REU 2022: Donaldson's Diagonalization Theorem and χ-slice links

Mentors: Jonathan Simone and Hannah Turner TA: Weizhe Shen Students: Sophia Fanelle, Evan Huang, and Ben Huenemann The goal of this REU is to learn about Donaldson's Diagonalization Theorem and to use it to obstruct links from bounding Euler characteristic 1 surfaces in the 4-ball (such links are called χ-slice). In particular, the REU participants have able to prove various results regarding the χ-sliceness of positive pretzel links, 3-stranded pretzel links, and 4-stranded pretzel links. Our results can be found in the following paper: On χ-slice pretzel links Here are slides of a talk that gives a gentle introduction to χ-slice links.

Sophia, Evan, and Ben at the Poster Session See their poster here

Lectures

• 5/30: Lattices and lattice embeddings
  
• 5/31: Diagonalization
  
• 6/1: Manifolds
  
• 6/2: Knots, Links, and Spanning Surfaces
  
• 6/3: Ribbon Surfaces
• 6/7: Handlebody diagrams
  
• 6/8: Intersection forms of 4-manifolds and Donaldson's Diagonalization Theorem
  
• 6/9: Obstructing χ-sliceness
• 6/13: Signatures of knots and links
  
• 6/14: Dehn surgery
  
• 6/15: Handlebody diagrams for 4-manifolds vs surgery diagrams for 3-manifolds, slam dunk, blowups/blowdowns, and Rolfsen twists
• 6/17: Cellular homology
  
• 6/21: Homology continued and branched covers
  
• 6/24: d-invariant obstruction

Relevant Research Articles

• The slice-ribbon conjecture for 3-stranded pretzel knots (Greene and Jabuka)
  
• Lens spaces, rational balls, and the ribbon conjecture (Lisca)
  
• On the slice-ribbon conjecture for pretzel knots (Lecuona)
  
• Concordance groups of links (Donald and Owens)