# A dequantized metaplectic knot invariant

## TL;DR

This paper introduces a new quadratic polynomial invariant for knots derived from their diagrams, which is sensitive to knot chirality and can distinguish certain knots with identical classical polynomial invariants.

## Contribution

It constructs a dequantized metaplectic knot invariant based on quadratic polynomials associated with knot diagrams, extending the toolkit for knot distinction beyond classical invariants.

## Key findings

- The invariant detects knot chirality, exemplified by the knot 10_71.
- It distinguishes knots 7_4 and 9_2 with identical Alexander polynomials.
- It differentiates knots 9_2 and K11n13 with the same Jones polynomial.

## 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.