Information theory via knot theory

[Twodogs]

A mathematician friend told me that the major tenets of information theory can be established through knot theory. Is that the case?
Thanks.

 

[fresh_42]

A mathematician friend told me that the major tenets of information theory can be established through knot theory. Is that the case?
Thanks.

An interesting idea!

Here is what I found searching for it.

Knot Theory and Error-Correcting Codes (Eindhoven, May 2024)

This paper builds a novel bridge between algebraic coding theory and mathematical knot theory, with applications in both directions. We give methods to construct error-correcting codes starting from the colorings of a knot, describing through a series of results how the properties of the knot translate into code parameters. We show that knots can be used to obtain error-correcting codes with prescribed parameters and an efficient decoding algorithm.

https://arxiv.org/pdf/2307.14882

Knot theory and quantum computing (Toronto, Jan. 2019)

This paper explores the interactions between knot theory and quantum computing. On oneside, knot theory has been used to create models of quantum computing, and on the other, it is a source of computational problems. Knot theory is often used to introduce topological idea to people without a formal mathematical background, and we are building on this tradition to discuss some of the deeper ideas of quantum computing.

https://arxiv.org/pdf/1901.03186

 

[martinbn]

A mathematician friend told me that the major tenets of information theory can be established through knot theory. Is that the case?
Thanks.

Why don't you ask him?

 

[Twodogs]

Why don't you ask him?

That would be helpful, thanks.
This was 40 years ago and I have lost touch. I do remember the comment though.

 

[Twodogs]

An interesting idea!

Here is what I found searching for it.

Knot Theory and Error-Correcting Codes (Eindhoven, May 2024)

https://arxiv.org/pdf/2307.14882

Knot theory and quantum computing (Toronto, Jan. 2019)

https://arxiv.org/pdf/1901.03186

Appreciate your for checking it out. I will look at these.

 

[mad mathematician]

I've taken the past year courses in Communcation Systems and Signal Systems from my EE route, I started contemaplating combining Quantum Field Theory and Digital Communications.... How Bizzare.

 

[fresh_42]

I've taken the past year courses in Communcation Systems and Signal Systems from my EE route, I started contemaplating combining Quantum Field Theory and Digital Communications.... How Bizzare.

I think this is less bizarre than developing information theory from knot theory.

 

[mad mathematician]

I think this is less bizarre than developing information theory from knot theory.

Adler had some ideas of incorporating Topology with EE.
https://robert.net.technion.ac.il/topological-methods-in-engineering-networks-and-data-analysis/

Well teaching EE students Topology can be a little hard; depends how good they in math.

 

[fresh_42]

Adler had some ideas of incorporating Topology with EE.
https://robert.net.technion.ac.il/topological-methods-in-engineering-networks-and-data-analysis/

Well teaching EE students Topology can be a little hard; depends how good they in math.

Ok, topology is so comprehensive that it may even play a role in information theory. Zomorodian looks promising. But that is far from

that the major tenets of information theory can be established through knot theory

 

[Twodogs]

anyons

Ok, topology is so comprehensive that it may even play a role in information theory. Zomorodian looks promising. But that is far from

 

[Twodogs]

"that the major tenets of information theory can be established through knot theory"

Well, I remember who made this statement and it was Chris Hillman who was post doc at University of Washington. I find on checking that he was a contributor to Physics Forums until sometime just before 2022.
My use of the term 'friend' was a bit overstated. We corresponded early on in the bionet.info-theory group for a couple of years. His statement was off hand and as he noted speculative.

 

[thomsj4]

Let $\text{Hom}(\pi_1(S^3 \setminus K), G)$ be the set of group homomorphisms from the fundamental group of a knot $K$'s complement into a finite group $G$. We define the "information content" of $K$ with respect to $G$ as

$$I(K;G) = \ln |\text{Hom}(\pi_1(S^3 \setminus K), G)|.$$

$|\text{Hom}(\pi_1(S^3 \setminus K), G)|$ intuitively counts the ways to assign elements of $G$ to loops in the knot complement, respecting the fundamental group relations.

For a connected sum $K_1 \# K_2$, the Seifert–van Kampen theorem often allows us to approximate $\pi_1(S^3 \setminus (K_1 \# K_2))$ as an amalgamated free product of $\pi_1(S^3 \setminus K_1)$ and $\pi_1(S^3 \setminus K_2)$. In favorable cases, we get an isomorphism

$$\text{Hom}(\pi_1(S^3 \setminus (K_1 \# K_2)), G) \cong \text{Hom}(\pi_1(S^3 \setminus K_1), G) \times \text{Hom}(\pi_1(S^3 \setminus K_2), G),$$

or at least a near-surjective map, implying

$$|\text{Hom}(\pi_1(S^3 \setminus (K_1 \# K_2)), G)| \approx |\text{Hom}(\pi_1(S^3 \setminus K_1), G)| \times |\text{Hom}(\pi_1(S^3 \setminus K_2), G)|.$$

Taking logarithms,

$$I(K_1 \# K_2; G) \approx I(K_1;G) + I(K_2;G).$$

This resembles the additivity of Shannon entropy. This may fail due to additional relations in the amalgamated group. However, under certain conditions (e.g., $G$ abelian or specific geometric constraints), the factorization is exact or approximately multiplicative.

Since $\pi_1(S^3 \setminus K)$ reflects geometric data, we can sometimes bound or compute $|\text{Hom}(\pi_1(S^3 \setminus K), G)|$ geometrically.

For the $n$-fold connected sum $K^{\# n}$,

$$I(K^{\# n}; G) = \ln |\text{Hom}(\pi_1(S^3 \setminus (K^{\# n})), G)|.$$

As $n \to \infty$, if $I(K^{\# n}; G) \approx n I(K; G)$, then

$$\lim_{n\to\infty} \frac{I(K^{\# n};G)}{n}$$

behaves like a channel capacity. In reality, a subexponential correction or constant multiplier $C$ may be needed:

$$|\text{Hom}(\pi_1(S^3 \setminus (K_1 \# K_2)), G)| \le C |\text{Hom}(\pi_1(S^3 \setminus K_1), G)| |\text{Hom}(\pi_1(S^3 \setminus K_2), G)|.$$

We can view knot operations as "topological channels." Restricting knot families reduces "information" while increasing the complexity of $G$ increases it. Counting homomorphisms is more like measuring the "state space." The combinatorial approach, counting representations and their products, parallels Shannon entropy's additivity.

For $G = \mathbb{Z}/m\mathbb{Z}$, homomorphisms sometimes correspond to $\text{mod}(m)$-valued linking numbers or pairings. If $K$'s Alexander polynomial has factors with diverse $\text{mod}(m)$ behavior, $|\text{Hom}(\pi_1(S^3 \setminus K), \mathbb{Z}/m\mathbb{Z})|$ can be large, and connected sums can be nearly multiplicative.

Defining

$$I(K;G) = \ln|\text{Hom}(\pi_1(S^3 \setminus K), G)|,$$

gives an entropy-like quantity for knots. Under suitable conditions, connected sums yield an approximate additive law for $I$, mirroring Shannon entropy.

 

[Twodogs]

We can view knot operations as "topological channels." Restricting knot families reduces "information" while increasing the complexity of G increases it.

It is interesting to see what the math might look like, thanks.
So, would increasing the complexity of G be equivalent to adding symbols to your alphabet, increasing
bits/symbol?
I am thinking that your "topological channels" is kindred to communication channel rather than river channel. I am attracted to the latter because knots do channel mechanical force in a particular pattern. I am looking for a general bit-from-it proposition wherein the turnings in a flux has an information-theoretic measure.
Does this make sense at all?

 

[Twodogs]

Knotty Knits are Tangles on Tori
Shashank G. Markande1 and Elisabetta A. Matsumoto2
https://archive.bridgesmathart.org/2020/bridges2020-103.pdf

This is a less abstract & more hands-on approach that does note the mechanical responses of various knit patterns.

 

[WWGD]

Adler had some ideas of incorporating Topology with EE.
https://robert.net.technion.ac.il/topological-methods-in-engineering-networks-and-data-analysis/

Well teaching EE students Topology can be a little hard; depends how good they in math.

The link is dead. At least not functioning in my end.

 

[WWGD]

Just to note, in an intro/ superficial way, that braids formed from Anyons; where the braids are quotiented out and turned into $S^1$ knots ( collapsing the sides) , are used as Logic Gates ( not reversible ones like the Toffoli gates) . Maybe @thomsj4 can elaborate on my superficial knowledge here?
Re $\mathbb R^n-\mathbb Q^n$ being path-connected , take $p1(x_1, y_1), p2(x_2, y_2)$ we just go horizontally first , then vertically between the two. Then we just slightly disturb the right angle. Only tricky thing is to show the path remains in the complement.