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Critical Shapes of Knots: From Pretty Pictures to DNA Gel-Electrophoresis.

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Dr. Jonathan Simon, Univ. of Iowa

  • Colloquium
When Thu, Apr 15, 1999
from 01:10 PM to 02:00 PM
Where RH 128
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Abstract: Given a knot K in space, we can compute "energies", numbers E(K), that somehow measure how complicated or tangled the given knot is. The motivation for people developing this idea has been a combination of basic mathematics together with a desire to model "physical knots", such as flux tubes in plasmas, DNA loops, or just plain ropes. Definitions are based on the knot trying to avoid or exclude itself, some kind of self-repelling or rope thickness. We can see in a brief video that the energies are suprisingly effective in simplifying tangled figures. For smooth knots, energies and crossing numbers are connected: Ignoring multiplicative constants, we have (crossing number) < (repelling energies) < (4/3) power of "rope length energy" The latter being an energy based on self-exclusion. The exponent (4/3) is the right one. Computer experiments flowing knots to minimum energy levels reveal: (1) Critical conformations often are beautiful, typically the kinds of elegant pictures that in the past were drawn for knot tables. (2) With polygons, it is possible to "get stuck"; that is, there may be several local minima, even for the unknot! (3) The minimum energy numbers observed for the different knot types correlate well with the relative velocities of those knot types in gel-electrophoresis experiments for DNA knots. When DNA of equal length are tied in different types of knots, and run in gel-electrophoresis experiments, different knots are separated.We do not yet fully understand this phenomenon, just that it happens in reliable and predictable ways. We can offer the beginning of a model to account for the correlation between gel velocity and knnot energy, but it is just a beginning.

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