Useful facts about tangles:

(1) Both N(a/b) and D(a/b) are 4-plats. The knot/link N(a/b) is the 4-plat S(a, −b). The knot/link D(a/b) is the 4-plat S(b, a).

(2) N((c_{1}, ..., c_{n})) = N((c_{n}, ..., c_{1})) = N((c_{1}, ..., c_{n} − 1, 1)) = N((1, c_{1} − 1, ..., c_{n}))

(3) The tangle corresponding to \frac{a_{1}}{b_{1}} is the same as the tangle corresponding to \frac{a_{2}}{b_{2}} if and only if \frac{a_{1}}{b_{1}} = \frac{a_{2}}{b_{2}} .

(4) The 4-plats N(a_{1}/b_{1}) and N(a_{2}/b_{2}), a_{i} ≥ 0, are are the same if and only if a_{1} = a_{2} and b_{1}b_{2}^{±1} ≅ 1 (mod a_{1}). i.e. b_{1} = b_{2} (mod a_{1}) or b_{1}b_{2} = 1 (mod a_{1}) or equivalently, b_{1} = b_{2} (mod a_{1}) or b_{1} = b_{2}^{−1} (mod a_{1})

(5) The mirror image of N(\frac{a}{b}) is N(−\frac{a}{b})

(6) N(\frac{a}{b}) is achiral if and only if b^{2} ≅ −1 (mod a) or 2b = 1 (mod a).

(7) N(\frac{a_{1}}{b_{1}} + \frac{a_{2}}{b_{2}} ) = N(\frac{a_{1}b_{2}+a_{2}b_{1}}{a_{1}b′_{2}+a′_{2}b_{1}} ) where a′_{2}b_{2} − a_{2}b′_{2} = 1 [5].

(8) N(A + C) = N(C + A)

(9) A + C ≠ C + A (10) \frac{a_{1}}{b_{1}} + \frac{a_{2}}{b_{2}} ≠ a rational tangle unless either b_{1} = ±1 or b_{2} = ±1.

(11) \frac{a}{b} + t = \frac{a+bt}{b}

(12) D(A + C) = D(A)#D(C).

(13) N(\frac{a_{1}}{b_{1}} + ... + \frac{a_{n}}{b_{n}}) is a Montesinos knot which is not a 4-plat unless at most two of the tangles \frac{a_{i}}{b_{i}} are non-integral.

(14) N(A + C) = 4-plat implies at least one of A or C is rational or locally knotted [8][9].

(15) If A is a prime tangle and C is locally unknotted and not the ∞-tangle, then A + C is a prime tangle [9][10].

The tangle, C, is locally unknotted if every 3-ball which intersects the arcs of C in exactly two points contains an unknotted spanning arc. A tangle is locally knotted if it is not locally unknotted. A tangle is prime if it is neither rational nor locally knotted.

(16) R is a rational tangle if and only if its double branch cover is a solid torus.

(17) M a ambient isotopic to a sum of rational tangles if and only if its double branch cover is a Seifert fiber space with orbit surface a disk and n > 0 exceptional fibers [11].

[5] C. Ernst, D. W. Sumners, A calculus for rational tangles: applications to DNA recombination, Math. Proc. Camb. Phil. Soc. 108 (1990), 489–515.

[7] J. R. Goldman, L. H. Kauffman, Rational tangles, Adv. in Appl. Math. 18 (1997), no. 3, 300–332.

[8] S. Bleiler, Knots prime on many strings, Trans. Amer. Math. Soc. 282 (1984), no. 1, 385–401.

[9] W. B. R. Lickorish, Prime knots and tangles, Trans. Amer. Math. Soc. 267 (1981), no. 1, 321–332.

[10] C. Ernst, D. W. Sumners, Solving tangles equations arising in a DNA recombination model, Math. Proc. Camb. Phil. Soc. 124 (1998).

[11] C. Ernst, Tangle equations, J. Knot Theory Ramifications 5 (1996), no. 2, 145–159.