| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | September 5, 2013 |
| Latest Amendment Date: | September 5, 2013 |
| Award Number: | 1317942 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Joanna Kania-Bartoszynska
jkaniaba@nsf.gov (703)292-4881 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | September 15, 2013 |
| End Date: | August 31, 2017 (Estimated) |
| Total Intended Award Amount: | $142,234.00 |
| Total Awarded Amount to Date: | $142,234.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
202 HIMES HALL BATON ROUGE LA US 70803-0001 (225)578-2760 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
LA US 70803-4918 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): | TOPOLOGY |
| Primary Program Source: |
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| Program Reference Code(s): |
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| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
The Volume Conjecture claims a deep relationship between the Jones polynomial of cablings of a knot on one side and the hyperbolic volume of the knot complement on the other side. This project is inspired by the Volume Conjecture. The scope is to gain a better understanding of the colored Jones polynomial, and its relations to the geometry of the knot complement. In earlier works of the principal investigator and his collaborators it was shown that bounds for the hyperbolic volume of certain classes of knots can be read off from coefficients of the colored Jones polynomial. This made it interesting to study the leading and trailing coefficients of the colored Jones polynomial. Under certain conditions on the knot there is a power series assigned to the knot that determines the first k coefficients of its colored Jones polynomial, for every fixed k and sufficiently large color. The investigator will study the geometric and number theoretic properties of these power series. Moreover, in earlier work of the investigator and his collaborators the Jones polynomial was interpreted as a state sum over subgraphs of a graph, embedded on an oriented surface, that is assigned to each knot diagram. The relation of the genera of those graphs and their subgraphs to properties of the colored Jones polynomial will be studied.
When studying objects that are embedded in three dimensional space, e.g. knots, via their projections on a plane information about the original object is lost and additional information is needed to indicate which arc of the knot is farther away from the projection plane. By projecting on other surfaces more information about the original object can be preserved. These projections will be used to gain understanding of the topological and geometrical properties of knot invariants like the Jones polynomial.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
Knots and links are important objects in a variety of areas in mathematics and physics. When studying knots that are embedded in three-dimensional space via their projections on a plane, information about the original knot is lost and additional information is needed to indicate, for example, which arc of the knot is farther away from the projection plane. By projecting knots on other surfaces, such as tori, i.e.surfaces of objects that are shaped like donuts, more information about the original knot can be preserved. The project studied various characteristics of a knot, and it determined how natural projections on the plane or other surfaces can recover those characteristics. As an example, one can measure in a well-defined way the volume of the surrounding space of a knot. The project showed that data that can be read from the projection of a knot onto the plane gives an upper bound on that volume. The project also showed how the property that a knot admits certain projections on a torus affects measures that are assigned to the knot.
The project influenced the work of multiple undergraduate and graduate students. The undergraduate students went on to Ph.D. programs in Mathematics or in Applied Mathematics at leading universities. It also influenced the outreach activities of the PI to increase the interest of high school students in STEM areas.
Last Modified: 02/08/2018
Modified by: Oliver T Dasbach
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