Handle friendship seminar 2025 since 2013

Place:online or University of Tokyo (Komaba campus) or Online
japanese version

If you want the Zoom address, please register here.
If you register there, I send the Zoom URL to your email address.

2025 Spring

• The third break 7/9 (Wed.) (Zoom meeting)
  Hajime Kubota (Kyoto University) 10:00-12:30 (GMT +09:00)
  [Title] On a diagonal knot
  [Abstract] Grid homology is a purely combinatorial reformulation of knot Floer homology. Grid homology is computed from a diagram called a grid diagram, and knots that can be represented by a specific type of grid diagram are called diagonal knots. In this talk, I will explain the following three points:
  (1) That diagonal knots are, in a certain sense, easier to handle in grid homology;
  (2) What kinds of geometric properties of diagonal knots are captured by grid homology;
  (3) A comparison between diagonal knots and some well-known classes of knots.
  
  
• The third break 6/20 (Fri.) (Zoom meeting)
  Masaki Taniguchi (Kyoto University) 10:00-12:30 (GMT +09:00)
  [Title] Knot Concordance and Real Seiberg-Witten Theory
  [Abstract] One well-known conjecture in knot concordance is the slice-ribbon conjecture (Fox, 1962), for which no counterexamples have been found to date. A prominent class of potential counterexamples is the cables of the figure-eight knot (and their connected sums), which have been considered strong candidates (Miyazaki, 1994). Recently, the non-sliceness of the (2,1)-cable of the figure-eight knot, the leading candidate in this class, was shown using involutive Heegaard Floer theory by Da-Kang-Mallick-Park. On the other hand, in joint work with Kang and Park using the real Seiberg-Witten theory, which has been developed by several groups in recent years, we proved that the (2n,1)-cable of the figure-eight knot has infinite order in the smooth concordance group. Combining this result with existing work, it is now known that any nontrivial cable of the figure-eight knot has infinite order in the smooth concordance group. The core of this proof lies in the real relative 10/8 inequality of Konno-Miyazawa-Taniguchi and the construction of knot cobordisms embedded in 4-manifolds by Castro-Miller-Park-Stipsicz. In collaboration with Fukumoto, we also explored not only non-sliceness but cases in which the stabilizing number can be estimated. In this talk, I will give an overview of these developments. The method of using knot cobordisms as a computational tool in real Seiberg-Witten theory is expected to remain a key approach in the future. I will explain under what conditions a knot cobordism allows for the computation of concordance invariants derived from real Seiberg-Witten theory, and discuss future directions.
  


• The second break 6/13 (Fri.) (Zoom meeting)
  Jin Miyazawa (RIMS) 10:00-12:30 (GMT +09:00)
  [Title] A satellite formula for real Seiberg-Witten Floer homotopy types
  [Abstract] Real Seiberg-Witten theory extracts differential topological information from 3- and 4-dimensional manifolds equipped with an involution, by considering the fixed-point set of the involution's twisted lift acting on the space of solutions to the Seiberg-Witten equations or the Seiberg-Witten flow. By focusing on the part fixed under the twisted lift, one gains access to information that cannot be obtained through ordinary gauge theory. One example of this is the detection of exotic ${\Bbb R}P^2$-knots: embeddings of into $S^4$ that are topologically isotopic but not smoothly isotopic. In this talk, after giving an overview of real Seiberg-Witten theory and a sketch of how exotic ${\Bbb R}P^2$-knots can be detected, I will describe a joint work with JungHwan Park and Masaki Taniguchi, in which we develop invariants from real Seiberg-Witten theory for knots in 3-manifolds. I will also explain a satellite formula for these invariants and discuss applications to knot theory.


• The first break 6/11 (Wed.) (Zoom meeting)
  Tatsumasa Suzuki (Meiji University) 10:00-12:30 (GMT +09:00)
  [Title] The non-simply connected Price twist for the 4-sphere
  [Abstract] A cut-and-paste operation along a ${\Bbb R}P^2$-knot $S$ in a 4-manifold is called a Price twist. Applying a Price twist to the 4-sphere $S^4$ yields, up to diffeomorphism, at most three distinct 4-manifolds: the standard 4-sphere $S^4$, another homotopy 4-sphere $\Sigma_S(S^4)$, and a non-simply connected 4-manifold $\tau_S(S^4)$. In this talk, I will discuss some properties of $\tau_S(S^4)$ arising from tree-like ${\Bbb R}P^2$-knots $S$, and present results on the classification of its diffeomorphism types. In particular, I will introduce a simplification of handle diagrams that we developed to determine the smooth structure of $\tau_S(S^4)$, and explain how to perform handle calculus on these diagrams. I will also mention related results on pochette surgery that arise in the context of this work. This talk is based on joint research with Tsukasa Isoshima.
  [Reference]
  T. Isoshima and T. Suzuki, The non-simply connected Price twist for the 4-sphere, arXiv:2505.09332.