[Paper Review] Longitudinal Mapping Knot Invariant for SU(2)

W. Edwin Clark, Masahico Saito|arXiv (Cornell University)|Feb 24, 2018

Geometric and Algebraic Topology11 references3 citations

TL;DR

This paper introduces the longitudinal mapping invariant for knots using SU(2) as a topological group, generalizing Eisermann's knot coloring polynomial to infinite groups. It establishes a connection between this invariant and quandle colorings of 1-tangles via generalized Alexander quandles, and computes the invariant explicitly for torus knots T(2,n), their mirrors, and the figure eight knot, showing it captures non-trivial topological information beyond the knot group.

ABSTRACT

The knot coloring polynomial defined by Eisermann for a finite pointed group is generalized to an infinite pointed group as the longitudinal mapping invariant of a knot. In turn this can be thought of as a generalization of the quandle 2-cocycle invariant for finite quandles. If the group is a topological group then this invariant can be thought of a topological generalization of the 2-cocycle invariant. The longitudinal mapping invariant is based on a meridian-longitude pair in the knot group. We also give an interpretation of the invariant in terms of quandle colorings of a 1-tangle for generalized Alexander quandles without use of a meridian-longitude pair in the knot group. The invariant values are concretely evaluated for the torus knots $T(2,n)$, their mirror images, and the figure eight knot for the group $SU(2)$.

Motivation & Objective

• To generalize Eisermann's finite group knot coloring polynomial to infinite topological groups, particularly SU(2), as a longitudinal mapping invariant.
• To provide a topological analogue of the quandle 2-cocycle invariant using homomorphisms from the knot group to SU(2) with fixed meridian image.
• To establish an equivalence between the longitudinal mapping invariant and quandle colorings of 1-tangles using generalized Alexander quandles, eliminating dependence on meridian-longitude pairs.
• To compute the invariant explicitly for key knots: torus knots T(2,n), their mirrors, and the figure eight knot.

Proposed method

• Define the longitudinal mapping invariant $\mathcal{L}_{G}^{x}(K)$ as a map from group homomorphisms $\rho: \pi_1(S^3 \setminus K) \to G$ with $\rho(m_K) = x$ to the centralizer $C(x) \cap G'$, where $G = \mathrm{SU}(2)$.
• Use the Eisermann quandle $\mathrm{Eis}(G,x)$ and its isomorphism to the generalized Alexander quandle $\mathrm{GAlex}(G', f_x)$ with $f_x(g) = x^{-1}gx$ to reframe the invariant in terms of tangle colorings.
• Leverage the lifting property of quandle coverings to relate colorings of the knot via $Q = x^G$ to colorings of the 1-tangle via $\tilde{Q} = \mathrm{Eis}(G,x)$.
• Apply the inner representation and automorphism group structure to ensure the invariant is well-defined and independent of basepoint choices.
• Use the fact that $\mathrm{SU}(2)$ is perfect (equal to its commutator subgroup) and that non-central elements generate the group to ensure the invariant is non-degenerate.
• Evaluate the invariant via explicit computation for $T(2,n)$, $m(T(2,n))$, and the figure eight knot using the tangle coloring framework and the relation $\tilde{C}(n) = (x, \mathcal{L}(C))$.

Experimental results

Research questions

• RQ1 Can the knot coloring polynomial for finite groups be extended to infinite topological groups like SU(2) in a meaningful way?
• RQ2 How does the longitudinal mapping invariant relate to quandle 2-cocycle invariants in the infinite group setting?
• RQ3 Can the invariant be computed without relying on meridian-longitude pairs in the knot group?
• RQ4 What are the explicit values of the longitudinal mapping invariant for torus knots $T(2,n)$ and the figure eight knot in $\mathrm{SU}(2)$?
• RQ5 Is the invariant strong enough to distinguish knots that are indistinguishable by the knot group or standard quandle invariants?

Key findings

• The longitudinal mapping invariant $\mathcal{L}_{\mathrm{SU}(2)}^{x}(K)$ is well-defined for $K = T(2,n)$, $m(T(2,n))$, and the figure eight knot, with values in $\mathrm{SU}(2)$.
• For the torus knot $T(2,n)$ with odd $n \geq 3$, the invariant takes values in the conjugacy class of $x$, and the image is non-trivial when $x \neq \pm 1$.
• The invariant distinguishes $T(2,n)$ from its mirror image $m(T(2,n))$ when $n$ is odd, showing it is sensitive to chirality.
• For the figure eight knot $4_1$, the invariant is non-trivial and distinct from that of torus knots, indicating it captures unique topological data.
• The invariant is equivalent to the quandle coloring map $\Psi_Q^e(K)$ for the generalized Alexander quandle $\mathrm{GAlex}(\mathrm{SU}(2)', f_x)$, providing a tangle-based computation method independent of meridian-longitude pairs.
• The construction confirms that the longitudinal mapping invariant is a topological generalization of the 2-cocycle invariant, valid even when group homomorphism coefficients are infinite.