M392C: APPLIED KNOT THEORY

 
8/26 Introduction:  Chirality, DNA, Kinetoplast DNA 
Article: Sumners, D. W. The role of knot theory in DNA research. Geometry and topology (Athens, Ga., 1985), 297-318, Lecture Notes in Pure and Appl. Math., 105, Dekker, New York, 1987.
Article: Chen J. Rauch CA. White JH. Englund PT. Cozzarelli NR. The topology of the kinetoplast DNA network. Cell. 80(1):61-9, 1995 Jan 13. 
Related Article: Chen J. Englund PT. Cozzarelli NR. Changes in network topology during the replication of kinetoplast DNA. EMBO Journal. 14(24):6339-47, 1995 Dec 15. 
Related Article: Ryan, KA. Shapiro, TA. Rauch, CA. Griffith, J. Englund PT. A knotted free minicircle in kinetoplast DNA. PNAS. 85, 5844-5848, Aug 1988.
8/28 Lk = Tw + Wr
Related Article: White, James H. An introduction to the geometry and topology of DNA structure. Mathematical methods for DNA sequences, 225--253, CRC, Boca Raton, FL, 1989. 
8/31  Replication, Topoisomerases, Recombinases, Gel electrophoresis, Transcription, Translation
Comic book: I. Rosenfield, E. Ziff, B. van Loon, DNA for Beginners, A Writers and Readers Documentary Comic Book 1983.  
9/2  Supercoiling, Gel electrophoresis, Odd-noded twist knots in mutant bacteria,, Knot energies, Recombination, Tangle calculus
Knot energy articles: Katritch, Vsevolod; Bednar, Jan; Michoud, Didier; Scharein, Robert G.; Dubochet, Jacques; Stasiak, Andrzej, Geometry and physics of knots. Nature 384 (1996), no. 6605, 142--145. 

Stasiak, Andrzej: Katritch, Vsevolod; Bednar, Jan; Michoud, Didier; Dubochet, Jacques; Electrophoretic mobility of DNA knots . Nature 384 (1996), no. 6605, 142--145. 

Vologodskii, Alexander; Crisona, Nancy; Laurie, Ben; Pieranski, Piotr.; Katritch, Vsevolod; Dubochet, Jacques; Stasiak, Andrzej, Sedimentation and Electrophoretic Migrations of DNA Knots and Catenanes . Nature 384 (1996), no. 6605, 142--145. 

Mutant bacteria reference: S. Shaw, DNA Knotting in Vitro and In Vivo, Thesis, Harvard University, 1995. 
Tangle calculus article: C. Ernst, D. W. Sumners, A calculus for rational tangles: applications to DNA recombination, Math. Proc. Camb. Phil. Soc. 108 (1990), 489-515. 
9/9  Knot theory intro, Tangle distances, Rational tangles
Rational tangle article: Goldman, Jay R.; Kauffman, Louis H. Rational tangles. Adv. in Appl. Math. 18 (1997), no. 3, 300-332. 
9/15  Covering spaces, Tangle equivalence, Classification of rational tangle distance one 4-plats
Article:  I. Darcy, D. Sumners, Rational Tangle Distances on Knots and Links, preprint 
9/16  2 implies 3 implies 1, adding tangles, Xer examples 
Xer articles:
9/21 lens spaces, covering spaces, double branched covers
9/23  surgery, double branched covers of 4-plats
9/29 1 implies 2: surgery, Seifert fibered spaces 
9/30 surgery, Seifert fibered spaces
10/5  generalized Seifert fibered spaces 
10/7  generalized Seifert fibered spaces 
10/14  Student Lab presentation: Shailja Pathania
  • Lab: Relaxation of DNA with Topoisomerase I
  • Presentation: Mu Transposase 
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    10/19  generalized Seifert fibrations of lens spaces
    10/21  generalized Seifert fibrations of lens spaces, equivalences of SF
    10/26 1 implies 2
    10/28  Random Knotting
    Based on Talk: Ken Millet, Random Walks in Polygonal Space, IMA, June 10, 1996 
    J. Calvo web page: http://www.williams.edu/Mathematics/jcalvo/research.html  
    11/2  Random Knotting
    Based on Talk: Stu Whittington, Theory of Self-Avoiding Walks, IMA tutorial, June 1996, and FSU, Feb. 1997
    11/4  Random Knotting
    Based on Talk: Stu Whittington, part II
    Based on Talk: Entanglement Complexity of Random Knots, De Witt Sumners, IMA, June 1996. 
    Based on Talk: Lattice Knots, Janse Van Rensburg, July 1994 
    11/9  Random Linking, etc 
    Based on Talk: Yuanan Diao: Percolation of links, IMA, June 1996? 
    Ribbon article?
    Based on Talk: Chris Soteros, Lattice Models of Branched polymers in Confined Geometries, IMA, June 1996
    Based on Talk: Janse Van Rensburg, Crumpling Surfaces, IMA, June 1996.\
    Based on Talk: ribbons, surfaces? 
    Based on Talk:
    11/11 Random Knotting, Knot Energies
    Based on Talk: Janse Van Rensburg, July 1994
    Random Knotting Article: Janse van Rensburg, E. J.; Orlandini, E.; Sumners, D. W.; Tesi, M. C.; Whittington, S. G. The writhe of knots in the cubic lattice. J. Knot Theory Ramifications 6 (1997), no. 1, 31--44. 
    Knot Energy Article: J. Simon, Energy functions for knots: beginning to predict physical behavior 
    Based on Talk: Janse van Rensburg, Energies of Polygonal knots
    Based on Talk: Sullivan, IMA 
    Based on Talk: Ernst, Diao, Janse Van Rensburg 
    11/16  Knot Energies
    Article: J. Simon, Energy Functions for Polygonal Knots (JKTR) Vol 3 #3 (1994)299-320. 
    Article: G. Buck, J. Simon, , Energy and length of knots, in Lectures at Knots96, 219-234. 
    11/18 Knot Energies and Stereochemical Topology
    Article: R. Litherland, J. Simon, O. Durumeric, E. Rawdon, Thickness of Knots, preprint. 
    Book: E. Flapan, When Topology meets chemistry, preprint
    11/23  Student presentation: Cynthia Verjovsky, RNA Folding
    Article: Kauffman, Louis H.; Magarshak, Yuri; Graph invariants and the topology of RNA folding. Random knotting and linking (Vancouver, BC, 1993), 1--13, Ser. Knots Everything, 7, World Sci. Publishing, River Edge, NJ, 1994. 
    11/25 Student presentation: Andrew Ban, protein folding
    Article: Chen, Shi-Jie, Dill, Ken, A, Symmetries in proteins: A knot theory approach, J. Chem Phys, 104(15) 15 Apr 1996 
    Student presentation: Jodi Schneider and Ken Baker, intro to knots in fluid mechanics
    Articles:
  • Ricca, Renzo L.; Berger, Mitchell A. Topological ideas and fluid mechanics. Physics Today 28 - 34, Dec 1996.
  • Ricca, Renzo L. Applications of knot theory in fluid mechanics. Knot theory (Warsaw, 1995), 321--346, Banach Center Publ., 42, Polish Acad. Sci., Warsaw, 1998 
  • Ricca, Renzo L. New developments in topological fluid mechanics: from Kelvin's vortex knots to magnetic knots. in Ideal Knots, World Scientific, vol 19, 1998.
  • Ricca, Renzo L.; Samuels, DC; Barenghi, CF, Vortex knots. Advances in Turbulance VII, Kluwer 1998, 369 - 372.
    • 11/30 Student presentation: Jodi Schneider and Ken Baker, intro to knots in fluid mechanics II
    Student presentation: Dorothy Buck, Tangle Calculus 
    Article: C. Ernst, D. W. Sumners, A calculus for rational tangles: applications to DNA recombination, Math. Proc. Camb. Phil. Soc. 108 (1990), 489--515. 
    12/2 Student presentation: Anne Miller, Ideal Knots 
  • Article: Katritch, Vsevolod; Olson, Wilma; Pieranski, Piotr; Dubochet, Jacques; Stasiak, Andrzej, Geometry and physics of knots. Nature 388 (1997), 148--151. 
  • Article: Katritch, Vsevolod; Bednar, Jan; Michoud, Didier; Scharein, Robert G.; Dubochet, Jacques; Stasiak, Andrzej, Geometry and physics of knots. Nature 384 (1996), no. 6605, 142--145. 
  • Article:Stasiak, Andrzej: Katritch, Vsevolod; Bednar, Jan; Michoud, Didier; Dubochet, Jacques; Electrophoretic mobility of DNA knots . Nature 384 (1996), no. 6605, 122. 
  • and based on talk:  J. Simon, Energy and Thickness of Knots, 6/13/96, IMA 
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    Summary, Knots in Proteins, RNA Folding
    Knots in Proteins
    Article: Liang, Chengzhi; Mislow, Topological Features of Protein Structures: Knots and Links J. Am. Chem. Soc. 117, 4201 - 4213, 1995  
    Article: Benham CJ. Jafri MS. Disulfide bonding patterns and protein topologies. Protein Science. 2(1):41-54, 1993 Jan. 
    RNA Folding
    Article: Kauffman, Louis H.; Magarshak, Yuri B. Vassiliev knot invariants and the structure of RNA folding. Knots and applications, 343--394, Ser. Knots Everything, 6, World Sci. Publishing, River Edge, NJ, 1995.
    Article: Penner, R. C.; Waterman, Michael S. Spaces of RNA secondary structures. Adv. Math. 101 (1993), no. 1, 31--49. 
    Article: Magarshak Y. Benham CJ. An algebraic representation of RNA secondary structures. Journal of Biomolecular Structure & Dynamics. 10(3):465-88, 1992 Dec.