Stability of the Cauchy Additive Functional Equation on Tangle Space and Applications

We introduce real tangle and its operations, as a generalization of rational tangle and its operations, to enumerating tangles by using the calculus of continued fraction and moreover we study the analytical structure of tangles, knots, and links by using new operations between real tangles which need not have the topological structure. As applications of the analytical structure, we prove the generalized Hyers-Ulam stability of the Cauchy additive functional equation f(x ⊕ y) = f(x) ⊕ f(y) in tangle space which is a set of real tangles with analytic structure and describe the DNA recombination as the action of some enzymes on tangle space.


Introduction
In 1970, Conway introduced rational tangles and algebraic tangles for enumerating knots and links by using Conway notation.The rational tangles are defined as the family of tangles that can be transformed into the trivial tangle by sequence of twisting of the endpoints.Given a tangle, two operations, called the numerator and denominator, by connecting the endpoints of the tangle produce knots or 2component links.To enumerating and classifying knots, the theory of general tangles has been introduced in [1].
Moreover the rational tangles are classified by their fractions by means of the fact that two rational tangles are isotopic if and only if they have the same fraction [1].This implies the known result that the rational tangles correspond to the rational numbers one to one.It is clear that every rational number can be written as continued fractions with all numerators equal to 1 and that every real number  corresponds to a unique continued fraction, which is finite if  is rational and infinite if  is irrational.Thus the continued fractions give the relationship between the analytical structure and topological structure under a certain restricted operator.See [2], for example.There are some operations that can be performed on tangles as the sum, multiplication, rotation, mirror image, and inverted image.
Topologically, the sum and multiplication on tangles are defined as connecting two endpoints of one tangle to two endpoints of another.However they are not commutative and do not preserve the class of rational tangles.Furthermore the sum and multiplication of two rational tangles are a rational tangle if and only if one of two is an integer tangle [3].Thus the set of rational tangles is not a group because it was discovered that not all rational tangles form a closed set under the sum and multiplication.Considering a braid of rational tangles, a series of strands that are always descending, the set of braids is a group under braid multiplication.
In 1940, Ulam introduced the stability problem of functional equations during talk before a Mathematical Colloquium at the University of Wisconsin [4]: Given a group  1 , a metric group ( 2 , ) and a positive number , does there exist a number  > 0 such that if a function  :  1 →  2 satisfies the inequality ((), ()()) <  for all ,  ∈  1 , there exists a homomorphism  :  1 →  2 such that ((), ()) <  for all  ∈  1 ?
Analytically, the stability problem of functional equations originated from a question of Ulam concerning the stability of group homomorphisms.The functional equation is called the Cauchy additive functional equation.In particular, every solution of the Cauchy additive functional equation is said to be an additive mapping.In [5], Hyers gave the first affirmative partial answer to the question of Ulam for Banach spaces.In [6], Hyers' theorem was generalized by Aoki for additive mappings and by Rassias for linear mappings by considering an unbounded Cauchy difference in [7].In [8], a generalization of the Rassias theorem was obtained by Gȃvrut ¸a by replacing the unbounded Cauchy difference by a general control function in Rassias' approach.There are many interesting stability problems of several functional equations that have been extensively investigated by a number of authors.See [9][10][11][12][13][14].
In recent years, new applications of tangles to the field of molecular biology have been developed.In particular, knot theory gives a nice way to model DNA recombination.The relationship between topology and DNA began in the 1950s with the discovery of the helical Crick-Watson structure of duplex DNA.The mathematical model is the tangle model for site-specific recombination, which was first introduced by Sumners [15].This model uses knot theory to study enzyme mechanisms.Therefore rational tangles are of fundamental importance for the classification of knots and the study of DNA recombination.In this paper, we introduce new tangles called real tangles to apply the stability problem and DNA recombination on tangles.
In Section 2, we introduce real tangles and operations between tangles which can be performed to make up tangle space and having analytical structure.Moreover we show that the operations together with two real tangles will always generate a real tangle.In Section 3, we prove the Hyers-Ulam stability of the Cauchy additive functional equation in tangle space and study the DNA recombination on real tangles, as applications of knots or links.

Continued Fractions and Tangle Space
A rational tangle is a proper embedding of two unoriented arcs (strings)  1 and  2 in 3-ball  3 so that the endpoints of the arcs go to a specific set of 4 points on the equator of  3 , usually labeled NW, NE, SW, SE.This is equivalent to saying that rational tangles are defined as the family of tangles that can be transformed into the trivial tangle by a sequence of twisting of the endpoints.Note that there are tangles that cannot be obtained in this fashion: they are the prime tangles and locally knotted tangles.For example, see Figure 1.
By Conway [1], rational tangles are classified by fractions by fact of the following: two rational tangles are isotopic if and only if they have the same fraction.For example, (2, −2, 3) and (1, 2, 2) represent the same tangles up to isotopy because they have a fraction 7/5.Therefore the rational tangles ( 1 ,  2 , . . .,   ) with the exception of {(0), (±1), (∞)} are said to be in canonical form if |  | > 1,   ̸ = 0 for 2 ≤  ≤ .Note that all nonzero entries have the same sign and every rational tangle has a unique canonical form.The canonical form of the example above is (1, 2, 2) and the following corollary, which is a direct result of Conway's theorem [1], will give us a means of classifying rational tangles by way of fractions.

Note that 𝛼 ∈ 𝑅 − 𝑄 is quadratic irrational if and only if it is of the form
where , ,  ∈ ,  > 0,  ̸ = 0, and  is not the square of a rational number.Thus an irrational number is called quadratic irrational if it is a solution of a quadratic equation  2 +  +  = 0, where , ,  ∈  and  ̸ = 0. Moreover  is eventually periodic of the form where the bar indicates the periodic part with  terms.Thus an infinite tangle is said to be periodic if it has eventually periodic of the form See Figure 5 for  = 1 and  = 2.

Corollary 4.
There is a one-to-one correspondence between infinite periodic tangles and quadratic irrational numbers.
Finally, tangles are said to be real if it is rational tangles or infinite tangles, and so the real tangles are finite if it is rational tangles and infinite if it is infinite tangles.Thus continued fractions of finite real tangles are rational and continued fractions of infinite real tangles are irrational.Moreover infinite real tangle is periodic if it is infinite periodic tangles, and so continued fractions of infinite periodic real tangles are quadratic irrational numbers.Let  be a real number that corresponds to finite or infinite continued fractions.Then, by corollaries, the fact that () has a unique real tangle can be proved.The following is examples of the corollaries above.
(3) Considering a quadratic irrational number ) is an infinite periodic tangle of quadratic irrational number (5+ √ 3)/2.See Figure 5.In Figure 5, the boxes mean periodic parts as (2, 1).Now we introduce the operations on real tangles with analytical structure, which need not have the topological structure.However, on rational tangles, our operations are applicable to geometrical results obtained from topological structure.Our operations need to discuss the generalized Hyers-Ulam stability of the Cauchy additive functional equation (+) = ()+() and DNA recombinations in next section.
For the remainder, the identity element is the trivial tangle (0) ∈  and the inverse of  ∈  is −.See Lemma 6.Thus the set  forms a group with respect to ⊕.
In order to determine a group (generally, a vector space) from the set of rational tangles (generally, real tangles), two binary operators ⊕ and ⊗ are necessary.For other operators, restricted on rational tangles, addition (denote by #) and multiplication (denote by * ) of horizontal and vertical rational tangles are considered in [1].In detail, the multiplication of two rational tangles is defined as connecting the top two ends of one tangle to the bottom two endpoints of another, and the addition of two rational tangles is defined as connecting the two leftmost endpoints of one tangle with the two rightmost points of the other as shown in Figure 6.
However the addition of two rational tangles is not necessarily rational, but it can be algebraic tangle [1].For example, it can be easily seen that the sum of (1/2) and (1/2) is not a rational tangle.As the results, in [3], the multiplication (resp., addition) of two rational tangles will be rational tangle if one of two is a vertical (resp., horizontal) tangle.Note that, as a special case of rational tangles, the set of braids is a group under the multiplication.Therefore two operators ⊕ and ⊗ on rational tangles are a generalization of operators # and * introduced in [1].For two operators ⊕ and ⊗ on real tangles, we do not know yet whether it has a topological or geometrical structure.
In Section 3, we will study some applications for two operators ⊕ and ⊗ on real tangles.In this paper, the set (, ) of the real tangles with a metric  is called the tangle space.for all ,  ∈  and for some  > 0. Then there exists a unique additive mapping  :  →  such that ‖() ⊖ ()‖ <  for all  ∈ .

Some Applications on Tangle Space
Proof.Suppose that  :  →  is a mapping such that for all ,  ∈  and for some  > 0. Then we have for some (  ) = .
Generally, for each , the real tangles are as the following: and so the additive mapping () is real tangle as the following: We note that two operations () and () by connecting the endpoints of  produce knots or 2-component links, called rational knot or link if  is a rational tangle, and that every 2-bridge knot is a rational knot because it can be obtained as the numerator or denominator closure of a rational tangle.See Figure 8.Let   ⊂  be the set of rational tangles and  the set of rational knots or links.Then for given  ∈   , it allows defining a function  :   →  in order that () is the numerator closure.The following theorem discusses equivalence of rational knots or links obtained by taking the numerator closure of rational tangles.We call this theorem the tangle classification theorem Theorem 16 (see [16]).Let (/) and (  /  ) be the rational tangles with reduced fractions / and   /  , respectively.Then ((/)) and ((  /  )) are topologically equivalent if and only if  =   and  ± ≡   mod .Proof.Let (/) and (  /  ) be the rational tangles with reduced fractions / and   /  , respectively.Then  =   and  =   because (/) and (  /  ) are isotopic.
However there is a counterexample for the converse of Corollary 17 as follows.
Define the numerator closure of the sum of two rational tangles as the following: where gcd( A tangle equation is an equation of the form ( ⊕ ) = , where ,  ∈   and  ∈ .Solving equations of this type will be useful in the tangle model and gaining a better understanding of certain enzyme mechanisms [15].If one of the tangles in the equation is unknown and the other tangle and the knot  are known, then there is one tangle as the solution of equation, but it is not unique.In fact, let  be known rational tangle and  rational knot or link.Then there are two different rational tangles as the solution of the equation ( ⊕ ) =  which is the topological equivalent under numerator operation in Theorem 16.Corollary 21.Let  and  be known rational tangle in   and rational knot or link in , respectively.Then there exist two solutions  ∈   of the equation ( ⊕ ) = .

3.3.
Tangle Space and DNA.Suppose that tangles , , and  below are rational.As discussed in the introduction of Section 1, DNA must be topologically manipulated by enzymes in order for vital life processes to occur.The actions of some enzymes can be described as site-specific