Mini-workshop on knot concordance

Purposes

Speakers

Capmpus, Building and Lecture room

• W1008 (9/17, 9/19) (It will be open in 8:30am.)
• W809 (9/18)
• H206 (9/20)

Time Table

Titles and Abstracts

• 9/17(Tue.)
  • Tetsuya Abe  15:00-16:00 Resume
    [Title]: The construction of smoothly slice knots 1
    [Abstract]: We review the construction of Cappell-Shaneson's (homotopy) 4-spheres and Gompf's 4-spheres. Then we explain how to obtain (possibly non-ribbon) slice knots from these 4-spheres.
    
    
• 9/18(Wed.)
  • Tetsuya Abe  9:30-10:30
    [Title]: The construction of smoothly slice knots 2
    [Abstract]: We review the construction of (possibly non-ribbon) slice knots via annulus twists. We also explain some sufficient conditions for given slice knots to be ribbon. (arXiv:1305.7492v2)
    
    
  • Hye Jin Jang  11:00-12:00 Note
    [Title]: Introduction to (h)-solvable filtration on the knot concordance group
    [Abstract]: We review the definition of the (h)-solvable filtration on the knot concordance group defined by Cochran, Orr, and Teichner. We discuss the development of obstructions for detecting (h)-solvability of knots, and known results about (h)-solvable filtration. Finally, we provide new 2-torsion elements in the knot concordance group related to this filtration.
    
    
  • Motoo Tange  13:00-14:00 Note
    [Title]: Knot concordance invariants coming from Floer homology
    [Abstract]: We introduce some knot concordance invariants τ(K), δ(K), and dS_-1(K), originated by Heegaard Floer homology. We discuss about the relations of those invariants and the computation.
    
    
  • Min Hoon Kim  14:30-15:30 Note
    [Title]: Introduction to bipolar filtration
    [Abstract]: We review the definition of the bipolar filtration of topologically slice knots and survey the recent results of Cochran-Harvey-Horn, Cochran-Horn and Cha-Powell. Also, we discuss open problems.
    
    
• 9/19(Thu.)
  • Min Kyoung Song  9:30-10:30 Note
    [Title]: Introduction to Conant-Schneiderman-Teichner's filtrations on (string) links
    [Abstract]: We review the Whitney tower filtration of order on the set of concordance classes of (string) links. The filtrations are completely classified by Milnor invariants together with new higher-order Sato-Levine invariants and higher-order Arf invariants. Additionally, we see the relationship with Johnson filtrations and Goussarov-Habiro Y-filtrations on the group of concordance classes of string links and the group of homology cobordism classes of homology cylinders.
    
    
  • Hye Jin Jang  11:00-12:00 Note
    [Title] Kauffman conjecture
    [Abstract]: In 1982, Kauffman conjectured that sliceness of a knot K can be read off from the information of curves on any Seifert surfaces of K. Recently, Cochran and Davis proved that the Kauffman conjecture is false for knots. (arXiv:1303.4418) We prove that link version Kauffman conjecture is also false. This is a joint work with Minhoon Kim and Mark Powell.
    
    
  • Min Hoon Kim  13:00-14:00 Note
    [Title]: Whitney towers, grope and Casson-Gordon style invariants of links
    [Abstract]: S. Friedl and M. Powell introduced a Casson-Gordon style invariant for 2-component links with linking number 1. Motivated from the seminal work of Cochran-Orr-Teichner, they conjectured that their invariant is an obstruction to being height 3.5 Whitney tower/grope concordant to the Hopf link. We give a proof of this conjecture. If time allows, we give generalizations of the classical results of J. Hillman and A. Kawauchi.
    
    
  • Min Kyoung Song  14:30-15:30 Note
    [Title]: The homology cobordism group of homology cylinders with new homomorphism invariants.
    [Abstract]: The homology cobordism group of homology cylinders is enlargement of both the mapping class group and the concordance group of string links in homology D²× I. We see the structure of the group via the filtration of extended Milnor invariants. We also obtain Hirzebruch-type intersection form defect invariants, which can detect infinitely many elements invisible by existing homomorphism invariants.
    
    
• 9/20(Fri.)
  • Tetsuya Abe  10:00-11:00
    [Title]: Introduction to Rasmussen's s-invariant
    [Abstract]: We briefly review Rasmussen's s-invariant for knots.
    
    
  • Keiji Tagami  11:20-12:00 Note
    [Title]: Rasmussen invariants of almost positive knots
    [Abstract]: It is known that the Rasmussen invariants of positive knots are computed from their positive diagrams. We compute the Rasmussen invariants of almost positive knots from almost positive diagrams.
    
    
  • Extra talks, discussion, or excursion. 13:00-

• Min Hoon Kim (POSTECH)
• Hye Jin Jang (POSTECH)
• Min Kyoung Song (POSTECH)

• Tetsuya Abe (Tokyo Tech)
• Hisaaki Endo (Tokyo Tech)
• Tomomi Kawamura (Nagoya)
• Takahiro Oba (Tokyo Tech)
• Kouki Sato (Tokyo Gakugei)
• Keiji Tagami (Tokyo Tech)
• Toshifumi Tanaka (Gifu)
• Motoo Tange (Tsukuba)
• Yuichi Yamada (The University of Electro-Communications )