1. Tangle analysis of protein-DNA complexes.
Isabel K. Darcy, University of Iowa
Joint work with John Luecke & Mariel Vazquez
This work is partially supported by NIH DMS/NIGMS GM76242

2. Solving tangle equations

3. Shailja Pathania, Makkuni Jayaram and
Path of DNA within the Mu Transpososome Transposase Interactions Bridging Two Mu Ends and the Enhancer Trap Five DNA Supercoils
Shailja Pathania, Makkuni Jayaram and
Rasika M Harshey

4. Shailja Pathania, Makkuni Jayaram and
Path of DNA within the Mu Transpososome Transposase Interactions Bridging Two Mu Ends and the Enhancer Trap Five DNA Supercoils
Shailja Pathania, Makkuni Jayaram and
Rasika M Harshey

5. Zhiqi Yin, Asaka Suzuki, Zheng Lou,
Interactions of Phage Mu Enhancer and Termini that Specify the Assembly of a Topologically Unique Interwrapped Transpososome
Zhiqi Yin, Asaka Suzuki, Zheng Lou,
Makkuni Jayaram and Rasika M. Harshey

7. A difference topology experiment:

12. (assuming a particular handedness of the products)
Theorem [DLV]: If T is a solution to the Mu-Cre equations, and if either
1.) T is rational or split or has parallel strands or
2.) If T has fewer than 9 crossings,
Then
T =
(assuming a particular handedness of the products)

14. By Hirasawa and Shimokawa plus tangle calculus
implies
By Hirasawa and Shimokawa plus tangle calculus

18. n2 n3 n1

20. Theorem (Thompson): Let G be an abstractly planar graph embedded in S3
Theorem (Thompson): Let G be an abstractly planar graph embedded in S3. G is planar if and only if
1.) Every proper subgraph of G is planar, and
2.) X(G)=S3 - int(N(G)) has compressible boundary.

21. A surface F in a 3-manifold M is compressible if there is a disk D embedded in M such that
intersection of D with F = the boundary of D.
2.the boundary of D does not bound a disk in F.
F is incompressible otherwise
(where F is not a 2-sphere).

23. Substrate = circular component containing two Mu binding sites and two Cre binding sites PLUS a linear component containing the third Mu binding site.

24. Thm: A system of four deletion experiments must include all three in cis experiments to conclude that a rational tangle satisfying this system is standard

25. Corollary: Suppose G is a tetrahedral graph with the following properties:
1.) There exists three edges such that removing any one of these three edges results in a planar graph.
2.) These three edges share a common vertex.
3.) There exists two additional edges such that removing either of these edges results in a graph whose exterior has compressible boundary
4.) X(G) has compressible boundary.
Then G is planar.

26. Extension to 4-string tangles:
Theorem (Soojeong Kim, D) Suppose T is 4-string tangle which has less than 8 crossings up to freely isotopy and T is solution tangle, then T is R-standard.