Symmetric Union Diagrams and Refined Spin Models
Canadian mathematical bulletin, Tome 62 (2019) no. 3, pp. 451-468
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An open question akin to the slice-ribbon conjecture asks whether every ribbon knot can be represented as a symmetric union. Next to this basic existence question sits the question of uniqueness of such representations. Eisermann and Lamm investigated the latter question by introducing a notion of symmetric equivalence among symmetric union diagrams and showing that non-equivalent diagrams can be detected using a refined version of the Jones polynomial. We prove that every topological spin model gives rise to many effective invariants of symmetric equivalence, which can be used to distinguish infinitely many Reidemeister equivalent but symmetrically non-equivalent symmetric union diagrams. We also show that such invariants are not equivalent to the refined Jones polynomial and we use them to provide a partial answer to a question left open by Eisermann and Lamm.
Collari, Carlo; Lisca, Paolo. Symmetric Union Diagrams and Refined Spin Models. Canadian mathematical bulletin, Tome 62 (2019) no. 3, pp. 451-468. doi: 10.4153/S0008439518000115
@article{10_4153_S0008439518000115,
author = {Collari, Carlo and Lisca, Paolo},
title = {Symmetric {Union} {Diagrams} and {Refined} {Spin} {Models}},
journal = {Canadian mathematical bulletin},
pages = {451--468},
year = {2019},
volume = {62},
number = {3},
doi = {10.4153/S0008439518000115},
url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439518000115/}
}
TY - JOUR AU - Collari, Carlo AU - Lisca, Paolo TI - Symmetric Union Diagrams and Refined Spin Models JO - Canadian mathematical bulletin PY - 2019 SP - 451 EP - 468 VL - 62 IS - 3 UR - http://geodesic.mathdoc.fr/articles/10.4153/S0008439518000115/ DO - 10.4153/S0008439518000115 ID - 10_4153_S0008439518000115 ER -
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