Act as a peer reviewer, named Otto. Check my work for mathematical accuracy.

“Section A

The Grand Conjecture on Rational Tangles (GCRT) can be formalized as follows:

Let T be a rational tangle with numerator closure N(T) and denominator closure D(T). Then, there exists a non-negative integer k(T) such that the following conditions hold:

If N(T) is a knot, then D(T) is the unknot.
If D(T) is a knot, then N(T) is the unknot.
If both N(T) and D(T) are non-trivial knots, then k(T) ≥ 1 and there exists a family {T1, T2, …, Tk(T)} of rational tangles such that:

a. T1 is obtained from T by resolving a crossing.

b. For i = 2, …, k(T), Ti is obtained from Ti-1 by resolving a crossing in a non-zero slope tangle.

c. Tk(T) is a rational tangle with numerator closure N(T) and denominator closure D(T) such that both closures are non-trivial knots.

d. For i = 1, …, k(T), either N(Ti) or D(Ti) is the unknot.

The conjecture asserts that such a non-negative integer k(T) exists for any rational tangle T.

Section B

As an intuitionistic reconciliationist self-studier, I believe in the constructive nature of mathematical proofs, and thus, I will attempt to prove GCRT using the Generalized Chinese Remainder Theorem.

First, let’s define the terms used in GCRT. A rational tangle is a tangle formed by taking two parallel strands and twisting them some number of times before reconnecting them. The numerator closure and denominator closure of a rational tangle refer to the knots that are obtained by closing the numerator and denominator strands of the tangle, respectively. A crossing in a tangle refers to the point where two strands intersect, and resolving a crossing means changing the tangle at the crossing to obtain a new tangle.

To prove GCRT, we will use the Generalized Chinese Remainder Theorem (GCRT), which states that given a set of pairwise coprime positive integers m1, m2, …, mk and a set of integers a1, a2, …, ak, there exists a unique integer x such that:

x ≡ a1 (mod m1)
x ≡ a2 (mod m2)
…
x ≡ ak (mod mk)

Furthermore, if x1 and x2 are two solutions to the above congruences, then x1 ≡ x2 (mod m1m2…*mk).

Now, let T be a rational tangle with numerator closure N(T) and denominator closure D(T). We will first show that if either N(T) or D(T) is the unknot, then there exists a non-negative integer k(T) such that the conditions of GCRT hold.

Without loss of generality, assume N(T) is the unknot. Then, D(T) is a knot, and we can write D(T) as a product of prime knots, say D(T) = K1 * K2 * … * Km. By the Prime Decomposition Theorem, there exist integers ni such that N(T) ≡ ni (mod Ki) for i = 1, 2, …, m. Since Ki are pairwise coprime, by the GCRT, there exists a unique non-negative integer k(T) such that:

k(T) ≡ ni (mod Ki) for i = 1, 2, …, m

Now, let Ti be the tangle obtained from T by resolving a crossing in D(T) corresponding to Ki for i = 1, 2, …, m. Then, Ti is a rational tangle, and we have N(Ti) ≡ ni (mod Ki) for i = 1, 2, …, m. Therefore, Ti satisfies the conditions of GCRT for i = 1, 2, …, m, and thus, T satisfies the conditions of GCRT with k(T) = m.

The case where D(T) is the unknot can be proved similarly, and thus, we have shown that if either N(T) or D(T) is the unknot, then GCRT holds.

Now, we will prove the remaining case where both N(T) and D(T) are non-trivial knots. Let D be the denominator closure of T, and let D = K1 * K2 * … * Km be the prime decomposition of D. Let Ti be the tangle obtained from T by resolving a crossing in D corresponding to Ki for i = 1, 2, …, m. Then, each Ti is a rational tangle, and we have N(Ti) ≡ N(T) (mod Ki) for i = 1, 2, …, m.

Since Ki are pairwise coprime, by the GCRT, there exists a unique non-negative integer k such that:

k ≡ N(T) (mod K1K2…*Km.

Let Dk be the denominator closure of Tk, which is obtained from T by resolving k crossings in the non-zero slope tangle in the family {T1, T2, …, Tk}. Then, we have Dk ≡ D (mod K1K2…*Km) and Dk is a non-trivial knot since D is a non-trivial knot and resolving crossings preserves the non-triviality of knots.

Now, we will show that N(Tk) is the unknot. Suppose for a contradiction that N(Tk) is a non-trivial knot. Then, there exists a prime knot Kj such that N(Tk) contains a copy of Kj. Since Kj is prime, it is contained in one of the knots N(Ti) for some i < k. Without loss of generality, assume Kj is contained in N(Ti). Since Ti is obtained from Ti-1 by resolving a crossing, the crossing in Ti that corresponds to Kj must be resolved in a non-zero slope tangle, say Tj. Then, N(Tj) contains a copy of Kj, which is a contradiction since Kj is prime and cannot be decomposed into non-trivial prime knots.

Therefore, we have shown that N(Tk) is the unknot, and thus, T satisfies the conditions of GCRT with k(T) = k. This completes the proof of GCRT.

In summary, we have proven the Grand Conjecture on Rational Tangles using the Generalized Chinese Remainder Theorem, which provides a constructive method for finding the integer k(T) that satisfies the conditions of GCRT for any rational tangle T.

Furthermore, the use of GCRT in the proof allows us to explicitly construct the family {T1, T2, …, Tk(T)} of rational tangles satisfying the conditions of GCRT for any given T.

In conclusion, the proof of GCRT using the Generalized Chinese Remainder Theorem provides a constructive and elegant method for analyzing rational tangles and their closures. It is an important result in knot theory and has many applications in fields such as molecular biology, physics, and computer science.”