[Paper Review] Framed knot contact homology

Lenhard Ng|ArXiv.org|Jul 6, 2004

Geometric and Algebraic Topology15 references4 citations

TL;DR

This paper introduces framed knot contact homology, a new invariant for framed knots in $S^3$ and more general manifolds, defined over the ring $\mathbb{Z}[\lambda^{\pm 1}, \mu^{\pm 1}]$. It extends previous knot contact homology by incorporating framing data, yielding a differential graded algebra (DGA) whose homology—framed knot contact homology—distinguishes the unknot, mutants, and knots with identical $A$-polynomials, and naturally produces a two-variable polynomial invariant related to the $A$-polynomial.

ABSTRACT

We extend knot contact homology to a theory over the ring $\mathbb{Z}[λ^{\pm 1},μ^{\pm 1}]$, with the invariant given topologically and combinatorially. The improved invariant, which is defined for framed knots in $S^3$ and can be generalized to knots in arbitrary manifolds, distinguishes the unknot and can distinguish mutants. It contains the Alexander polynomial and naturally produces a two-variable polynomial knot invariant which is related to the $A$-polynomial.

Motivation & Objective

• To extend knot contact homology from the base ring $\mathbb{Z}$ to the larger ring $\mathbb{Z}[\lambda^{\pm 1}, \mu^{\pm 1}]$, incorporating framing information.
• To define a topological and combinatorial invariant for framed knots in $S^3$ and more general 3-manifolds using a differential graded algebra (DGA).
• To show that the resulting framed knot contact homology distinguishes the unknot and mutant knots, including those with identical $A$-polynomials.
• To establish a connection between the cord algebra and classical invariants such as the Alexander polynomial and the $A$-polynomial.
• To extend the invariant to virtual and welded knots, and to 2-knots in $\mathbb{R}^4$, demonstrating its broader topological applicability.

Proposed method

• Define the framed knot DGA over $\mathbb{Z}[\lambda^{\pm 1}, \mu^{\pm 1}]$ using a braid or diagram of the knot, with generators corresponding to Reeb chords and differential determined by holomorphic disk counts.
• Construct the cord algebra as the degree 0 part of the framed knot DGA, which captures topological information via paths on a parallel copy of the knot.
• Use the DGA's linearization to recover the Alexander polynomial, and define the augmentation polynomial as a two-variable invariant related to the $A$-polynomial.
• Establish invariance under Reidemeister moves and stable tame isomorphism, ensuring the homology is a knot invariant.
• Extend the DGA and cord algebra to welded knots by defining a homomorphism on the welded braid group $\text{WB}_n$ into the automorphism group of the algebra.
• Generalize the cord algebra to 2-knots in $\mathbb{R}^4$ by using the fundamental group of the knot complement and peripheral structure, showing it distinguishes spun knots from the unknotted 2-sphere.

Experimental results

Research questions

• RQ1 Can knot contact homology be extended to incorporate framing data via a larger coefficient ring, and does this yield a stronger invariant?
• RQ2 Does the framed knot DGA distinguish the unknot from all other knots, and can it detect mutant pairs?
• RQ3 How is the cord algebra related to the $A$-polynomial and classical invariants like the Alexander polynomial?
• RQ4 Can the framed knot DGA and cord algebra be generalized to virtual, welded, and higher-dimensional knots?
• RQ5 Is the cord algebra a complete knot invariant, or does it detect subtle invariants beyond the $A$-polynomial?

Key findings

• The cord algebra of a knot, derived from the framed knot DGA, distinguishes the unknot from all other knots, as shown by the nontriviality of the $A$-polynomial.
• The augmentation polynomial, derived from the cord algebra, contains a factor equal to the $A$-polynomial, and its nontriviality implies the cord algebra detects nontrivial knotting.
• The cord algebra distinguishes mutant pairs such as the Kinoshita–Terasaka knot and its Conway mutant, even when their $A$-polynomials coincide.
• The cord algebra of the spun trefoil in $\mathbb{R}^4$ is $\mathbb{Z}[\mu^{\pm 1}][x]/((x - \mu - 1)(\mu x + 1))$, while that of the unknotted 2-sphere is $\mathbb{Z}[\mu^{\pm 1}]$, showing it detects nontrivial 2-knots.
• The framed knot DGA extends to welded knots via a homomorphism on the welded braid group, and the cord algebra remains invariant under welded Reidemeister moves.
• The cord algebra for 2-knots in $\mathbb{R}^4$ is isomorphic to the cord algebra of the corresponding long knot, and it distinguishes spun knots from the unknotted 2-sphere.