Mathematics – Geometric Topology
Scientific paper
2011-02-07
Mathematics
Geometric Topology
9 pages, 4 figures
Scientific paper
Filtered chain complexes and their associated spectral sequences and exact triangles are standard tools of Homological Algebra that have found numerous applications in the deep theories of categorification. A major advantage of Knot Homology is its functorial nature, which numerical TQFT invariants (such as the Jones polynomial) lack. We argue that the notions of {\em holonomy} and {\em $q$-holonomy} play the role of functoriality in TQFT. We illustrate this principle with two independent results. The first result concerns the behavior of the $A$-polynomial and quantum invariants (such as the Jones or Alexander polynomials) under 1-parameter fillings of a 2-cusped manifold. In the $A$-polynomial case, it divides a holonomic sequence of 2-variables, and in the quantum invariant case, it is a holonomic sequence of one variable. In our second result, we prove that the Newton polygon $N_n$ of a holonomic sequence of polynomials is quasi-linear, complementing results of \cite{CW} and \cite{Ga4}.
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