can be described in terms of the Alexander polynomial of $N$. C. Taubes formulated the conjecture that, if $S^{1} \times N$ is symplectic, then $N$ fibers over the circle. P. Kronheimer studied the case where $N$ is obtained as $0$--surgery along a knot $K \subset S^{3}$ and showed that the aforementioned constraints on the Alexander polynomial $\Delta_{N}$ give evidence to Taubes' conjecture, i.e. $\Delta_{N}$ must be monic and its degree must coincide with the genus of the knot. Still, these conditions are short of characterizing fibered knots. In this talk we discuss how to extend these ideas to the case of a general $3$--manifold and how these conclusions can be strengthened by taking into account the twisted Alexander polynomials associated to an epimorphism of $\pi_{1}(N)$ into a finite group. This way we get new evidence to Taubes' conjecture and, practically, new obstructions to the existence of symplectic structures on $S^1 \times N$, even in the case of $0$--surgery along a knot. As an application o! f these results we show that if $N$ is the $0$--surgery along the pretzel knot $(5,-3,5)$, a case that cannot be decided with the use of the Alexander polynomial, $S^1 \times N$ is not symplectic: this answers a question of Kronheimer. In a similar way, we show that Taubes' conjecture holds for knots up to $12$ crossings. (\textit{Joint work with Stefan Friedl of Rice University}) TuesdayFeb 14 Fumikazu Nagasato Tokyo Institute of Technology, Japan Title: Algebraic varieties via a filtration of the Kauffman bracket skein module Abstract: In this talk, we define an algebraic variety in an affine space $\mathbb{C}^N$ for a knot in 3-sphere $S^3$, called the reduced character variety (RCV) of a knot, which is constructed via a filtration of the Kauffman bracket skein module (KBSM) of a knot exterior. For a braid $\sigma$, we can in fact construct a polynomial map $f_{\sigma}$ from $\mathbb{C}^N$ to itself by using a representation of the braid group into the automorphism group of the KBSM of a handlebody at $t=-1$, which representation is closely related to the non-linear Magnus representation of the braid group into a polynomial ring over $\mathhbb{Z}$. The RCV of a knot is defined as a subvariety of the set of fixed points of $f_{\sigma}$ for a braid presentation $\sigma$ of the knot. We can show that the RCV is an invariant of knots in $S^3$. (We note that a quantization of the non-linear Magnus representation and the RCV can be derived from the above system at general $t$.) We try to get a better understanding of the RCV by focusing on the number of its irreducible components (*)''. First, we observe numerically a relationship of the quantity (*) with so-called the Casson-Lin invariant defined by X-.S-. Lin, which in fact inspired the RCV. Moreover we look into the quantity (*) and see that the quantity (*) is closely related to the knot determinant and moreover the highest degree of the A-polynomial $A_K(M,L)$ of a knot $K$ in terms of $L$. TuesdayFeb 21 Jim HostePitzer College postponed to later in the semester TuesdayFeb 28 Jesse Johnson UC Davis Title: Heegaard Splittings and the Pants Complex Abstract: John Hempel defined a distance" for Heegaard splittings based on the complex of curves. I will describe two similar types of distance, using Hatcher and Thurston's pants complex and the dual graph to the curve complex, respectively. These distances prove to be more global notions of complexity for a Heegaard splitting in the sense that they behave nicely under stabilization and lead to non-trivial 3-manifold invariants. TuesdayMar 7 Thomas MattmanCSU Chico Title: Boundary slopes (nearly) bound cyclic slopes. Abstract: Let $r_m$ and $r_M$ be the least and greatest finite boundary slopes of a hyperbolic knot $K$ in $S^3$. We show that any cyclic surgery slopes of $K$ must lie in the interval ${[} r_m - 1/2, r_M + 1/2 {]}$. TuesdayMar 14 no meetingSpring Break TuesdayMar 21 Jim HostePitzer College Title: Boundary Slopes of 2-Bridge Links Determine the Crossing Number Albstract: A {\it diagonal} surface in a link exterior $M$ is a properly embedded, incompressible, boundary incompressible surface which furthermore has the same number of boundary components and same slope on each component of $\partial M$. We derive a formula for the boundary slope of a diagonal surface in the exterior of a 2-bridge link which is analogous to the formula for the boundary slope of a 2-bridge knot found by Hatcher and Thurston. Using this formula we show that the {\it diameter} of a 2-bridge link, that is, the difference between the smallest and largest finite slopes of diagonal surfaces, is equal to the crossing number. TuesdayMar 28 Sam NelsonUC Riverside Title: A quandle-theoretic definition of linking number. Abstract: The study of knot and link invariants defined in terms of finite quandles has generally focused on connected quandles and single-component links, i.e. knots. In this talk, we'll see how a particular kind of finite quandle can be used to give a purely algebraic definition of linking number, which can then be applied to arbitrary quandles. This is joint work with Natasha Harrell, an undergraduate at UCR. TuesdayApr 4 no meeting TuesdayApr 11 no meeting TuesdayApr 18 Mohamed Ait Nouh Cal State Univ Channel Islands Title: A new invariant of knots via Kirby calculus. Abstract: A twisted knot is a knot obtained from the unknot by grabbing a handful of the unknot strands, cutting then all, then after twisting one set, reglue. This operation is equivalent to doing a certain type of Dehn surgery on an unknot in the complement of the original unknot. By using Kirby calculus, a twisted knot bounds a nice disk in a standard 4-manifold. We introduce the notion of characteristic twisting of knots in the 3-sphere, motivated by characteristic classes. We also define a new invariant of knots corresponding to the minimal number of characteristic twisting disks of a knot K. Parts of the proofs are inspired by Kirby calculus and old guage theory. TuesdayApr 25 Marta Asaeda UC Riverside Title: Skein Modules Based on Incompressible Surfaces TuesdayMay 2 Vin de Silva Pomona College Title: Topological approximation by small simplicial complexes. Abstract: How does one obtain topological information about a data set? An obvious strategy is to attempt to approximate the configuration of data points by a suitable simplicial complex. I will explain one particular construction, "the witness complex" (due to Gunnar Carlsson and myself), which has been found to be useful and effective. There are two flavours, "strong" and "weak", and a central part of this work involves understanding the relationship between the two flavours. I will discuss the "weak witnesses theorem" which underpins this understanding.