New York Journal of Mathematics
Volume 22 (2016) 1039-1053


Susan M. Abernathy and Patrick M. Gilmer

Even and odd Kauffman bracket ideals for genus-1 tangles

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Published: September 21, 2016
Keywords: Tangles, tangle embedding, determinants, Kauffman bracket skein module
Subject: 57M25


This paper refines previous work by the first author. We study the question of which links in the 3-sphere can be obtained as closures of a given 1-manifold in an unknotted solid torus in the 3-sphere (or genus-1 tangle) by adjoining another 1-manifold in the complementary solid torus. We distinguish between even and odd closures, and define even and odd versions of the Kauffman bracket ideal. These even and odd Kauffman bracket ideals are used to obstruct even and odd tangle closures. Using a basis of Habiro's for the even Kauffman bracket skein module of the solid torus, we define bases for the even and odd skein module of the solid torus relative to two points. These even and odd bases allow us to compute a finite list of generators for the even and odd Kauffman bracket ideals of a genus-1 tangle. We do this explicitly for three examples. Furthermore, we use the even and odd Kauffman bracket ideals to conclude in some cases that the determinants of all even/odd closures of a genus-1 tangle possess a certain divisibility.


The first author was supported as a research assistant by NSF-DMS-1311911
The second author was partially supported by NSF-DMS-1311911

Author information

Susan M. Abernathy:
Department of Mathematics, Angelo State University, ASU Station #10900, San Angelo, TX 76909, USA

Patrick M. Gilmer:
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA