This paper presents a summary of the content of the article titled On Tangle Decompositions of Twisted Torus Knots. The main objective of the paper was to demonstrate that for any integer n>0, there are various twisted torus knots having n-string vital tangle decompositions, and that these knots have tori in the exteriors. These twisted knots were presented in details and examples were also provided.
Marimoto (2013) began by giving an example expressed in the following form: Let P, Q, R, S be integers. In this case, the value of P is greater than the value of R and R is greater than 1. Moreover, Q is greater than zero; gcd (P, Q) is equivalent to 1. Furthermore, Marimoto (2013) added to the equation by stating the expression let T (P, Q) serve as the torus knot of type (P, Q) in the S3. By adding S time full twists on r-strands in T (P, Q), a twisted torus knot is obtained by the operation. Such twisted torus knot is denoted by T (P, Q, R, and S).
Through time, there were other twists made on the torus knots. For instance, in theorem 1.1, it is stated that e > 0, k1 > 1, k2 > 1 are integers, and p0 = (e + 1)(k1 + k2) + 1, q0 = e(k1 + k2) + 1,r0 = p0 − k1 and s0 = −1. The equivalent equation for this is T (p0, q0; r0, s0) which is the associated sum of the two torus knots T (k1, ek1+1) and T (K2, − (e + 1) K2 − 1). In theorem 1.2, Marimoto (2013) expressed the following equation: e > 0, k1 > 1, k2 > 1, x1 > 0, x2 > 0 are integers with gcd(x1, x2) = 1. Then, p = ((e + 1)(k1 + k2 − 1) + 1)x1 + (e + 1)x2,q = (e(k1 + k2 − 1) + 1)x1 + ex2,r = ((e + 1)(k1 + k2 − 1) − k1 + 2)x1 + ex2 and s = −1. Out of this expression, the following assumptions were produced. First, T (p, q; r, s) possesses an x1-string vital tangle decomposition. Second, the decomposition is obtained by the x1-string fusion of the torus knot T ((k1−1) x1+x2, e((k1−1)x1+x2)+x1) and the torus link T (k2x1,−((e+1)k2+1)x1). Lastly, T (p, q; r, s) possesses a vital torus in the exterior whose cohort is the torus knot T (k2, − (e + 1) k2 − 1). Thus, for any integer n > 0, by placing x1 = n, an infinitely numerous twisted torus knots with n-string critical tangle decompositions are obtained (MARIMOTO 2013).
Marimoto (2013) also discussed two other important topics namely the “Parallelized Torus Knots and Parallelized Twisted Torus Knots.” In this aspect, Marimoto (2013) presented the following: Let T (p0, q0) serve as the torus knot of type (p0, q0), in which p0 and q0 are considered as positive co-prime integers having p0 > 1, and x1 and x2 serve as positive integers. Four points are taken in this situation namely P1, P2, P3 and P4 situated on the contiguous two strands in T (p0, q0). In addition, P1 and P3 are replaced with x1 parallel strings while P2 through P4 are replaced with x2 parallel strings. Moreover, the rectangle P1, P2, P3 and P4 are replaced with x1 + x2 strands. From this equation, the output is a torus link represented by T (P. Q) for some P, Q. Marimoto (2013) demonstrated the detection of P and Q. Initially, the P0 strings situated under the (p0, q0)-torus braid 0, 1, 2. . . p0 − 2, p0 – 1 are numbered accordingly. Moreover, the arc that starts at P1 goes to the braid at P0-1 and exits at Q0-1. Following the first round, the arc goes back to the braid and exits at 2q0 − 1. Following this, the arc exits the braid at 3q0 – 1. By proceeding, it exits at aq0 − 1 ≡ p0 − 2 (mod p0) for some a, until finally, the arc meets point P3. Therefore, the output is aq0 ≡ −1 (mod p0).
Marimoto (2013) made some propositions concerning the event. The first proposition is stated as follows: For co-prime positive integers p0 and q0, there exclusively occur positive integers a, b, c, d which fulfill the following conditions:
The second proposition is stated as follows: Let x1 and x2 serve as positive integers, and p = ax1 + bx2 and q = cx1 + dx2, gcd (p, q) = gcd(x1, x2). Specifically, T (p, q) is a torus knot if and only if gcd(x1, x2) = 1. The third proposition states T (p0, q0) is the torus knot of type (p0, q0) with p0 > 1,q0 > 0, gcd(p0, q0) = 1, and x1, x2 are positive integers. Hence, by the parallelization of T (p0, q0), a torus knot or a link T (p, q) with p = ax1 +bx2 and q = cx1+ dx2, in which (a, b, c, d) are distinctively determined by the conditions in the first proposition. Several other propositions were made concerning parallelized torus knots and twisted knots.
References
MORIMOTO, KANJI. 2013. 'ON TANGLE DECOMPOSITIONS OF TWISTED TORUS KNOTS'.Journal Of Knot Theory And Its Ramifications 22 (09): 1350049. doi:10.1142/s0218216513500491.
