Friday Seminar on Knot Theory (2012)
| Date | February 1 (Fri.) 16：00～17：00 |
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| Speaker | Naoyuki Monden (Kyoto University) |
| Title | On stable commutator length of a Dehn twist |
| Place | Dept. of Mathematics, General Research Bldg., 401 |
| Abstract | Let $[G,G]$ be the commutator subgroup of a group $G$. For $x\in [G,G]$, we will denote by ${\rm cl}(x)$ the smallest number of commutators in G whose product is equal to $x$. We call ${\rm cl}(x)$ the commutator length of $x$. The stable commutator length of $x$, denoted by ${\rm scl}(x)$, is the limit $${\rm scl}(x)=\lim_{n\rightarrow \infty}\frac{{\rm cl}(x^n)}{n}.$$ In generally, computing (stable) commutator length is difficult. In this talk, we will present some background results of stable commutator length in mapping class groups. And we will give an upper bound of the stable commutator length of a Dehn twist. This is joint work with Danny Calegari and Masatoshi Sato. |
| Date | February 1 (Fri.) 15：00～16：00 |
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| Speaker | Kumi Kobata（OCAMI） |
| Title | On enumeration of edge colored graphs (joint work with Yasuo Ohno (Kinki University)) |
| Place | Dept. of Mathematics, General Research Bldg., 401 |
| Abstract | We previously gave a generalization of Ohno's theorem which gives a formula for enumeration of cyclic automorphism graphs. We analogously consider the enumeration of edge colored graphs under transposition. |
| Date | January 25 (Fri.) 16：00～17：00 |
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| Speaker | Yeonhee Jang (Nara Women's University) |
| Title | Heegaard splittings of distance $n$ |
| Place | Dept. of Mathematics, General Research Bldg., 401 |
| Abstract | Hempel introduced the concept of distance of Heegaard splitting by using
curve complex, and showed that there exist Heegaard splittings of closed
orientable 3-manifolds with distance $>n$ for any integer $n$. In this
talk, we construct pairs of curves with distance exactly $n$ for any non-negative
integer $n$, and use them to show that there exist Heegaard splittings
of 3-manifolds with distance exactly $n$. (This is a joint work with Ayako Ido and Tsuyoshi Kobayashi.) |
| Date | December 14 (Fri.) 16：00～17：00 |
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| Speaker | Ayaka Shimizu (Hiroshima University) |
| Title | The reducivity of spherical curves |
| Place | Dept. of Mathematics, General Research Bldg., 401 |
| Abstract | We show that we can obtain a reducible spherical curve from any non-trivial spherical curve by three or less inverse-half-twisted splices, i.e., the reducivity, which represents how reduced a spherical curve is, is three or less. We also discuss some unavoidable sets of tangles for spherical curves. |
| Date | December 7 (Fri.) 16：00～17：00 |
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| Speaker | Cheng Zhiyun（Beijing Normal University） |
| Title | A polynomial invariant of virtual knots and links |
| Place | Dept. of Mathematics, General Research Bldg., 401 |
| Abstract | Virtual knot theory was introduced by Professor Louis Kauffman in 1990s.
One simple invariant of virtual knots, the odd writhe was defined by Kauffman
himself in 2004. I will discuss the generalization of this invariant with
two viewpoints: one comes from V. O. Manturov's parity axioms, the other
approach was inspired by Professor Akio Kawauchi and Ayaka Shimizu's warping polynomial. If time permits, I will also give a similar generalization of the linking number of 2-component links. |
| Date | November 30 (Fri.) 16：10～17：10 |
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| Speaker | Sang Youl Lee（Pusan National University） |
| Title | Studying knots and links via net diagrams |
| Place | Dept. of Mathematics, General Research Bldg., 401 |
| Abstract | A net diagram is a knot or link diagram obtained from a quasitoric braid
diagram by replacing each positive (respectively, negative) crossing with
positive (respectively, negative) half-twists. Due to some recent works,
net diagrams for knots and links turned out to have various ramifications
and applications to study invariants for knots and links, including the
Casson invariant, genus, delta unknotting number, Alexander polynomial,
Jones polynomial and so on. In this talk, we will discuss some recent contributions
towards MFW inequality and the Jones conjecture on a minimal braid representation,
the Tripp conjecture on the canonical genus for Whitehead doubles of alternating
knots and the Hoste's conjecture on the Alexander polynomial of alternating
knots via net diagrams. This is partly a joint work with H. J. Jang and M. Seo. |
| Date | November 30 (Fri.) 15：30～16：00 |
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| Speaker | Jieon Kim（Pusan National University） |
| Title | The Alexander biquandles for oriented surface links diagrams of types A and D |
| Place | Dept. of Mathematics, General Research Bldg., 401 |
| Abstract | A biquandle is an algebraic structure consisting of a set with four binary operations satisfying axioms derived from the oriented Reidemeister moves, where generators of the algebra are identified with semi-arcs in an oriented link diagram. This relationship between the biquandle axioms and the Reidemeister moves makes biquandles a natural source of (virtual) knot and link invariants. The Alexander biquandle is an example of a biquandle that gives rise to the generalized Alexander polynomial for oriented virtual knots and links. In this talk, we will discuss a construction of the Alexander biquandle for oriented surface links via marked graph diagrams. We will show that the elementary ideals for a presentation matrix for the biquandle are invariants for the oriented surface link. To do this we first give a minimal generating set of Yoshikawa's moves and then investigate the behavior of presentation matrices under Yoshikawa's moves. We also compute the invariants for oriented surface links represented by marked graph diagrams with triangle-type and square-type ch-graphs. This is a joint work with Y. Joung and S. Y. Lee. |
| Date | November 30 (Fri.) 15：00～15：30 |
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| Speaker | Yewon Joung（Pusan National University） |
| Title | Obstructions for Yoshikawa's moves on marked graph diagrams for surface links |
| Place | Dept. of Mathematics, General Research Bldg., 401 |
| Abstract | A surface link is a closed, possibly orientable or nonorientable, surface
F smoothly embedded in the oriented 4-space R^4 or S^4. If F is a connected
surface, then it is called a surface knot. If F is oriented, then we call
it an oriented surface link. A surface link can be represented by a marked
graph diagram, that is, a knotted regular 4-valence rigid vertex graph
diagram in which each 4-valence vertex has a marker. Two marked graph diagrams
represent the same surface link if and only if they are transformed into
each other by a finite sequence of Yoshikawa's moves. In this talk, we
will discuss some obstructions for Yoshikawa's moves derived from a polynomial
defined by a state model analogous to the Kauffman's state model for the
Jones polynomial of classical knots and links, and calculated by using
a skein relation based on marked graph diagrams. We also discuss some applications
of these obstructions. This is a joint work with J. Kim and S. Y. Lee. |
| Date | November 9 (Fri.) 16：00～17：00 |
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| Speaker | Kenta Hayano (Osaka University) |
| Title | Vanishing cycles and homotopies of wrinkled fibrations |
| Place | Dept. of Mathematics, General Research Bldg., 401 |
| Abstract | Wrinkled fibrations are fibration structures on four-manifolds, which have been studied recently for the purpose of understanding some invariants of four-manifolds. In this talk, we will explain how vanishing cycles of wrinkled fibrations are changed by several homotopies. As an application, we also give new examples of surface diagrams of four-manifolds, which are combinatorial descriptions of four-manifolds introduced by Williams. Part of the results in this talk is joint-work with Stefan Behrens (The Max Plank Institute for Mathematics). |
| Date | November 2 (Fri.) 16：00～17：00 |
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| Speaker | Kenta Okazaki (Kyoto University, RIMS) |
| Title | On the state-sum invariants of 3-manifolds constructed from the E_6 and E_8 linear skeins |
| Place | Dept. of Mathematics, General Research Bldg., 401 |
| Abstract | The state-sum invariants of 3-manifolds, introduced by Turaev and Viro and generalized by Ocneanu, are formulated as a state-sum on triangulations of 3-manifolds derived from certain 6j-symbols. In this talk, we give elementary and self-contained constructions of the state-sum invariants of 3-manifolds derived from 6j-symbols of the E_6 and E_8 subfactors. We give such constructions by introducing E_6 and E_8 linear skeins, motivated by the E_6 and E_8 planar algebras. |
| Date | November 2 (Fri.) 15：00～16：00 |
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| Speaker | Atsuhide Mori（OCAMI） |
| Title | Essential dichotomy in contact topology |
| Place | Dept. of Mathematics, General Research Bldg., 401 |
| Abstract | A contact structure is a completely non-integrable hyperplane field usually
defined by a global Pfaff equation. However, in saying so, I suspect that
1) three dimensional contact topology is just a three dimensional topology for an imaginary person who does not know a mirror, and 2) higher dimensional contact topology is just an analogue of three dimensional contact topology. Indeed a contact structure fixes the orientation of the manifold even locally, while it has no moduli even globally. (A person who knows a mirror must wonder about the fact that two closed braids present the same contact knot iff they are related by only right-handed stabilizations/destabilizations.) Though a 3-manifold admits infinitely many contact structures, most of them (i.e., overtwisted ones) are spoiled and uniquely determined by homotopy data of plane fields. Contrastingly, the topology of the other decent structures (i.e., tight ones) adds a few extra data to the topology of the 3-manifold itself. I will talk about this dichotomy (overtwisted vs. tight) and its tentative generalizations. |
| Date | October 26 (Fri.) 16：00～17：00 |
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| Speaker | Shin'ya Okazaki（OCAMI） |
| Title | Bridge genus and braid genus of lens space |
| Place | Dept. of Mathematics, General Research Bldg., 401 |
| Abstract | The bridge genus and the braid genus are invariants of an oriented closed connected $3$-manifold which are introduced by Kawauchi. In this talk, we calculate the bridge genus and braid genus for some lens spaces $L(p,q)$, and we give an upper bound of $L(p,q)$ such that $p$ is an even number. |
| Date | October 19 (Fri.) 16：00～17：00 |
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| Speaker | Atsushi Ishii (University of Tsukuba) |
| Title | On some properties of handlebody-knots |
| Place | Dept. of Mathematics, General Research Bldg., 401 |
| Abstract | A handlebody-knot is a handlebody embedded in the 3-sphere, and a handlebody-link is a disjoint union of handlebodies embedded in the 3-sphere. Two handlebody-links are equivalent if one can be transformed into the other by an isotopy of the 3-sphere. I will explain the basics of handlebody-knots, which include fundamental moves on diagrams. Then I will talk about some properties of handlebody-knots. This talk consists of short topics. |
| Date | October 19 (Fri.) 15：00～16：00 |
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| Speaker | Javier Arsuaga（San Francisco State University） |
| Title | Using knot theory to model the formation of minicircle networks on trypanosomatida |
| Place | Dept. of Mathematics, General Research Bldg., 401 |
| Abstract | Trypanosomatida parasites, such as trypanosoma and lishmania, are the cause of deadly diseases in many third world countries. A distinctive feature of these organisms is the three dimensional organization of their mitochondrial DNA into maxi and minicircles. In some of these organisms minicircles are confined into a small disk shaped volume and are topologically linked, forming a gigantic linked network. The origins of such a network as well as of its topological properties are mostly unknown. In this paper we quantify the effects of the confinement on the topology of such a minicircle network. We introduce a simple mathematical model in which a collection of randomly oriented minicircles are spread over a rectangular grid. We present analytical and computational results showing that a positive critical percolation density exists, that the probability of formation of a highly linked network increases exponentially fast when minicircles are confined, and that the mean minicircle valence (the number of minicircles that a particular minicircle is linked to) increases linearly with density. When these results are interpreted in the context of the mitochondrial DNA of the trypanosome they suggest that confinement plays a key role on the formation of the linked network. This hypothesis is supported by the agreement of our simulations with experimental results that show that the valence grows linearly with density. Our model predicts the existence of a percolation density and that the distribution of minicircle valences is more heterogeneous than initially thought. |
| Date | October 5 (Fri.) 16：00～17：00 |
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| Speaker | Teruhisa Kadokami (East China Normal University) |
| Title | Switching scheme and switching complex |
| Place | Dept. of Mathematics, General Research Bldg., 401 |
| Abstract | Motivated by a notion, region crossing change, which is defined by A. Shimizu, we define generalized notions, switching scheme and switching complex. |
| Date | July 20 (Fri.) 16：00～17：00 |
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| Speaker | Michael Yoshizawa（University of California, Santa Barbara） |
| Title | Generating Examples of High Distance Heegaard Splittings |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | Given a closed orientable 3-manifold M, a surface S in M is a Heegaard surface if it separates the manifold into two handlebodies of equal genus. This decomposition is called a Heegaard splitting of M. The Hempel distance of this splitting is the length of the shortest path in the curve complex of S between the disk complexes of the two handlebodies. In 2004, Evans developed an iterative process to construct a manifold that admits a Heegaard splitting with arbitrarily high distance. We first provide an introduction to Heegaard splittings and Hempel distance and then improve on Evans' results. |
| Date | July 13 (Fri.) 16：00～17：00 |
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| Speaker | Greg McShane（Universite Joseph Fourier） |
| Title | Orthospectra and identities |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | The orthospectra of a hyperbolic manifold with geodesic boundary consists of the lengths of all geodesics perpendicular to the boundary. We discuss the properties of the orthospectra, asymptotics, multiplicity and identities due to Basmajian, Bridgeman and Calegari. We will give a proof that the identities of Bridgeman and Calegari are the same. |
| Date | July 13 (Fri.) 15：00～16：00 |
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| Speaker | Yoshiro Yaguchi (Gunma National College of Technology) |
| Title | On the extended 1-st Johnson homomorphism of the braid group |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | The braid group $B_m$ of degree $m$ is regarded as a mapping class group $M(D_m)$ of the $m$-puncterd disk $D_m$. There exists a natural surjective homomorphism from $B_m$ to the symmetric group $S_m$ of degree $m$, which is regarded as a homomorphism from $M(D_m)$ to the automorphism group of the 1-st homology group $H_1(D_m)$. By using an analogy of the Johnson's theory for mapping class groups of compact oriented surfaces, we construct a homomorphism from $B_m$ to a group extension of $S_m$. We call it the extended 1-st Jhonson homomorphism of $B_m$. We also study a way to caluculate the extended 1-st Johnson homomorphism by using braid diagrams. This is a joint work with Yusuke Kuno. |
| Date | July 6 (Fri.) 16：00～17：00 |
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| Speaker | Sakie Suzuki (RIMS, Kyoto University) |
| Title | Bing doubling and the colored Jones polynomials |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | The colored Jones polynomial is a quantum invariant associated with the quantized enveloping algebra of the Lie algebra sl2. We are interested in the relationship between topological properties of links and algebraic properties of the colored Jones polynomial. Bing doubling is an operation which gives a satellite of a knot. Habiro defined a certain series of the colored Jones polynomials to construct the unified WRT invariant of 3-manifolds. In this talk, we derive Habiro's colored Jones polynomials of the Bing double of a knot K from these of K. We aim to apply the result to study the unified WRT invariant. |
| Date | July 6 (Fri.) 15：00～16：00 |
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| Speaker | Tsukasa Yashiro (Sultan Qaboos University) |
| Title | On surface-knots with cross-exchangeable cycles |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | A surface-knot is a closed oriented surface embedded in 4-space. A surface diagram of a surface-knot is the projected image in 3-space under the orthogonal projection with crossing information. It is not known whether or not any surface diagram can be obtained from a trivial surface diagram by applying crossing changes along double curves. For some surface diagrams, there exist special double curves along which crossing information can be changed so that the surface diagram is deformed into a trivial surface diagram. In this talk, we present a construction to obtain a family of surface-knots with exchangeable cycles. This research consists of collaboration projects with A. Mohamad and also with A. AlKharusi. |
| Date | June 29 (Fri.) 15：00～16：00 |
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| Speaker | Takahiro Miura (Kobe University) |
| Title | On the flat braidzel length of links |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | It is known that any link has a flat braidzel surface as a Seifert surface. We introduce the flat braidzel length of a link defined as the minimal length of all braids which represent flat braidzel surfaces for the link. In this talk, we study relationships between the flat braidzel length and the Alexander-Conway polynomial and give a lower bound for the flat braidzel length. If time allows, we will show that, for any integer n greater than or equal to three, there exists a knot whose flat braidzel length is n. |
| Date | June 22 (Fri.) 16：00～17：00 |
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| Speaker | Naoko Kamada (Nagoya City University) |
| Title | Surface bracket polynomials of twisted links |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | Bourgoin defined the notion of a twisted link. Twisted links are equivalent to abstract links on non-orientable surfaces. Dye and Kauffman defined the surface bracket polynomials of virtual links. In this talk I introduced those of twisted links. We discuss a relationship between the multivariable invariant and the surface bracket polynomials of twisted links. |
| Date | June 15 (Fri.) 16：00～17：00 |
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| Speaker | Tetsuya Abe (RIMS, Kyoto University) |
| Title | The knot 12a990 is ribbon |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | Herald, Kirk and Livingston showed that the knot 12a990 is slice. Indeed,
they showed that the connected sum of 12a990 with T(2,3) and T(2,-3) is
ribbon, where T(2,3) and T(2,-3) are right- and left-handed trefoils. We
observe that 12a990 is ribbon and generalize this fact. The rest of the
time, we study homotopically slice knots obtained from unknotting number
one ribbon knots by applying annulus twists. This is a joint work with In Dae Jong and Motoo Tange. |
| Date | June 8 (Fri.) 16：00～17：00 |
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| Speaker | Ikuo Tayama（OCAMI） |
| Title | Tabulation of 3-manifolds of lengths up to 10 |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | This is a joint work with A. Kawauchi. A well-order was introduced on the set of links by A. Kawauchi. This well-order also naturally induces a well-order on the set of prime link groups and eventually induces a well-order on the set of closed connected orientable $3$-manifolds. With respect to this order, we enumerated the prime links, the prime link groups and 3-manifolds with lengths up to 10. In this talk, we show a list of the enumeration of $3$-manifolds and discuss a relationship between link exteriors and link groups. |
| Date | June 1 (Fri.) 16：00～17：00 |
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| Speaker | Toshihiro Nogi（OCAMI） |
| Title | On extendibility of a map induced by Bers isomorphism |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | Let $T(S)$ be the Teichmuller space of a closed Riemann surface $S$ of
genus $g(>1)$. Denote by $U$ the universal covering of $S$, that is,
the upper half-plane and denote by $\dot{S}$ the surface obtained by removing
a point from $S$. By Bers isomorphism theorem, we have a map from $T(S)
\times U$ to $T(\dot{S})$. It is known that the Teichmuller space $T(\dot{S})$
is embedded in $(3g-2)$-dimensional complex vector space. Thus the boundary
$\partial T(\dot{S})$ of $T(\dot{S})$ is naturally defined. Let $A$ be a subset of $\partial U$ consisting of all points filling $S$. In this talk, we show that the map of $T(S) \times U$ to $T(\dot{S})$ has a continuous extension of $T(S)\times (U \cup A)$ into $T(\dot{S}) \cup \partial T(\dot{S})$. This is a joint work with Hideki Miyachi (Osaka University). |
| Date | May 25 (Fri.) 16：00～17：00 |
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| Speaker | Takefumi Nosaka (RIMS, Kyoto University) |
| Title | Mochizuki's quandle 3-cocycles and Inoue-Kabaya chain map |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | In 2003, T. Mochizuki has determined the third quandle cohomologies of all Alexander quandles $X$ over finite fields; further he listed polynomials-presentations of their basis, as solutions of differential equations. I show that all the Mochizuki's 3-cocycles are derived from certain group 3-cohomologies via Inoue-Kabaya chain map. For example, if $X$ is the dihedral quandle, the Mochizuki's 3-cocycle is deduced with an easy computation. |
| Date | May 11 (Fri.) 16：00～17：00 |
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| Speaker | Ayaka Shimizu (Hiroshima University) |
| Title | The half-twisted splice operation on reduced knot projections |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | We show that any nontrivial reduced knot projection can be obtained from a trefoil projection by a finite sequence of half-twisted splice operations and their inverses without becoming a reducible projection. This is a joint work with Noboru Ito. |
| Date | April 27 (Fri.) 16：00～17：00 |
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| Speaker | Teruhisa Kadokami (East China Normal University) |
| Title | Integrality of Seifert surgery coefficient of twist knot, and Reidemeister torsion (joint work with Tsuyoshi Sakai (Nihon University)) |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | M. Brittenham and Y. Wu determined exceptional surgeries along every $2$-bridge knot by using a lamination structure of the knot complement. In particular, a $2$-bridge knot producing Seifert fibered spaces is a twist knot, which is denoted by $C(2n, 2)$\ $(n\in \mathbb{Z})$ in Conway's notation up to mirror images, and its Seifert surgery coefficients are $1$, $2$ and $3$ (and more for $n=0, \pm 1$). The speaker proved that the Alexander polynomial of a twist knot for $n\ne 0, -1$ restricts the positive numerators of Seifert surgery coefficients into $1$, $2$ or $3$. We try to prove that the denominators of Seifert surgery coefficients are $\pm 1$ (i.e.\ integrality) by using the Alexander polynomial of the knot and an invariant deduced from the Reidemeister torsion of the branched covering over the knot. We obtained necessary conditions as Diophantine equations, and partial answers for some cases. |
| Date | April 20 (Fri.) 16：00～17：00 |
|---|---|
| Speaker | Chad Musick (Nagoya University) |
| Title | Drawing Minimal Bridge Projections of Links |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | A method is given to achieve a minimal-bridge projection of a knot. This method relies on maze-solving and minimal path finding to shift all over-crossings off of extra underpasses. The method is proven correct by considering the knot as a fiber bundle from one end of a cylinder to the other end. |