## Knot Arithmetic: Bridge Number and Factorization

Like the genus the bridge number has an additive property with knot addition. The general formula for the addition of bridge numbers can be written as follows:

b(K + L) = b(K) + b(L) - 1

This formula can lead to some interesting applications in finding the prime factors of knots. In the case of a 2-bridge knot, one can try substituting 2 as the sum of the bridge numbers in the above equation and obtaining:

2 = b(K) + b(L) - 1

and therefore

b(K) + b(L) = 3

Now, because the bridge number of a knot must be a whole number greater than or equal to 1, b(K) and b(L) can only be 2 and 1 respectively or 1 and 2 respectively. Therefore, since one of the factor bridge numbers is 1, one of the factor knots must be the unknot. The definition of the prime knot is explicitly a knot which, given any two factor knots K1 and K2, can be obtained through the sum of K1 and K2, but one of either K1 or K2 must be the unknot. This proves that any knot with a bridge number of 2 is prime.

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