Topology Seminar- Knots with cyclotomic Jones polynomials

Greater Washington Topology Seminar

The following is a list of confirmed speakers and dates for the Greater (George) Washington Topology Seminar seminar: 

1. March 5, 2021: Maciej Mroczkowski (University of Gdańsk)

2. March 12, 2021: Victoria Lebed (Université de Caen - Normandie )

3. March 26, 2021: Dirk Schütz (University of Durham)

4. April 2, 2021: Peter Samuelson (University of California, Riverside)

5. April 9, 2021: Krzysztof Putyra (Universität Zürich)

6. April 16, 2021: Michal Jablonowski (University of Gdańsk)

7. April 23, 2021: Nathan Geer (Utah State University)

8. May 7, 2021: William Goldman (University of Maryland, College Park) 


The next talk will be by Maciej Mroczkowski and will take place on March 5, 2021 from 1pm to 2pm EST. 

 

Title: Knots with cyclotomic Jones polynomials

 

Abstract: We construct infinite families of knots with very special Jones polynomials: their coefficients are alternatingly $1$ and $-1$, starting and ending with $1$. This extends the well known examples of the Jones polynomials of the knots $4_1$ and $9_{42}$. Their spans are equal to $2n^2-2$ and $n^2+n-2$ for any $n\ge 2$. Such polynomials have Mahler measure equal to $1$ and they are products of cyclotomic polynomials. In particular, we show that infinitely many roots of unity occur as zeros of Jones polynomials. We also show that some infinite families of roots of unity cannot occur as zeros of Jones polynomials. We also study some properties of the constructed knots: they are non alternating (except for $4_1$) and they are $(1,1)$-knots (hence prime).

 

The Zoom information is listed below: 

Topic: GW Topology Seminar 

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