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.
