On the quantum sl_2 invariants of knots and integral homology spheres

By Kazuo Habiro. 2002

83 citations recorded by the source corpus.

We will announce some results on the values of quantum sl2 invariants of knots and integral homology spheres. Lawrence’s universal sl2 invariant of knots takes values in a fairly small subalgebra of the center of the h-adic version of the quantized enveloping algebra of sl2 . This implies an integrality result on the colored Jones polynomials of a knot. We define an invariant of integral homology spheres with values in a completion of the Laurent polynomial ring of one variable over the integers which specializes at roots of unity to the Witten-Reshetikhin-Turaev invariants. The definition of our invariant provides a new definition of Witten-ReshetikhinTuraev invariant of integral homology spheres. AMS Classification 57M27; 17B37

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/017b28af4f3ef6de1830d1f70ecca48fc6b6dc1f

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Citations in the corpus

Fivebranes and 3-manifold homology (2016, background): “he a = 0 term) is an element of Q[[q 1]] and is known as Ohtsuki series (perturbative RTW invariant) in the math literature (see e.g. [86]). For integer homology spheres, it is an element of Z[[q 1]] [87,88]. Let us stress an important dierence between decompositions (6.75) and (6.32) of the Chern-S”
A unified Witten–Reshetikhin–Turaev invariant for integral homology spheres (2006, background): “T U R A EV IN VA R IA N T FO R IN T EG R A L H O M O LO G Y SP H ER ES K A ZU O H A BIRO A bstract.”
DIFFERENCE EQUATION OF THE COLORED JONES POLYNOMIAL FOR TORUS KNOT (2004, methodology): “By applying a twisting formula given in [15], we have another expression of the colored Jones polynomial in the form of cyclotomic expansion in a sense of [8] as follows.”
Bottom tangles and universal invariants (2005, background): “1 q// studied in [24], which unifies the quantum g invariants at all roots of unity, as announced in [23], [67, Conjecture 7.”
Quantum modular forms (2010, background): “We observe in passing that the function of this example, like those of Examples 4 and 5, belongs to the Habiro ring of “analytic functions of roots of unity” [10].”
On the characteristic and deformation varieties of a knot (2003, background): “This can be converted into a recursion relation using forward shifts by: In[6]:= ForwardShifts[% ] 3 + 2 n 2 2 + n q (q - q ) SUM[n] Out[6]= ----------------------------- + 3 4 + 2 n q - q -2 - n 2 + n 2 + n > (q (q - q ) (q + q ) 4 5 + n 6 + 2 n 7 + 2 n 7 + 3 n 8 + 4 n Geometry & Topology Monograph”