Introduction to disoriented knot theory

Abstract: This paper is an introduction to disoriented knot theory, which is a generalization of the oriented knot and link diagrams and an exposition of new ideas and constructions, including the basic de nitions and concepts such as disoriented knot, disoriented crossing and Reidemesiter moves for disoriented diagrams, numerical invariants such as the linking number and the complete writhe, the polynomial invariants such as the bracket polynomial, the Jones polynomial for the disoriented knots and links.


Introduction
This paper gives an introduction to the subject of disoriented knot theory. We introduce the notion of a disoriented crossing and explain how to adapt the fundamental concepts and invariants of the knot theory to a setting in which we have disoriented crossing.
The disoriented knot and link diagrams arise when calculating the Jones [1,2], HOMFLY [3] polynomials etc., using the oriented diagram structure of the state summation for the oriented knot and link diagrams. When we split a crossing of an oriented knot diagram using Kau man's bracket model [4,5], one of the occurring diagram is a disoriented diagram. Since the disoriented diagrams acquire orientations outside the category of knot and link diagrams, only unoriented and oriented knot and link diagrams have previously been considered in the literature for the development of knot theory.
In this paper, we put disoriented knot and link diagrams on the same footing as oriented knot and link diagrams. We de ne the concepts of a disoriented diagram and a disoriented crossing and give the Reidemeister moves for disoriented diagrams. We see that most classic basic concepts of knot theory are not invariant for the disoriented diagrams of a knot or link. Here, we rede ne the linking number for the disoriented link diagrams and prove that the linking number is a link invariant of the link. We de ne a new concept called complete writhe instead of the classic writhe that is not a regular isotopy invariant for disoriented link diagrams. Thus, the normalized bracket polynomial (the Jones polynomial) by the complete writhe for disoriented link diagrams will be an invariant of the link.
The paper is organized as follows. Section 2 gives the de nitions of the disoriented knot and link diagrams and disoriented crossing. In this section, we also give the brief description about of disoriented diagrams and the Reidemeister moves on the disoriented diagrams.
Sections 3 and 4 contain some numerical invariants. In Section 3, we de ne the linking number for the disoriented link diagrams, prove that the linking number is a link invariant and give two examples. In Section 4, we adapt the writhe for the disoriented knot and link diagrams. We de ne a new concept called complete writhe for the disoriented knot and link diagrams and prove that the complete writhe is invariant under the Reidemeister moves RII and RIII for the disoriented diagrams.
Section 5 presents a review of the bracket polynomial [4][5][6] and the Jones polynomial for the disoriented knot and link diagrams. We expand the bracket polynomial for disoriented diagrams of a knot (or link). So, the normalized bracket polynomial can also be extended for the disoriented knot and link diagrams. Since the normalized bracket polynomial (the Jones polynomial) is invariant under the rst Reidemeister move for oriented knot diagrams, it is a stronger invariant than regular isotopy invariant for the disoriented diagrams. However, the Jones polynomial is not invariant under all the moves on disoriented diagrams. Therefore, it is not an invariant for the disoriented knot and link diagrams. In this section, we rede ne the Jones polynomial for the disoriented knot and link diagrams by using complete writhe. This polynomial called the complete Jones polynomial is invariant under all the moves on disoriented diagrams. Thus it is a knot and link invariant. Moreover, the complete Jones polynomial of each disoriented diagram of a knot is equal to the original Jones polynomial of the knot. The last part of this section contains a few examples.

De ning disoriented knots and links
A knot is an embedding of a circle in three dimensional space R (or S ). An oriented knot is a knot diagram which has been given an orientation. Taking into account the facts about the embedding, we can also de ne an oriented knot as an embedding of an oriented circle in three dimensional space. In a similar way, we de ne a disoriented knot.

De nition 2.1. A disoriented circle is a circle upon which we have chosen two points, and have chosen an orientation of both of the arcs between those two points. We allow that the orientation of one of the arcs is the reverse of the orientation of the other.
A few simple disoriented diagrams, a disoriented circle and their replacements have been illustrated in Fig. 1. The basic reduction move in Fig. 1 corresponds to elimination of two consecutive cusps on a simple loop. Note that it is allowed to the cancelation of consecutive cusps along a loop where the cusps both points to the same local region of the two local regions delineated by the loop but not allowed to the cancelation of a "zig-zag" where two consecutive cusps point to opposite local sides of the loop. A zig-zag is represented an oriented or disoriented virtual crossing of the oriented or disoriented diagram of a knot and link. Since we work on the classical knot and link diagrams in this paper, we do not encounter a zig-zag. Thus, for we can easily draw a disoriented knot or link diagram, we use the local disoriented curve of the form instead of the cusp with two points of the form . Due to our present disclosure, a disoriented curve can be replaced with an oriented curve. Likewise, a disoriented circle can be replaced with an oriented circle. For detailed information about disoriented con gurations, replacements and disoriented relations can be seen in [7][8][9][10][11].

De nition 2.2. A disoriented knot is an embedding of a disoriented circle in three dimensional space R (or S ). A disoriented link of k-components is an embedding of a disjoint union of k circles in R , where at least one of circles is disoriented. Two disoriented knots equivalent if there is a continuous deformation of R taking one to the other.
If K a disoriented knot (or link) in R , its projection is π(K) ⊂ R , where π is the projection along the z-axis onto the xy-plane. The projection is said to be regular projection if the preimage of a point of π(K) consists of either one or two points of K. If K has a regular projection, then we can de ne the corresponding disoriented diagram D by redrawing it with an arc near crossing (the place with two preimages in K) to incorporate the overpass/underpass information.

De nition 2.3.
A crossing of a disoriented knot K is disoriented if the overpass and underpass arcs of the crossing have opposite orientations. In other words, if A and A are the arcs of the disoriented circle of which K is an embedding, then one of the underpass and overpass arcs is A , and the other is A . If a crossing of a disoriented knot K is not disoriented, we say that it is oriented. An oriented knot is a disoriented knot with zero disoriented crossing. For example, see Fig. 3.

De nition 2.4. Let L be a link with exactly two components, K and K . Choose a disorientation of both K and K . Denote the two arcs of K by A and A , and denote the two arcs of K by A and A . We say that a crossing of L is disoriented if either of the following holds: 1. One of the underpass and overpass arcs of the crossings is A , and the other is A or A . 2. One of the underpass and overpass arcs of the crossings is A , and the other is A or A .
Otherwise, we say that the crossing is oriented.
Remark 2.5. Similar considerations apply to disoriented links with more than two components. We have not discussed here to avoid going beyond the purpose of the paper.
Although the underpass and overpass of an oriented crossing are the same with the underpass and overpass of a disoriented crossing, the sings of these crossings are not the same. Then, a disoriented crossing can not be replaced with an oriented crossing. We illustrate oriented and disoriented crossings in Fig. 2. The Reidemeister moves on disoriented diagrams generalize the Reidemeister moves for the oriented knot and link diagrams. We illustrate the Reidemeister moves on disoriented diagrams in Fig. 4, 5 and 6. The Reidemeister moves of types II and III for oriented diagrams are expanded on disoriented diagrams. A new disoriented curled move is added to Reidemeister move of type I for disoriented diagrams. Disoriented knot and link diagrams that can be connected by a nite sequence of these moves and their inverse moves are said to be equivalent. We list the equivalence of these Reidemesiter moves illustrated in Fig. 4, 5 and 6 as below: Note that there are many other possible choices of the orientation of the arcs in Fig. 6.
We call regular isotopy to the equivalence relation generated by the moves RII and RIII (and the planar moves), ambient isotopy to the equivalence relation generated by the moves RI, RII and RIII, and complete ambient isotopy to the equivalence relation on disoriented diagrams that is generated by all the moves in Fig. 4, 5 and 6. Since there is no disoriented crossing of an oriented diagram, the complete ambient isotopy is equivalent to the ambient isotopy on the oriented diagrams. But, the complete ambient isotopy is more powerful than the ambient isotopy for the disoriented knot and link diagrams.

Linking number
In this section, we de ne the linking number for disoriented links and prove that the linking number of a disoriented link is its invariant.

De nition 3.1. Let L be a disoriented link with two components K and K . The linking number lk(L) is de ned by formula
where K ⊓ K denotes the set of crossings of K with K (no self-crossings), where the rst sum runs over the oriented crossings of K ⊓ K , the second sum runs over the disoriented crossings of K ⊓ K , and ε(o) and ε(d) denote the sign of an oriented crossing and the sign of a disoriented crossing of belonging to K ⊓K , respectively.
The following theorem gives that the linking numbers of all the disoriented diagrams of a link are equal.

Theorem 3.2. The linking number lk(L) is an invariant of the link L.
Proof. Let D be a disoriented regular diagram of the link L with two components. We suppose that D ′ is another regular diagram of L. From our discussions so far, we know that we may obtain D ′ by performing, if necessary several times, the Reidemeister moves in Fig. 4, 5 and 6. Therefore, in order to prove the theorem, it is su cient to show that the value of the linking number remains unchanged after each of the Reidemeister moves is performed on D.
The move RI: At the crossings of D at which we intend to apply the move RI, every section (edge) of such a crossing belongs to the same component. Therefore, applying the move RI does not a ect the calculation of the linking number. In the same way, the move RI ′ does also not a ect the calculation of the linking number.
The moves RII: We shall only examine the e ects of the cases J ′ , J ′ , L ′ and L ′ of the moves RII in Fig.  5 to the calculation of linking number. The remaining cases of the moves RII can be examined in a similar way. An application of the moves RII on D only has an e ect on the linking number if A and B belong to di erent components, see Fig. 7. In the cases (a) and (c), the crossing c is disoriented and the crossing c is oriented. Also, the crossings c and c have the same signs. By De nition 3.1, we get ε(c ) − ε(c ) = . In the cases (b) and (d), both crossings are disoriented. Since the crossings c and c have opposite signs, we get ε(c ) + ε(c ) = . So, in each case, the linking number is unchanged under the second Reidemeister moves.
To be identical the e ects of the cases T i and S i , the following equations should always be hold: If all crossings are oriented or disoriented, If one of the crossings c and c ′ is oriented while other is disoriented or one of the crossings c and c ′ is oriented while other is disoriented If one of the crossings c and c ′ is oriented while other is disoriented and one of the crossings c and c ′ is oriented while other is disoriented, If A, B and C belong to the same component for the only case in Fig. 8, then the linking number is una ected. So we suppose that A belong to a di erent component than B and C. Then the parts that have an e ect on linking number is the sum of the signs of the crossings c , c and c ′ , c ′ . Since the crossings are oriented in the case (a), we have by De nition 3.1. Since the crossing c ′ is disoriented and others are oriented in the case (b), we have Since the crossing c ′ is oriented and others are disoriented in the case (c), we have Since the crossing c and c ′ are disoriented and others are oriented in the case (d), we have Thus, none of the sums does cause any change to the linking number. The other cases, (i.e. the various possibilities for the components that A, B and C belong to) can be treated in a similar manner. The remaining cases of the moves RIII can be examined in a similar way. Hence, the linking number remains unchanged when we apply the moves RIII.
We suppose now that L is a disoriented link with n components, K , K , ..., K n With regard to two components, K i and K j , i < j, we may de ne as an extension of the linking number lk(L) = lk(K i , K j ), ≤ i < j ≤ n This approach will give us n(n − ) linking numbers, and their sum, is called the total linking number of K. One can show that, in fact, the total linking number of K is an invariant of K.

Writhe
Recall that the classical writhe w(D) of a regular diagram D of a knot (or a link) is the sum of the signs of all the crossings of D. This de nition of the writhe can be also adapted to disoriented diagram D of a knot. The classical writhe is not a knot invariant. However, regularly isotopic oriented knot diagrams have the same writhe. It can be easily understood that the classical writhe is not invariant under the moves RII and RIII for disoriented diagrams. Now, we de ne a new writhe called complete writhe for a disoriented diagram of a knot or link and prove that the complete writhes of all the disoriented diagram of a knot are equal.

De nition 4.1. Let D be a disoriented regular diagram of a knot (or link) K. The complete writhe of D, cw(D), is de ned by formula
where the rst sum runs over the oriented crossings of D, the second sum runs over the disoriented crossings of D and ε(o) and ε(d) denote the sign of an oriented crossing and the sign of a disoriented crossing of belonging to D, respectively.

Theorem 4.2. The complete writhe cw(D) is a regular isotopy invariant of the disoriented diagram D.
Proof. If the proof of Theorem 3.2 is adopted for all crossings of the disoriented diagram D, it is easy to see that the e ects of the moves RII and RIII on the linking number are identical to those on the complete writhe cw(D).
The complete writhe cw(D) of a disoriented diagram D is not invariant under the moves RI and RI ′ .  As shown in the following example, all the disoriented diagrams of a knot have the same complete writhe, although each disoriented diagram of it has a di erent writhe.

Polynomial invariants for disoriented knot diagrams
In this section, we give the bracket polynomial and the normalized bracket polynomial for disoriented knots and links. We show that the bracket polynomial is a regular isotopy invariant for disoriented knots and links. But, the normalized bracket polynomial is not an invariant for disoriented knots and links. For the normalized bracket polynomial to be an invariant of the disoriented knot we rede ne the original normalized bracket polynomial by considering the notion of disoriented crossing that we call the complete normalized polynomial or the complete Jones polynomial. The construction of the complete normalized polynomial invariant begins with the disoriented summation of the bracket polynomial. This means that each local smoothing is either an oriented smoothing or a disoriented smoothing as illustrated in Fig. 11. The su cient information about these smoothings can be found in Kau man's works [7,9]. The bracket expansion for the crossings with both positive and negative sign in an oriented knot and link diagram can be written as an oriented bracket state model: where K + , K − , K and K ∞ are diagrams in Fig. 11, and D is an oriented diagram with zero-crossing of unknot and an oriented knot or link diagram, respectively and ⊔ denotes disjoint union. Proof. The proof is the same as oriented ones. The proof for oriented link diagrams given in [7].
The extended bracket polynomial is not an invariant of the moves RI and RI ′ for the disoriented knot and link diagrams. Its behavior under these moves is examined in the following lemma.
This de nition is a direct extension to the disoriented knot and link category of the state sum model for the original Jones polynomial. It is straightforward to verify the invariances stated above, see [4,5]. In this way, we have the Jones polynomial for the disoriented knot and link diagrams. The original Jones polynomial is not an invariant for the disoriented links. Indeed, it is not invariant under the RI ′ move, see Example 5.7. We now rede ne the normalized bracket polynomial for the disoriented knot and link diagrams that we call the complete normalized polynomial and show that the complete normalized polynomial is invariant under all the moves of the disoriented knot and link diagrams. Proof. Reserving all crossings exchanges the roles of A and A − in the de nition of ⟨K⟩, f K and T K .
Next, we show that T K is the Jones polynomial via the complete writhe. We call this polynomial as complete Jones polynomial for the disoriented knot and link diagrams and denote V K .
Theorem 5.6. The complete normalize polynomial T K yields the complete Jones polynomial, V K (t). That is, Proof. Since the complete writhes of all the disoriented diagrams of K are equal by Theorem 4.3, we have that the complete normalized polynomial are equal for all choice of disorientation. From Remark 4.4, we obtain that the complete normalized polynomial of K is equal to the Kau man polynomial of K. By Theorem 5.2. in [5], T K (t − ) = V K (t). With A = t − , V = t − and V = . Example 5.8. For the disoriented diagrams of the right hand trefoil in Fig. 3,

Discussion
In this paper, we have given an introduction to the subject of the disoriented knot theory and we have explained how to adapt the Reidemeister moves, the linking number, the writhe, the bracket polynomial and the Jones polynomial to the disoriented knot and link diagrams. We have based our work on disorientation.
We have de ned a new writhe called complete writhe and rede ned the Jones polynomial for the disoriented knots and links diagrams. This paper lays the foundation for future works on these ideas.