On the Volume Conjecture for Cables of Knots

TL;DR

This paper proves the volume conjecture for certain cables of the figure 8 knot, revealing parity-dependent behaviors and conditions under which the conjecture holds or fails.

Contribution

It establishes the volume conjecture for (m,2)-cables of the figure 8 knot when m is odd and explores parity effects for general knots, clarifying when the conjecture is valid.

Findings

Proved the volume conjecture for (m,2)-cables of the figure 8 knot with odd m.

Showed that the limit depends on the parity of the color for even m.

Identified conditions where the conjecture fails or holds based on color restrictions.

Abstract

We establish the volume conjecture for (m,2)-cables of the figure 8 knot, when m is odd. For (m,2)-cables of general knots where m is even, we show that the limit in the volume conjecture depends on the parity of the color (of the Kashaev invariant). There are many cases when the volume conjecture for cables of the figure 8 knot is false if one considers all the colors, but holds true if one restricts the colors to a subset of the set of positive integers.