In this paper, we define a t wo-variable p olynomial i nvariant o f r egular i sotopy, M K f or a... more In this paper, we define a t wo-variable p olynomial i nvariant o f r egular i sotopy, M K f or a d isoriented link diagram K. By normalizing the polynomial MK using complete writhe, we obtain a polynomial invariant of ambient isotopy, NK , for a disoriented link diagram K. The polynomial NK is a generalization of the expanded Jones polynomial for disoriented links and is an expansion of the Kauffman p olynomial F to the disoriented l inks. Moreover, the polynomial MK is an expansion of the Kauffman p olynomial L t o t he d isoriented links.
This paper is a work on elementary soft (𝜖-soft) compact spaces. We first define the cofinite 𝜖-s... more This paper is a work on elementary soft (𝜖-soft) compact spaces. We first define the cofinite 𝜖-soft compact space and prove that the image of an 𝜖-soft compact space under a soft continuousmapping is 𝜖-soft compact space. We then examine the relationship between 𝜖-soft compactspace and classical compact space and give an illustrative example.
In this paper, some properties of the atoms in the lattice of a family τ(X) of all topologies on ... more In this paper, some properties of the atoms in the lattice of a family τ(X) of all topologies on a finite set X are proved. In addition, a geometric method is given to find the number of the atomic and the anti-atomic topologies, and it is observed that the number of elements of τ(X) satisfies the following expression; n(2 n-2) < |τ(X)| < (2 n-2) (n-1) where n = |X| (cardinality of X).
Bu calismada, izomorf esas gruplara sahip, kompakt yonlendirilebilir indirgenemez 3-boyutlu manif... more Bu calismada, izomorf esas gruplara sahip, kompakt yonlendirilebilir indirgenemez 3-boyutlu manifoldlarin homeomorf olduklari gosterildi.
In [1], we have constructed a polynomial invariant of regular isotopy, , for oriented knot and li... more In [1], we have constructed a polynomial invariant of regular isotopy, , for oriented knot and link diagrams L. From by multiplying it by normalizing factor, we obtained an ambient isotopy invariant, , for oriented knotsand links. In this paper, we give some properties of these polynomials. Wealso calculate the polynomials and of the knots through nine crossings and thetwo-component links through eight crossing
In this paper, we present a generalization of two variables of the Alexander polynomial for a giv... more In this paper, we present a generalization of two variables of the Alexander polynomial for a given oriented knot diagram. We define the Alexander polynomial of two variables by an easy method which will be achieved as a result of the interpretation of the crossing point as a particle with input-output spins in the mathematical physics. The classical Alexander polynomial is the case of one of the variables to be equal to 1 in the Alexander polynomial of two variables.
In this paper, we define the delta derivative in the integer group ring and we show that the delt... more In this paper, we define the delta derivative in the integer group ring and we show that the delta derivative is well defined on the free groups. We also define a polynomial invariant of oriented knot and link by carrying the delta derivative to the link group. Since the delta derivative is a generalization of the free derivative, this polynomial invariant called the delta polynomial is a generalization of the Alexander polynomial. In addition, we present a new polynomial called the difference polynomial of oriented knot and link, which is similar to the Alexander polynomial and is a special case of the delta polynomial.
In this paper, we apply computer algebra (MAPLE) techniques to calculate the N-colored Jones poly... more In this paper, we apply computer algebra (MAPLE) techniques to calculate the N-colored Jones polynomial of the trefoil knot and the figure eight knot. For this purpose, a computer program was developed. When an integer N P 2 is given, the program calculates the N-colored Jones polynomial of the trefoil and the figure eight knot.
In this article, we describe the concepts such as sequentially soft closeness, sequential compact... more In this article, we describe the concepts such as sequentially soft closeness, sequential compactness, totally boundedness and sequentially continuity in any soft cone metric space and prove their some properties. Also, we examine soft closed set, soft closure, compactness and continuity in an elementary soft topological cone metric space. Unlike classical cone metric space, sequential compactness and compactness are not the same here. Because the compactness is an elementary soft topological property and cannot be defined for every soft cone metric space. However, in the restricted soft cone metric spaces, they are the same. Additionally, we prove some fixed point theorems related to diametrically contractive mapping in a complete soft cone metric space.
This paper is an introduction to soft cone metric spaces. We define the concept of soft cone metr... more This paper is an introduction to soft cone metric spaces. We define the concept of soft cone metric via soft element, investigate soft converges in soft cone metric spaces and prove some fixed point theorems for contractive mappings on soft cone metric spaces.
Bu makale esnek koni metrik uzaylara bir giristir. Once esnek koni metrik uzaylari, esnek eleman ... more Bu makale esnek koni metrik uzaylara bir giristir. Once esnek koni metrik uzaylari, esnek eleman yardimiyla tanimladik ve onun temel ozelliklerini verdik. Sonra esnek koni metrik uzaylarda esnek yakinsaklik kavramini inceledik ve esnek koni metrik uzaylar uzerinde daralma donusumu icin bazi onemli sabit nokta teoremleri ispatladik
Bu calismada, bir sonlu X kumesi uzerine kurulan butun topolojilerin τ(X) ailesinin latisindeki a... more Bu calismada, bir sonlu X kumesi uzerine kurulan butun topolojilerin τ(X) ailesinin latisindeki atomlarin bazi ozellikleri ispat edildi. Ayrica, topolojilerin atom ve anti-atom sayilarini bullmak icin bir geometric yontem verildi ve τ(X) ailesinin elemanlarinin sayisi, |τ(X)| icin n(2n -2) < |τ(X)| < (2n -2) (n-1) ifadesinin saglandigi gozlemlendi.
In this paper, we define a polynomial invariant of regular isotopy, G L , for oriented knot and l... more In this paper, we define a polynomial invariant of regular isotopy, G L , for oriented knot and link diagrams L. From G L by multiplying it by a normalizing factor, we obtain an ambient isotopy invariant, N L , for oriented knots and links. We compare the polynomial N L with the original Jones polynomial and with the normalized bracket polynomial. We show that the polynomial N L yields the Jones polynomial and the normalized bracket polynomial. As examples, we give the polynomial G L of some knot and link diagrams and compute the polynomial G L for torus links of type (2, n), and applying computer algebra (MAPLE) techniques, we calculate the polynomial G L of torus links of type (2, n). Furthermore we give its applications to alternating links.
In this paper we work the bracket polynomial of (2,n)-torus link as a Fibonacci polynomial. We sh... more In this paper we work the bracket polynomial of (2,n)-torus link as a Fibonacci polynomial. We show that the bracket polynomial of (2,n)-torus link provides recurrence relation as similar to the Fibonacci polynomial and give its some fundamental properties. We also prove important identities, which are similar to the Fibonacci identities, for the bracket polynomial of (2,n)-torus link and prove Fibonacci-like identities of the Jones polynomial of (2,n)-torus link as a result of the bracket polynomial. Finally, we observe that the bracket polynomial of (2,n)-torus link and therefore its Jones polynomial can be derived from its Alexander-Conway polynomial or classical Fibonacci polynomial. MSC: 57M25 • 11B39 • 11C08
In this paper, we define the polynomial \(G_L\) by way of the braids. We construct the abstract t... more In this paper, we define the polynomial \(G_L\) by way of the braids. We construct the abstract tensor model of the polynomial \(G_L\) and we obtain the new solutions relevant with the state model of the polynomial \(G_L\) to the Yang-Baxter equation. We also construct the vacuum-vacuum expectation model of the polynomial \(G_L\) and we show that the studies performed using the Kaufmann bracket on the quantum group \(SL(2)_q\) with \(q = A^2\) are valid for the state model of the polynomial \(G_L\) without $q=A^2$.
In Simsek’s paper it was introduced a concept of soft cone metric space via soft elements and som... more In Simsek’s paper it was introduced a concept of soft cone metric space via soft elements and some fixed point theorems in soft cone metric space were provided. In this work, we examine topological structures such as open ball, soft neighbourhood and soft open set in soft metric spaces and their some properties, and prove that every soft cone metric space under some condition is a soft topological space according to elementary operations on soft sets.
In this paper, we define a t wo-variable p olynomial i nvariant o f r egular i sotopy, M K f or a... more In this paper, we define a t wo-variable p olynomial i nvariant o f r egular i sotopy, M K f or a d isoriented link diagram K. By normalizing the polynomial MK using complete writhe, we obtain a polynomial invariant of ambient isotopy, NK , for a disoriented link diagram K. The polynomial NK is a generalization of the expanded Jones polynomial for disoriented links and is an expansion of the Kauffman p olynomial F to the disoriented l inks. Moreover, the polynomial MK is an expansion of the Kauffman p olynomial L t o t he d isoriented links.
This paper is a work on elementary soft (𝜖-soft) compact spaces. We first define the cofinite 𝜖-s... more This paper is a work on elementary soft (𝜖-soft) compact spaces. We first define the cofinite 𝜖-soft compact space and prove that the image of an 𝜖-soft compact space under a soft continuousmapping is 𝜖-soft compact space. We then examine the relationship between 𝜖-soft compactspace and classical compact space and give an illustrative example.
In this paper, some properties of the atoms in the lattice of a family τ(X) of all topologies on ... more In this paper, some properties of the atoms in the lattice of a family τ(X) of all topologies on a finite set X are proved. In addition, a geometric method is given to find the number of the atomic and the anti-atomic topologies, and it is observed that the number of elements of τ(X) satisfies the following expression; n(2 n-2) < |τ(X)| < (2 n-2) (n-1) where n = |X| (cardinality of X).
Bu calismada, izomorf esas gruplara sahip, kompakt yonlendirilebilir indirgenemez 3-boyutlu manif... more Bu calismada, izomorf esas gruplara sahip, kompakt yonlendirilebilir indirgenemez 3-boyutlu manifoldlarin homeomorf olduklari gosterildi.
In [1], we have constructed a polynomial invariant of regular isotopy, , for oriented knot and li... more In [1], we have constructed a polynomial invariant of regular isotopy, , for oriented knot and link diagrams L. From by multiplying it by normalizing factor, we obtained an ambient isotopy invariant, , for oriented knotsand links. In this paper, we give some properties of these polynomials. Wealso calculate the polynomials and of the knots through nine crossings and thetwo-component links through eight crossing
In this paper, we present a generalization of two variables of the Alexander polynomial for a giv... more In this paper, we present a generalization of two variables of the Alexander polynomial for a given oriented knot diagram. We define the Alexander polynomial of two variables by an easy method which will be achieved as a result of the interpretation of the crossing point as a particle with input-output spins in the mathematical physics. The classical Alexander polynomial is the case of one of the variables to be equal to 1 in the Alexander polynomial of two variables.
In this paper, we define the delta derivative in the integer group ring and we show that the delt... more In this paper, we define the delta derivative in the integer group ring and we show that the delta derivative is well defined on the free groups. We also define a polynomial invariant of oriented knot and link by carrying the delta derivative to the link group. Since the delta derivative is a generalization of the free derivative, this polynomial invariant called the delta polynomial is a generalization of the Alexander polynomial. In addition, we present a new polynomial called the difference polynomial of oriented knot and link, which is similar to the Alexander polynomial and is a special case of the delta polynomial.
In this paper, we apply computer algebra (MAPLE) techniques to calculate the N-colored Jones poly... more In this paper, we apply computer algebra (MAPLE) techniques to calculate the N-colored Jones polynomial of the trefoil knot and the figure eight knot. For this purpose, a computer program was developed. When an integer N P 2 is given, the program calculates the N-colored Jones polynomial of the trefoil and the figure eight knot.
In this article, we describe the concepts such as sequentially soft closeness, sequential compact... more In this article, we describe the concepts such as sequentially soft closeness, sequential compactness, totally boundedness and sequentially continuity in any soft cone metric space and prove their some properties. Also, we examine soft closed set, soft closure, compactness and continuity in an elementary soft topological cone metric space. Unlike classical cone metric space, sequential compactness and compactness are not the same here. Because the compactness is an elementary soft topological property and cannot be defined for every soft cone metric space. However, in the restricted soft cone metric spaces, they are the same. Additionally, we prove some fixed point theorems related to diametrically contractive mapping in a complete soft cone metric space.
This paper is an introduction to soft cone metric spaces. We define the concept of soft cone metr... more This paper is an introduction to soft cone metric spaces. We define the concept of soft cone metric via soft element, investigate soft converges in soft cone metric spaces and prove some fixed point theorems for contractive mappings on soft cone metric spaces.
Bu makale esnek koni metrik uzaylara bir giristir. Once esnek koni metrik uzaylari, esnek eleman ... more Bu makale esnek koni metrik uzaylara bir giristir. Once esnek koni metrik uzaylari, esnek eleman yardimiyla tanimladik ve onun temel ozelliklerini verdik. Sonra esnek koni metrik uzaylarda esnek yakinsaklik kavramini inceledik ve esnek koni metrik uzaylar uzerinde daralma donusumu icin bazi onemli sabit nokta teoremleri ispatladik
Bu calismada, bir sonlu X kumesi uzerine kurulan butun topolojilerin τ(X) ailesinin latisindeki a... more Bu calismada, bir sonlu X kumesi uzerine kurulan butun topolojilerin τ(X) ailesinin latisindeki atomlarin bazi ozellikleri ispat edildi. Ayrica, topolojilerin atom ve anti-atom sayilarini bullmak icin bir geometric yontem verildi ve τ(X) ailesinin elemanlarinin sayisi, |τ(X)| icin n(2n -2) < |τ(X)| < (2n -2) (n-1) ifadesinin saglandigi gozlemlendi.
In this paper, we define a polynomial invariant of regular isotopy, G L , for oriented knot and l... more In this paper, we define a polynomial invariant of regular isotopy, G L , for oriented knot and link diagrams L. From G L by multiplying it by a normalizing factor, we obtain an ambient isotopy invariant, N L , for oriented knots and links. We compare the polynomial N L with the original Jones polynomial and with the normalized bracket polynomial. We show that the polynomial N L yields the Jones polynomial and the normalized bracket polynomial. As examples, we give the polynomial G L of some knot and link diagrams and compute the polynomial G L for torus links of type (2, n), and applying computer algebra (MAPLE) techniques, we calculate the polynomial G L of torus links of type (2, n). Furthermore we give its applications to alternating links.
In this paper we work the bracket polynomial of (2,n)-torus link as a Fibonacci polynomial. We sh... more In this paper we work the bracket polynomial of (2,n)-torus link as a Fibonacci polynomial. We show that the bracket polynomial of (2,n)-torus link provides recurrence relation as similar to the Fibonacci polynomial and give its some fundamental properties. We also prove important identities, which are similar to the Fibonacci identities, for the bracket polynomial of (2,n)-torus link and prove Fibonacci-like identities of the Jones polynomial of (2,n)-torus link as a result of the bracket polynomial. Finally, we observe that the bracket polynomial of (2,n)-torus link and therefore its Jones polynomial can be derived from its Alexander-Conway polynomial or classical Fibonacci polynomial. MSC: 57M25 • 11B39 • 11C08
In this paper, we define the polynomial \(G_L\) by way of the braids. We construct the abstract t... more In this paper, we define the polynomial \(G_L\) by way of the braids. We construct the abstract tensor model of the polynomial \(G_L\) and we obtain the new solutions relevant with the state model of the polynomial \(G_L\) to the Yang-Baxter equation. We also construct the vacuum-vacuum expectation model of the polynomial \(G_L\) and we show that the studies performed using the Kaufmann bracket on the quantum group \(SL(2)_q\) with \(q = A^2\) are valid for the state model of the polynomial \(G_L\) without $q=A^2$.
In Simsek’s paper it was introduced a concept of soft cone metric space via soft elements and som... more In Simsek’s paper it was introduced a concept of soft cone metric space via soft elements and some fixed point theorems in soft cone metric space were provided. In this work, we examine topological structures such as open ball, soft neighbourhood and soft open set in soft metric spaces and their some properties, and prove that every soft cone metric space under some condition is a soft topological space according to elementary operations on soft sets.
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