## Friday Seminar on Knot Theory（2017年度）

 日時 2017年10月6日（金）16：00〜17：00 講演者（所属） 河村 建吾（OCAMI） タイトル No immersed $2$-knot with one self-intersection point has triple point number two or three 場所 理学部 Ｆ棟 415号室（中講究室） アブストラクト The triple point number of an embedded/immersed $2$-knot is the minimum number of triple points needed for its diagram into a $3$-space. Satoh proved that no embedded $2$-knot has triple point number two or three. In this talk, we introduce diagrams of an immersed surface-knot and show that no immersed $2$-knot with one self-intersection point has triple point number two or three.
 日時 2017年7月28日（金）16：00～17：00 講演者（所属） Benjamin Bode（University of Bristol） タイトル Constructing links of isolated singularities of real polynomials 場所 理学部 Ｆ棟 415号室（中講究室） アブストラクト In order to implement knotted configurations in physical systems it is often very useful to have an explicit function, ideally a polynomial, $f:R^3\to C$ with a zero level set of given knot type. In this talk I will introduce an algorithm that for every link $L$ constructs a polynomial $f:R^4\to R^2$ whose zero set on the unit three-sphere is $L$. Applying stereographic projection then makes these functions applicable to physical systems. This constructive approach allows us to prove several results about the functions and their knotted zero level sets, for example under which conditions $f$ can be taken to have an isolated singularity or when $\arg f$ is a fibration of the link complement over $S^1$.
 日時 2017年7月14日（金）16：00～17：00 講演者（所属） 佐藤 光樹（東京工業大学） タイトル A partial order on $\nu^+$ equivalence classes 場所 理学部 Ｆ棟 415号室（中講究室） アブストラクト The $\nu^+$ equivalence is an equivalence relation on the knot concordance group. Hom proves that many concordance invariants derived from Heegaard Floer homology are invariant under $\nu^+$ equivalence. In this work, we introduce a partial order on $\nu^+$ equivalence classes, and study its algebraic and geometrical properties. As an application, we prove that any genus one knot is $\nu^+$ equivalent to one of the unknot, the trefoil and its mirror.
 日時 2017年7月7日（金）16：00～17：00 講演者（所属） 村尾 智 (筑波大学) タイトル The Gordian distance of handlebody-knots and Alexander biquandles 場所 理学部 Ｆ棟 415号室（中講究室） アブストラクト A handlebody-knot is a handlebody embedded in the 3-sphere $S^3$. Any two handlebody-knots of the same genus can be related by a finite sequence of crossing changes. Then we define their Gordian distance by the minimal number of crossing changes needed to be deformed each other. In particular, for any handlebody-knot $H$ and the trivial handlebody-knot of the same genus, we call their Gordian distance the unknotting number of $H$. In this talk, we give evaluation formulas of the Gordian distance and the unknotting number of handlebody-knots by using Alexander biquandles and some examples.
 日時 2017年6月30日（金）16：00～17：00 講演者（所属） 田中 利史（岐阜大学） タイトル On the region index and the unknotting number of a knot 場所 理学部 Ｆ棟 415号室（中講究室） アブストラクト It is known that, for any knot, there exists a diagram such that we have the unknot if we change all crossings on the boundary of some region of the diagram. The minimal number of the crossing changes over all such diagrams is called the region index of a knot. Clearly, the unknotting number is less than or equal to the region index. In this talk, we show that there exists a knot which has unknotting number $m$ and region index $m+1$ for any positive integer $m$(>1) by using the Goeritz invariant. We also show that there exists a knot which has unknotting number and region index that are equal to $n$ for any positive integer $n$(>1) by using the Rasmussen invariant.
 日時 2017年6月23日（金）16：00～17：00 講演者（所属） 屋代 司（Sultan Qaboos University） タイトル Covering diagrams of surface-knots 場所 理学部 Ｆ棟 415号室（中講究室） アブストラクト A surface-knot is a closed oriented surface embedded in 4-space. A surface-knot diagram of a surface-knot is the projected image in 3-space under the orthogonal projection with crossing information. Every non-trivial surface-knot diagram is equipped with the double decker set (the union of closures of upper and lower decker sets). From the lower decker set and quandle coloring, pseudo-cycles are defined. In this talk, we will introduce a covering diagram over a surface-knot diagram and we will show that a lower bound of the triple point number of the covering diagram is obtained.
 日時 2017年6月9日（金）16：00～17：00 講演者（所属） Sam Nelson（Claremont McKenna College） タイトル Biquandle Brackets 場所 理学部 Ｆ棟 415号室（中講究室） アブストラクト A biquandle bracket is a skein invariant of biquandle-colored oriented knots and links. We define an enhancement of the biquandle counting invariant by collecting the values of a biquandle bracket over the complete set of colorings of an oriented knot or link diagram. This class of invariants includes the classical skein invariants (HOMFLYpt, Jones, Alexander-Conway and Kauffman polynomials) as well as the biquandle 2-cocycle invariants as special cases, as well as new invariants.
 日時 2017年6月2日（金）16：00～17：00 講演者（所属） 和田 康載（早稲田大学） タイトル Invariants of welded links derived from multiplexing of crossings 場所 理学部 Ｆ棟 415号室（中講究室） アブストラクト For a link diagram $L$, we construct a virtual link diagram $L(m)$ which is obtained from $L$ by multiplexing of the crossings of $L$, where $m$ is a positive integer. If two link diagrams $L$ and $L'$ are equivalent, then $L(m)$ and $L'(m)$ are equivalent as welded links. Since an invariant of $L(m)$ is that of $L$, we try to find new invariants of $L$ via $L(m)$. This is a joint work with Haruko A. Miyazawa (Tsuda University) and Akira Yasuhara (Tsuda University).
 日時 2017年5月19日（金）16：00～17：00 講演者（所属） Anh Tran (The University of Texas at Dallas) タイトル Introduction to the AJ conjecture 場所 理学部 Ｆ棟 415号室（中講究室） アブストラクト The AJ conjecture was proposed by Garoufalidis about 15 years ago. It predicts a strong connection between two important knot invariants derived from very different background, namely the colored Jones function (a quantum invariant) and the A-polynomial (a geometric invariant). The colored Jones function is a sequence of Laurent polynomials which is known to satisfy a linear q-difference equation. The AJ conjecture states that by writing the linear q-difference equation into an operator form and setting q=1, one gets the A-polynomial. In this talk, I will give an introduction to this conjecture.
 日時 2017年5月12日（金）16：00～17：00 講演者（所属） 新國 亮 (東京女子大学) タイトル A regular isotopy between two spatial graphs which are ambient isotopic 場所 理学部 Ｆ棟 415号室（中講究室） アブストラクト Trace showed that two knot diagrams representing the same knot are transformed into each other by Reidemeister moves II and III if and only if they have the same writhe and rotation number, and this was extended to singular knots by Tsau. In this talk, for a graph each of whose components is the circle, theta graph or complete graph on four vertices, we give a necessary and sufficient condition for two diagrams of the graph representing the same spatial graph to be transformed into each other by Reidemeister moves II, III and IV. This is also extended to singular spatial graphs.
 日時 2017年4月28日（金）16：00～17：00 講演者（所属） 小鳥居 祐香 (理化学研究所/OCAMI) タイトル On link-homotopy classes of handlebody-links 場所 理学部 Ｆ棟 415号室（中講究室） アブストラクト A handlebody-link is a disjoint union of handlebodies embedded in $S^3$ and link-homotopy is an equivalence relation on handlebody-links generated by self-crossing changes. A. Mizusawa and R. Nikkuni classified the set of link-homotopy classes of 2-component handlebody-links completely using the linking numbers for handlebody-links. In this talk, by using Milnor's link-homotopy invariants, we construct an invariant for handlebody-links and give a bijection between the set of link-homotopy classes of $n$-component handlebody-links with some assumption and a quotient of the action of the general linear group on a tensor product of modules. This is joint work with Atsuhiko Mizusawa.
 日時 2017年4月21日（金）16：00～17：00 講演者（所属） 滝岡 英雄 (OCAMI) タイトル Infinitely many knots with the trivial $(2,1)$-cable $\Gamma$-polynomial 場所 理学部 Ｆ棟 415号室（中講究室） アブストラクト For coprime integers $p(>0)$ and $q$, the $(p,q)$-cable $\Gamma$-polynomial of a knot is the $\Gamma$-polynomial of the $(p,q)$-cable knot of the knot, where the $\Gamma$-polynomial is the common zeroth coefficient polynomial of both the HOMFLYPT and Kauffman polynomials. In this talk, we show that there exist infinitely many knots with the trivial $(2,1)$-cable $\Gamma$-polynomial, that is, the $(2,1)$-cable $\Gamma$-polynomial of the unknot. Moreover, we show that the knots have the trivial $\Gamma$-polynomial, the trivial first coefficient HOMFLYPT polynomial, and the distinct Conway polynomials.
 日時 2017年4月14日（金）16：00～17：00 講演者（所属） 橋爪 惠 (OCAMI) タイトル Link version of Inoue-Shimizu's result on region crossing 場所 理学部 Ｆ棟 415号室（中講究室） アブストラクト Recently, a new local transformation on link diagram called region freeze crossing change is proposed as a mutant of region crossing change. It is known that any change of crossings on any knot diagram can be realized as a region crossing change. Inoue-Shimizu showed there is a knot diagram such that some change of crossings can NOT be realized by region freeze crossing change. They showed necessary and sufficient condition for the exchangeability of any given crossing of the knot diagram via region freeze crossing change. In this talk, we discuss about a generalization of this result for links.