TWIST REGIONS AND COEFFICIENTS STABILITY OF THE COLORED JONES POLYNOMIAL

Area of Science:

• Knot theory
• Algebraic topology
• Low-dimensional topology

Background:

• The Jones polynomial is a significant invariant in knot theory.
• Alternating links are a well-studied class of knots and links.
• The colored Jones polynomial is a generalization of the Jones polynomial with additional structure.

Purpose of the Study:

• To investigate the behavior of coefficients of the colored Jones polynomial for alternating links.
• To determine if these coefficients exhibit a stabilization property.
• To establish a method for generating infinite families of q-power series from link invariants.

Main Methods:

• Analyzing the colored Jones polynomial for alternating link diagrams.
• Introducing and systematically increasing twists in link diagram regions.
• Observing the coefficients of the resulting polynomials.
• Developing a parametrization for the generated q-power series.

Main Results:

• Demonstrated that the coefficients of the colored Jones polynomial for alternating links stabilize as the number of twists increases.
• Established an infinite family of q-power series derived from the colored Jones polynomial.
• Parametrized these series by the color and twist regions of the alternating link diagram.

Conclusions:

• The stabilization of coefficients provides a new perspective on the colored Jones polynomial of alternating links.
• The infinite family of q-power series offers a novel tool for studying link invariants.
• This work opens avenues for further research in quantum topology and related fields.