# Alternating knots are rare

Harrison Chapman
Colorado State University

hchapman.github.io/talks/altknots

OIST Geometry and Topology of Manifolds Group
18th February 2019

## Alternating links

A knot or link type is alternating if it admits a diagram whose crossings alternate between over- and underpasses.

## Thistlethwaite '98

Alternating link types are exponentially rare among links.

Let $$\altlinkclass$$ be the set of all alternating link types,
with counts $$\altlinkcount$$ indexed by crossing number

If $$L_n$$ is the number of all link types, the theorem says; $\lim_{n\to\infty}{L_n^{1/n}} > \lim_{n\to\infty}{\altlinkcount^{1/n}}$

## Question (Classical)

Does the same result hold for alternating knot types?

## Alternating knots should be rare

Data of Hoste, Thistlethwaite, and Weeks '98 suggests
alternating knot types are rare

$$n$$ # Knots # Alternating % Alternating
633100%
777100%
8211886%
$$\vdots$$
15253 29385 26351%
161 388 705379 79938%

## Thistlethwaite '98

Alternating link types are exponentially rare among all link types.

## Weak & Insufficient Reasoning

Every $$n$$-crossing link diagram has $$2^n$$ crossing assignments, of which precisely two are alternating.

Must pay very close attention to equivalence classes of diagrams

## Simplification from Combinatorics

Objects with fewer symmetries are easier to enumerate
Sundberg and Thistlethwaite '98

Tangles can be closed to links by adding an additional crossing.

## Tait Flyping Conjecture

Ingredient 1: The Tait Flyping Conjecture holds for alternating links;

## Menasco and Thistlethwaite '91

Any two alternating link or tangle diagrams are related by a sequence of flypes if and only if they represent the same link or tangle type.

Zinn-Justin and Zuber '10

## Tait Flyping Conjecture & Knots

Tait Flyping Conjecture still holds for alternating knots

## Menasco and Thistlethwaite '91

Any two alternating knot diagrams are related by a sequence of flypes if and only if they represent the same knot type.

## Enumeration of alternating links

Ingredient 2: Alternating link and tangle types can be enumerated;

## Sundberg and Thistlethwaite '98

The numbers $$\tanglecount$$ of alternating tangle types have a well-understood generating function (g.f.), $\tanglegenfun{z} = \sum_{\color{tangle}\substack{\text{alternating}\\\text{link types } {{\color{black}L}}}}{z^{\mathop{cr}(L)}} = \sum_{n}^{\infty} {\tanglecount z^n}.$

## Sundberg and Thistlethwaite '98

The counts $$\tanglecount$$ grow asymptotically as,
$\tanglecount \sim \frac{3 \,c_1}{4 \sqrt \pi} n^{-5/2} \altlinkgrowth^{n-3/2},$
for known constants $$c_1$$ and $$\altlinkgrowth$$.

Zinn-Justin and Zuber '02 reprove this using
techniques from random matrix theory

## The generating function $$\tanglegenfun{z}$$

The g.f. of alternating tangle types $$\tanglegenfun{z}$$ satisfies: $\tanglegenfun{z} = \algtanglegenfun{z}{\bpotanglegenfun{\tanglegenfun{z}}}$ where;
• $$z$$ marks crossings,
• $$\zeta$$ marks slots for tangles (or templates)
• $$\bpotanglegenfun{\zeta}$$ is the g.f. of basic polyhedral templates
• $$\algtanglegenfun{z}{\zeta}$$ is the g.f. of flype-equivalence classes of algebraic templates,

## Basic polyhedral templates

Sundberg and Thistlethwaite '98 Basic polyhedral templates, counted by $$\bpotanglegenfun{\zeta}$$

Bijection with rooted $$c$$-nets studied by Tutte '63: $\bpotanglegenfun{\zeta} = {\scriptsize \frac{1}{2(\zeta+2)^3}\left( (1-4\zeta)^{3/2} + (2\zeta^2 - 10\zeta - 1) \right) - \frac{2}{1+\zeta} - \zeta + 2}$ (formula due to Sundberg and Thistlethwaite)

## Algebraic templates

Sundberg and Thistlethwaite '98 Flype equivalence classes of algebraic templates, counted by $$\algtanglegenfun{z}{\zeta}$$

## Key observation

Flyping can only occur within an algebraic template

## Algebraic templates

Sundberg and Thistlethwaite '98

Admit recursive decomposition into horizontal templates: $\algtanglegenfun{z}{\zeta} = \hortanglegenfun{z}{\zeta} + \hortanglegenfun{z}{\zeta} + z - \zeta,$ where $$\hortanglegamma$$ satisfies $\hortanglegamma = z\algtanglegenfun{z}{\zeta} + \frac{\hortanglegamma^2}{1-\hortanglegamma} + \zeta$

## Counts of tangles and links

Counts of alternating tangle types $$\tanglecount$$ and
alternating link types $$\altlinkcount$$ are related;

## Counts

$\frac{\tanglecountdec}{8(2n-3)} \le \altlinkcount \le \frac{\tanglecountdec}2$

## Exponential growth rates

$\lim_{n\to\infty}{\altlinkcount^{1/n}} = \lim_{n\to\infty}{\tanglecount^{1/n}} = \altlinkgrowth$

## Enumeration of alternating knots

Little is known about the counts $$\altknotcount$$ of alternating knot types;

## Schaeffer and Zinn-Justin '04

For some constants $$\tau$$ and $$\gamma$$, $\altknotcount \sim C \tau^n n^{\gamma-3}$

## Currently

The g.f. $$\sum{\altknotcount}z^n$$ is not well understood and the radius of convergence is not known

## Thistlethwaite '98

The set of tangles $$\superaltlinkclass$$ generated by introducing the above (non-alternating) tangle into the set of algebraic templates still satisfies the flyping conjecture

## Rarity of alternating links

The g.f. of this superclass of tangle types is obtained by replacing $\algtanglegenfun{z}{\zeta} \quad \text{with} \quad \beta(z,\zeta) = \algtanglegenfun{z}{\zeta} + z^{13}.$ This g.f. has strictly smaller radius of convergence than that of alternating tangle types $$\tanglegenfun{z}$$, so:

## Thistlethwaite '98

Alternating tangle (and link) types are exponentially rare.

## Observation

The above tangle can be found in knot types, and is contained in a superclass $$\superaltknotclass$$ of knot diagrams which still satisfy the flyping conjecture

Cannot quantify the ocurrences with g.f's
(not enough understanding of knots)

## We have:

Alternating knot types of $$\altknotclass$$ satisfy the flyping conjecture, as do a superclass $$\superaltknotclass$$.

## We lack:

Enumeration of alternating knot types, or knots in $$\superaltknotclass$$.

## Pattern theorem

Pattern theorems quantify the density of structure:

## C. '18

Let $$P$$ be a reduced alternating tangle admitting no interior flypes that may be found in alternating knot diagrams.

There exist constants $$\occurconst > 0$$, and $$N \in \NN$$ such that for all $$n \ge N$$, all but exponentially few alternating knot types contain $$\ge \occurconst n$$ copies of $$P$$.

## Pattern theorem implies rarity

$$R =$$ $$\overline R =$$

1. There is $$\occurconst$$ so that the tangle $$R$$ occurs $$\occurconst n$$ times in almost all alternating knot types.
2. In $$\superaltknotclass$$, the tangles $$R$$ and $$\overline R$$ are equiprobable.
3. Knot types which contain $$\overline R$$ are non-alternating.
4. To each alternating knot type in $$\altknotclass$$, there are at least $$2^{\occurconst n}-1$$ more non-alternating knot types in $$\superaltknotclass$$.

We conclude:

## C. '18

Alternating knot types are exponentially rare among all knot types.

## Requirements for a pattern theorem

Pattern theorem requires a way of adding
a pattern to alternating knot types that:

1. Only produces alternating knot types
2. Has a linear number ($$\propto n$$) of attachment sites, and
3. Can be un-done, yielding the original knot type

## The idea

$\text{# lone crossings} + \sum_{\substack{\text{all horiz.}\\\text{tangles }\gamma}}{\text{# sites in }\gamma} = n$

## Millett and Jablan '09

The probability that a minimal prime knot diagram contains a trefoil knotted segment goes to one as the crossing number goes to infinity

## C. '18

The probability that an alternating knot type has a minimal prime diagram containing a trefoil knotted segment goes to one as the crossing number goes to infinity

## Proof

The following pattern introduces a trefoil segment:

## Classical

The crossing number of knots is additive with respect to connected sum.

Of 1,388,705 prime knot types of 16 crossings,
1,388,694 are hyperbolic, leading to:

## Adams '94

Almost all prime knot types are hyperbolic.

## Malyutin '16

Either of these conjectures contradicts the other.

It's not unreasonable that Adams's conjecture is false and that current data is only representative of small crossing number

## Conjecture

Can we disprove Adams's conjecture, possibly using pattern-theoretic methods?

## Take-aways & conclusion

1. Alternating links have the flype conjecture and a complete enumeration
2. Alternating knots share the result on flypes but lack precise enumeration
3. Absent enumerations, pattern theorems can quantify presence of structure:
• Rarity of alternating knots
• Certainty of subknotting
4. Pattern theorems can likely be proven for other classes in topology, geometry, etc...

# Thank you!

C. 2018. On the structure and scarcity of alternating knots. Submitted. arXiv: 1804.09780