Algebraic Knot Theory - A Call for Action

Abstract

We knot theorists have lots of algebraically-defined knot invariants, but they tell us just little about knots. In this manifesto we suggest a reason for the failure and the means to rectify it.

Algebraic Knot Invariants: What They Do and What They Don't

There is now a highly developed theory of knot invariants defined (or that can be defined) by algebraic means. The list contains the Alexander-Conway polynomial, the Jones polynomial and its various generalizations depending on a choice of a Lie algebra and a representation thereof, finite type invariants and the Kontsevich integral, various knot homologies and more.

These invariants are quite good at telling knots apart. While it is not known if these invariants separate knots, in practice, for the first few million knots as enumerated by computers, these invariants are either separating or they come very close to that (which one it is also depends on the precise class of invariants under consideration).

But beyond knot separation, knot theorists are interested in many other questions. Let me list just a few, with some bias in favour of the questions our still-imaginary "Algebraic Knot Theory" seems more likely to address:

Forbidden and allowed for ribbon knots

1. How many crossing changes are required to unknot a given knot $K$?
2. What is the minimal genus of a Seifert surface whose boundary is $K$?
3. Is a given link $L$ a boundary link? (That is, is there a collection of disjoint Seifert surfaces for the components of $L$?)
4. Is $K$ a ribbon knot? (Recall that a ribbon knot is a knot that bounds a disk that is allowed to have "ribbon-type" singularities but is not allowed to have "clasp-type" singularities; see the image on the right).
5. Is $K$ a slice knot? That is, does $K$ bound a singularity-free disk in the four-ball?
6. Is $K$ fibered? (See an animation by Robert Barrington Leigh).
7. Does $K$ have some symmetries?
8. Is $K$ the closure of a braid on at most 6 strands?
9. Does $K$ have a projection with less than 23 crossings?
10. Does $K$ have an alternating projection?
11. Is $K$ algebraic?

With the exception of the Alexander polynomial (and its corresponding knot homology theory), currently algebraic knot invariants say very little on these questions. And while the Alexander polynomial is useful in answering some of these questions for some knots, it is simply not as strong as we wish it could be, and is not powerful enough to answer many of these questions for many other knots.

Thus as a whole, algebraically-defined knot invariants tell as very little about the knot properties we care about.

Definability and Knotted Trivalent Graphs

Why is this so?

Take for example the property of "being a ribbon knot". This property has a lovely definition in terms of of knotted trivalent graphs and some operations between such graphs. Let ${\mathcal {R}}$ denote the collection of all ribbon knots, and let ${\mathcal {K}}(\bigcirc )$, ${\mathcal {K}}(\bigcirc \bigcirc )$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal K}(\dumbbell)} denote the spaces of knotted circles, knotted pairs of circles, and knotted embeddings of the dumbbell graph Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\dumbbell)} into ${\mathbb {R} }^{3}$. Let $d$ and $u$ be the operations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d:{\mathcal K}(\dumbbell)\to{\mathcal K}(\bigcirc\bigcirc)} and Failed to parse (unknown function "\dumbbell"): {\displaystyle u:{\mathcal K}(\dumbbell)\to{\mathcal K}(\bigcirc)} defined by deleting and "unziping" the middle edge of the dumbbell graph, as illustrated in the following figure:

Finally, standing by themselves, let $\bigcirc \bigcirc$ denote the two unknotted and unlinked circles.

Almost Theorem. The collection of all ribbon knots is equal to the collection of all unzips of knotted dumbbell whose middle deletion is the two components unlink. In symbols, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal R}=\{u\gamma:\gamma\in{\mathcal K}(\dumbbell)\mbox{ and }d\gamma=\bigcirc\bigcirc\}} .

Semi Proof. The inclusion $\subset$ is very easy. The other inclusion is a bit harder and also a bit false. But it is not hard to fix, and for the purpose of this quick discussion, it is better to pretend that it is true.

Let us get back now to the failure of most algebraic knot invariants to detect the property of being ribbon. The property of being ribbon is best defined in terms of knotted dumbbells, edge deletions and edge unzips. But most algebraic knot invariants were hardly studied from this perspective. Only a few of those were generalized to knotted graphs, and even those were not studied with the operations $d$ and $u$ in mind.

To summarize and somewhat generalize: Many of the properties of interest in knot theory are definable using simple formulae (such as $\{u\gamma :d\gamma =\bigcirc \bigcirc \}$) involving knotted trivalent graphs, edge deletions and edge unzips (and also a certain binary "connected sum" operation which we ignore here). But the algebraically defined knot invariants we now know are largely incompatible with those operations and so we can hardly expect them to yield useful information about our properties of interest.

What We Seek - An "Algebraic Knot Theory"

To remedy this we seek more than merely knot invariants - we seek an Algebraic Knot Theory, which sees not only knots but also knotted trivalent graphs, and more importantly, which sees the edge deletion and edge unzip operations. More precisely, an Algebraic Knot Theory should consist of the following stucture:

• To every trivalent graph $\Gamma$ some algebraically-defined space ${\mathcal {A}}(\Gamma )$ and a knotted graph invariant $Z_{\Gamma }:{\mathcal {K}}(\Gamma )\to {\mathcal {A}}(\Gamma )$ (so $Z_{\Gamma }$ is an invariants of knottings of $\Gamma$ with values in ${\mathcal {A}}(\Gamma )$).
• For every pair $(\Gamma ,e)$, where $e$ is an edge of $\Gamma$, there is a pair of operations $d_{e}$ and $u_{e}$ defined on ${\mathcal {K}}(\Gamma )$ which delete (for $d_{e}$) or unzip (for $u_{e}$) the edge $e$ in any knotting of $\Gamma$. With the obvious definition of $d_{e}\Gamma$ and of $u_{e}\Gamma$, $d_{e}$ and $u_{e}$ take values in ${\mathcal {K}}(d_{e}\Gamma )$ and ${\mathcal {K}}(u_{e}\Gamma )$, respectively. In an algebraic knot theory there should be corresponding operations $d_{e}:{\mathcal {A}}(\Gamma )\to {\mathcal {A}}(d_{e}\Gamma )$ and $u_{e}:{\mathcal {A}}(\Gamma )\to {\mathcal {A}}(u_{e}\Gamma )$ that commute with $Z$ in the obvious sense.
• There should also be binary "connected sum" operations defined on pairs of spaces $({\mathcal {A}}(\Gamma _{1}),{\mathcal {A}}(\Gamma _{2}))$, and they too should be compatible with $Z$. But in the interest of simplicity, we omit the details here.