Background: Augmentation Varieties in Knot Contact Homology

Note to the reader. ¹

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Background

Knot contact homology, introduced by Lenhard Ng, ^{2 3} is an invariant of a knot ⁴ which, in a sense, has a non-linear history. The invariant was originally constructed by Ng to be a computable example of something coming from symplectic field theory (read about the related drama), but Ng’s construction and his proof of it being well-defined were independent from symplectic geometry. ⁵ The invariant itself is an unwieldy algebraic object (specifically: a non-commutative, differential graded algebra), considered up to an equivalence relation that is not a widely familiar (called: stable tame isomorphism). In a way, you might think of it as “too much to work with.” A different object, that depends on it, and is easier to study, is called the augmentation variety — the common solution set of a collection of certain polynomial equations.  You can write down these equations fairly easily when given a way of representing the knot. (Much to the algebraic geometer’s horror, this so-called variety need not be irreducible.)

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Research Output

Knot group representations and augmentations. One of my research achievements was that I clarified the relationship between the augmentation variety and another variety, one whose points are representations of the knot group into \(\text{SL}_2(\mathbb C)\)  — the knot group is the fundamental group of the complement of the knot in \(S^3\). More precisely, I found that augmentations exactly correspond to a special type of \(\text{GL}_n(\mathbb C)\)-representations of the knot group.⁶

Okay, that was quite a mouthful of words and I wouldn’t blame anyone for not wanting to go down all the rabbit holes to chase down the meaning. Roughly speaking, varieties (follow the link variety) are important in a lot of areas of math. I clarified a correspondence between a variety that symplectic & contact geometers care about to a variety that topologists care about, those that study 3-manifolds.

Papers:	Knot contact homology and representations of knot groups (2014)
	KCH representations, augmentations, and A-polynomials (2017)

The result was discovered around the time that others were finding out that “augmentations are sheaves,” and gave support to a big conjecture that related the symplectic geometry constructions to quantum invariants of knots. The conjectured relationship had a rough justification through some ideas from string theory. It appears that this conjecture has now been placed on solid mathematical footing by Ekholm and Shende.⁷

Augmentation rank. As a byproduct of my work on the connection between augmentations and representations of the knot group, I found a new numerical invariant of knots, called the augmentation rank. The augmentation rank of a knot provides a way to study a research question (Problem 11 from Rob Kirby’s list) that has been open for over 40 years — Is the bridge number of any knot (in \(S^3\)) the same as the number of meridians needed to generate its knot group? Using the augmentation rank, I made some progress towards an answer to this question. I also worked with an undergraduate student to show that, when the numbers are equal, that fact does not change under a commonly-studied operation on knots (called taking a satellite of the knot).

Papers:

Augmentation rank of satellites with braid pattern (with D.Hemminger\(^\ast\), 2016)

Branched double-covers and \(\text{SL}_2(\mathbb C)\)-characters. By focusing on a particular type of augmentation, one that has a natural symmetry in the context of the definition of Knot Contact Homology, I found a strong connection between those augmentations and characters of \(\text{SL}_2(\mathbb C)\) representations ⁸ of the fundamental group of the branched double-cover over the knot (one of the 3-manifolds that can be constructed from the knot).

Papers:

Character varieties of knot complements and branched double-covers via the cord ring (under review, 2019).

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future work

Bridge number vs. Number of Meridians.  As mentioned above, the question of the bridge number of a knot versus the number of needed meridians to generate its knot group has been around for a while, and the augmentation rank can help to study this question.  Since the papers that discussed the augmentation rank were written (Knot contact homology and representations of knot groups and Augmentation rank of satellites with braid pattern), the question has gained more attention.  Partial progress towards an answer has been made by other researchers since. ⁹

While I found in my work that augmentation rank cannot give a complete answer to this question, it does have promising properties that could allow one to attack the question from a kind of “macroscopic” approach. ¹⁰ Being able to do so would involve proving that a certain polynomial system of equations must have a solution.  Doing this is a rather hard problem, in general.  However, the polynomial system involved in this case is quite special, and experimental evidence suggests that some assumptions (which have a natural interpretation from the knot theory side) could keep enough control on the system to do the trick.

Characters of branched double-covers and “simple” hyperbolic manifolds.  A recent paper of Sivek and Zentner ¹¹ studied a class of 3-manifolds that are interesting because they are as simple as possible from the perspective of instanton gauge theory (one type of field theory in physics).  From that study, those which are the most challenging to classify are the ones that can be given a hyperbolic geometric structure.  One attempt the authors made to understand these manifolds (those with a hyperbolic structure) involved focusing on branched double-covers over knots.  As it turns out, two of the examples that featured in my Character varieties… paper were the only examples they could find that were hyperbolic and simple from this perspective!  These are the branched double-covers over the \(8_{18}\) knot and the \(10_{109}\) knot.

Two possible projects present themselves from this.  The first relates to the branched double-cover over the \(8_{18}\) knot.  This 3-manifold is known to have a fundamental group that is isomorphic to the Fibonacci group \(F(2,8)\), that has presentation

\(F(2,8) = \langle x_0,x_1,\ldots,x_7\ |\ x_ix_{i+1} \equiv x_{i+2}\ \forall\ i\in\mathbb Z/8\mathbb Z\rangle\).

The results mentioned imply that any homomorphism to the classical group \(SU(2)\), that is  \(F(2,8) \to SU(2)\), must have a cyclic image.  As Sivek and Zentner mention in the paper, it would be interesting to see a direct proof of this fact.

A second project that comes to mind involves extending the search that Sivek and Zentner begin.  They only find these two branched double-covers that are both hyperbolic and simple in this gauge theoretic way.  Their search went up to a certain complexity of knots, but a smart approach might be able to push that boundary farther.  With luck, this search would present a way to understand when such hyperbolic manifolds are simple in this gauge theoretic sense.