Problems on Assembly Graphs

This is one of the files posted at the web site
http://www.math.usf.edu/~saito/DNAweb/.
The web site contains information and results on assembly graphs,
that are mathematical models for DNA recombination processes.
See http://www.math.usf.edu/$\sim$saito/DNAweb/bachground.pdf
for background materials, which will be referred to as [BG].

This file contains problems that can lead to future research projects.
This will be updated often during 2009--2013.
Completed and on-going projects and their findings will be posted
in the web site above.

Tables of assembly graphs

• Find lower and upper bounds of numbers of irreducible/prime assembly graphs.
• Make tables of certain families of assembly graphs, similar to those that appear in knot theory.
  For example, analogues of closed braids, torus knots, pretzel knots, etc., may be useful.

Assembly numbers and polynomials

• Compute assembly polynomials for small assembly graphs in the table.
• Classify minimal Hamiltonian polygonal paths for small assembly graphs in the table.
• Combine the assembly polynomial and smoothings along the minimal Hamiltonian polygonal paths.
  For example, define a polynomial using only smoothings
  that are induced from minimal Hamiltonian polygonal paths.
• Investigate certain families of assembly graphs for assembly numbers and polynomials,
  such as those similar to torus knots, pretzel knots, etc.

Linking and writhe of assembly graphs

• Determine the linking numbers over all possible smoothings of assembly graphs in the tables.
• Determine those only for single Hamiltonian polygonal paths.
• Define polynomials and relate it to Other polynomials, such as Tutte polynomials.