TY - JOUR
T1 - Generalizing the relation between the Kauffman bracket and Jones polynomial
AU - Kaiser, Uwe
N1 - Publisher Copyright:
© 2023 World Scientific Publishing Company.
PY - 2023
Y1 - 2023
N2 - We generalize Kauffman's famous formula defining the Jones polynomial of an oriented link in 3-space from his bracket and the writhe of an oriented diagram [L. Kauffman, State models and the Jones polynomial, Topology 26(3) (1987) 395-407]. Our generalization is an epimorphism between skein modules of tangles in compact connected oriented 3-manifolds with markings in the boundary. Besides the usual Jones polynomial of oriented tangles we will consider graded quotients of the bracket skein module and Przytycki's q-analog of the first homology group of a 3-manifold [J. Przytycki, A q-analogue of the first homology group of a 3-manifold, in Contemporary Mathematics, Vol. 214 (American Mathematical Society, 1998), pp. 135-144]. In certain cases, e.g., for links in submanifolds of rational homology 3-spheres, we will be able to define an epimorphism from the Jones module onto the Kauffman bracket module. For the general case we define suitably graded quotients of the bracket module, which are graded by homology. The kernels define new skein modules measuring the difference between Jones and bracket skein modules. We also discuss gluing in this setting.
AB - We generalize Kauffman's famous formula defining the Jones polynomial of an oriented link in 3-space from his bracket and the writhe of an oriented diagram [L. Kauffman, State models and the Jones polynomial, Topology 26(3) (1987) 395-407]. Our generalization is an epimorphism between skein modules of tangles in compact connected oriented 3-manifolds with markings in the boundary. Besides the usual Jones polynomial of oriented tangles we will consider graded quotients of the bracket skein module and Przytycki's q-analog of the first homology group of a 3-manifold [J. Przytycki, A q-analogue of the first homology group of a 3-manifold, in Contemporary Mathematics, Vol. 214 (American Mathematical Society, 1998), pp. 135-144]. In certain cases, e.g., for links in submanifolds of rational homology 3-spheres, we will be able to define an epimorphism from the Jones module onto the Kauffman bracket module. For the general case we define suitably graded quotients of the bracket module, which are graded by homology. The kernels define new skein modules measuring the difference between Jones and bracket skein modules. We also discuss gluing in this setting.
KW - 3-manifold
KW - homology
KW - Skein module
UR - http://www.scopus.com/inward/record.url?scp=85165127873&partnerID=8YFLogxK
U2 - 10.1142/S0218216523400102
DO - 10.1142/S0218216523400102
M3 - Article
AN - SCOPUS:85165127873
SN - 0218-2165
JO - Journal of Knot Theory and its Ramifications
JF - Journal of Knot Theory and its Ramifications
M1 - 2340010
ER -