A dequantized metaplectic knot invariant

Abstract

Let $K\subset S^3$ be a knot, $X:= S^3\setminus K$ its complement, and $\mathbb{T}$ the circle group identified with $\mathbb{R}/\mathbb{Z}$. To any oriented long knot diagram of $K$, we associate a quadratic polynomial in variables bijectively associated with the bridges of the diagram such that, when the variables projected to $\mathbb{T}$ satisfy the linear equations characterizing the first homology group $H_1(\tilde{X}_2)$ of the double cyclic covering of $X$, the polynomial projects down to a well defined $\mathbb{T}$-valued function on $T^1(\tilde{X}_2,\mathbb{T})$ (the dual of the torsion part $T_1$ of $H_1$). This function is sensitive to knot chirality, for example, it seems to confirm chirality of the knot $10_{71}$. It also distinguishes the knots $7_4$ and $9_2$ known to have identical Alexander polynomials and the knots $9_2$ and K11n13 known to have identical Jones polynomials but does not distinguish $7_4$ and K11n13.