Presentation Title

P-15 Classifying Links Obtained by Strong Fusion

Presenter Status

Assistant Professor, Mathematics

Preferred Session

Poster Session

Start Date

25-10-2019 2:00 PM

Presentation Abstract

Band fusion modifies a link by fusing together two components of the link with a band. The effects of fusion on link invariants has been studied and the fusion pathways between links have been tabulated. We extended this work with an analysis of the effects of strong fusion, a modified form of fusion that preserves the number of components of a link by introducing an unknotted component around the fusion band. In particular, we determine the exact effect of strong fusion on several link invariants including the Alexander polynomial, Jones polynomial, and their generalization of the HOMFLY polynomial. We determine values for which of these polynomials remain unchanged by strong fusion. We also studied the Q-polynomial, signature, and the xi invariant. The work on the orientation independent invariants, the Q-polynomial and the xi invariant, is original; the effect of strong fusion on signature and HOMFLY has been studied by Kaiser (’91), but we extend and compliment his results. We also used link invariants⁠—including knottedness, linking number, and the dichromatic invariants of J. Hoste and M. Kidwell⁠—to classify which links are the result of strong fusion, tabulating all such links with up to 9 crossings.

Acknowledgments

Faculty Research Grant, Office of Research and Creative Scholarship, Andrews University
Undergraduate Research Scholarship, Office of Research and Creative Scholarship, Andrews University

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Oct 25th, 2:00 PM

P-15 Classifying Links Obtained by Strong Fusion

Band fusion modifies a link by fusing together two components of the link with a band. The effects of fusion on link invariants has been studied and the fusion pathways between links have been tabulated. We extended this work with an analysis of the effects of strong fusion, a modified form of fusion that preserves the number of components of a link by introducing an unknotted component around the fusion band. In particular, we determine the exact effect of strong fusion on several link invariants including the Alexander polynomial, Jones polynomial, and their generalization of the HOMFLY polynomial. We determine values for which of these polynomials remain unchanged by strong fusion. We also studied the Q-polynomial, signature, and the xi invariant. The work on the orientation independent invariants, the Q-polynomial and the xi invariant, is original; the effect of strong fusion on signature and HOMFLY has been studied by Kaiser (’91), but we extend and compliment his results. We also used link invariants⁠—including knottedness, linking number, and the dichromatic invariants of J. Hoste and M. Kidwell⁠—to classify which links are the result of strong fusion, tabulating all such links with up to 9 crossings.