# Knotoids, Braidoids and Applications

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Knotoids

#### 2.1. Knotoid Diagrams

#### 2.2. Moves on Knotoid Diagrams

## 3. Planar and Spherical Knotoids

#### 3.1. Knots via Knotoids

**Proposition**

**1**

**.**Two knotoid diagrams in ${S}^{2}$ or ${\mathbb{R}}^{2}$ represent the same classical knot if and only if they are related to each other by finitely many Ω-moves, swing moves and the forbidden ${\mathsf{\Phi}}_{-}$-moves.

#### 3.2. The Theory of Spherical Knotoids as an Extension of Knot Theory

#### A Geometric Interpretation of Planar Knotoids

## 4. Braidoids

#### 4.1. The Definition of a Braidoid Diagram

**Definition**

**1.**

#### 4.2. Braidoid Isotopy

#### 4.2.1. Moves on Segments of Strands

**Definition**

**2.**

#### 4.2.2. Moves of Endpoints

**Definition**

**3.**

**Definition 4**(Braidoid isotopy)

**.**

## 5. From Braidoids to Knotoids: The Closure Operation

**Definition**

**5.**

**Definition**

**6.**

**Remark**

**1.**

- 1.
- The endpoints of B do not participate in the closure, and they become the endpoints of $\widehat{B}$.
- 2.
- The reason that a joining arc is required to lie in an arbitrarily close distance to the line of the related corresponding ends is that, otherwise, forbidden moves may be an obstacle to an isotopy of $\widehat{B}$ between any two joining arcs.
- 3.
- The appearance of forbidden moves after closure is also the reason for requiring in a braidoid diagram that the endpoints are not aligned vertically with any braidoid ends. See Figure 13 for an illustrative example.
- 4.
- A joining arc can lie on the right or on the left of the line of the corresponding ends. It is a matter of preference that in this paper it is assumed to lie on the right. Clearly, both choices are isotopic since, by definition, the connection takes place far from the endpoints, so there is an isotopy between the two choices that avoids the forbidden moves.
- 5.
- A joining arc is oriented upward following the orientation of the connected strands.
- 6.
- The resulting (multi-)knotoid depends on the labeling of the braidoid ends. See Figure 14 for an example of two non-isotopic labeled braidoid diagrams with the same underlying braidoid diagram, which give rise to non-equivalent knotoids. To obtain a well-defined map, we define the closure on the set of labeled braidoid diagrams.

**Proposition**

**2.**

**Proof.**

## 6. From Knotoids to Braidoids: The Braidoiding Algorithm

**Theorem**

**1.**

#### 6.1. The Basic Idea of the Braidoiding

#### 6.2. Preparatory Notions for Braidoiding

#### 6.2.1. Up-Arcs and Free Up-Arc

#### 6.2.2. Subdivision

#### 6.2.3. Sliding Triangles and Cut-Points

#### 6.2.4. Defining General Position

**Definition**

**7.**

#### 6.3. The Proof of Theorem 1

#### 6.4. The Uniform Labeling

**Corollary**

**1.**

#### 6.5. An Example of Braidoiding

## 7. $\mathit{L}$-Equivalence on Braidoid Diagrams

**Definition**

**8.**

- 1.
- Cut a strand of the braidoid diagram B at some point, not vertically aligned with a braidoid end, or an endpoint, or a crossing. This can be ensured by small isotopies.
- 2.
- Pull the resulting ends away from the cut-point to the top and bottom of B, respectively, keeping them aligned with the cut-point, and so as to create a new pair of corresponding braidoid strands. See Figure 1 for an abstract illustration of an L-move.
- 3.
- There are two types of L-moves, namely ${L}_{over}$ and ${L}_{under}$-moves, denoted by ${L}_{o}$ and ${L}_{u}$, respectively. For an ${L}_{o}$-move, pull the resulting new strands entirely over the rest of the diagram. For an ${L}_{u}$-move, pull the new strands entirely under the rest of the diagram. See Figure 24.

**Remark**

**2.**

## 8. From Braidoids to Classical Braids

**Proposition**

**3.**

**Proof.**

## 9. Toward an Algebraic Structure for Braidoids

#### 9.1. Implicit Points and Indexing

**Definition**

**9.**

#### 9.2. Elementary Blocks of Combinatorial Braidoid Diagrams

- We call the ends of strands in elementary blocks usual ends if they are not the endpoints. Usual ends can be concatenated with usual ends only at the same position so that the resulting diagram contains strands emanating from the top to bottom row.
- An endpoint can be concatenated either with another endpoint or with implicit points. Two concatenated endpoints remain at their vertical position as two disjoint endpoints. See Figure 29. When an endpoint is concatenated with an implicit point, the endpoint remains as it is, while the implicit point is annihilated.
- An implicit point can be concatenated with an implicit points or with an endpoint. Two implicit points concatenated in the middle row annihilate each other. See Figure 29.

## 10. An Application of Braidoid Theory to the Study of Proteins

- Step 1:
- A protein is taken from the Protein Database (PDB) [27]. It is converted to $xyz$-coordinates, and its backbone is extracted (namely, the coordinates of the $C\alpha $ atoms).
- Step 2:
- The reconstructed protein chain is placed inside a large enough sphere, and it is projected to, say, 100 different planes perpendicular to corresponding fixed directions evenly distributed on the sphere. The projections are planar knotoid diagrams. We take the dominant knotoid diagram, which is the one with the highest probability of appearance. The steps so far have been implemented in [22].
- Step 3:
- We turn the dominant knotoid diagram into a dominant braidoid diagram using the most efficient braidoiding algorithm for the specific knotoid diagram. It would be very useful to automatize this step through a computer program. We then fill in the braidoid diagram with implicit points, and we index accordingly, obtaining the combinatorial dominant braidoid diagram.
- Step 4:
- We list all combinatorial braidoid diagrams that are obtained from the dominant braidoid diagram by braidoid isotopy and cyclic permutation (that preserves braidoid closure). These all have isotopic closures. We then note all words in elementary blocks for which our combinatorial braidoid diagrams stand. This list of words is associated with the original protein form.
- Step 5:
- Any given protein can be then identified by comparing its associated set of algebraic words against the tabulated lists that we produce. The last two steps can be easily implemented in a computer program. For example, with the dominant knotoid diagram of the protein 3KZN (recall Figure 30 and Figure 31) would be associated the set of words $\{{l}_{2}{\sigma}_{1}^{3}{h}_{2},{\sigma}_{1}^{3}{h}_{2}{l}_{2},{h}_{2}{l}_{2}{\sigma}_{1}^{3}\}$. The interested reader could exercise further with the example given in Figure 23.

## 11. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Turaev, V. Knotoids. Osaka J. Math.
**2012**, 49, 195–223. [Google Scholar] - Bartholomew, A. Andrew Bartholomew’s Mathematics Page: Knotoids. Available online: http://www.layer8.co.uk/maths/knotoids/index.htm (accessed on 14 January 2015).
- Gügümcü, N.; Kauffman, L.H. New invariants of knotoids. Eur. J. Comb.
**2017**, 65C, 186–229. [Google Scholar] [CrossRef] - Artin, E. Theorie der Zöpfe. In Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg; Springer: Berlin/Heidelberg, Germany, 1926; Volume 4, pp. 47–72. [Google Scholar]
- Artin, E. Theory of braids. Ann. Math.
**1947**, 48, 101–126. [Google Scholar] [CrossRef] - Kassel, C.; Turaev, V. Braid Groups. Volume 247 of Graduate Texts in Mathematics; Springer: New York, NY, USA, 2008. [Google Scholar]
- Lambropoulou, S.; Rourke, C.P. Markov’s theorem in 3-manifolds. Topol. Appl.
**1997**, 78, 95–122. [Google Scholar] [CrossRef] - Kauffman, L.H.; Lambropoulou, S. Virtual braids. Fundam. Math.
**2004**, 184, 159–186. [Google Scholar] [CrossRef] - Brunn, H. Über verknotete Kurven. Verh. Intern. Math. Kongr.
**1897**, 1, 256–259. [Google Scholar] - Alexander, J.W. A lemma on systems of knotted curves. Proc. Nat. Acad. Sci. USA
**1923**, 9, 93–95. [Google Scholar] [CrossRef] [PubMed] - Birman, J.S. Braids, links and mapping class groups. In Annals of Mathematics Studies; Princeton University Press: Princeton, NJ, USA, 1974; Volume 82. [Google Scholar]
- Morton, H.R. Threading knot diagrams. In Mathematical Proceedings of the Cambridge Philosophical Society; Cambridge University Press: Cambridge, UK, 1986; Volume 99, pp. 247–260. [Google Scholar]
- Yamada, S. The minimal number of Seifert circles equals the braid index of a link. Invent. Math.
**1987**, 89, 347–356. [Google Scholar] [CrossRef] - Vogel, P. Representation of links by braid: A new algorithm. Comment. Math. Helv.
**1990**, 65, 104–113. [Google Scholar] [CrossRef] - Lambropoulou, S. A Study of Braids in 3-Manifolds. Ph.D. Thesis, University of Warwick, Coventry, UK, 1993. [Google Scholar]
- Lambropoulou, S. Short Proofs of Alexander’s and Markov’s Theorems; Warwick University Preprint: Coventry, UK, 1990. [Google Scholar]
- Lambropoulou, S. Braid equivalences and the L-moves. In Introductory Lectures on Knot Theory, Proceedings of the Advanced School and Conference on Knot Theory and its Applications to Physics and Biology, ICTP, Trieste, Italy, 11–29 May 2009; Kauffman, L.H., Lambropoulou, S., Przytycki, J.H., Jablan, S., Eds.; Series on Knots and Everything; World Scientific Press: Singapore, 2011; Volume 46, pp. 281–320. [Google Scholar]
- Gügümcü, N.; Lambropoulou, S.; National Technical University of Athens, Athens, Greece. Unpublished work. 2017.
- Jones, V.F.R. Index for subfactors. Invent. Math.
**1983**, 72, 1–25. [Google Scholar] [CrossRef] - Jones, V.F.R. Hecke algebra representations of braid groups and link polynomials. Ann. Math.
**1987**, 126, 335–388. [Google Scholar] [CrossRef] - Goundaroulis, D.; Dorier, J.; Benedetti, F.; Stasiak, A. Studies of global and local entanglements of individual protein chains using the concept of knotoids. Sci. Rep.
**2017**, 7, 6309. [Google Scholar] [CrossRef] [PubMed] - Goundaroulis, D.; Gügümcü, N.; Lambropoulou, S.; Dorier, J.; Stasiak, A.; Kauffman, L.H. Topological models for open knotted protein chains using the concepts of knotoids and bonded knotoids. Polymers
**2017**, 9, 444. [Google Scholar] [CrossRef] - Reidemeister, K. Elementare Begründung der Knotentheorie. In Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg; Springer: Berlin/Heidelberg, Germany, 1927; Volume 5, pp. 24–32. [Google Scholar]
- Goundaroulis, D.; University of Lausanne, Lausanne, Switzerland. Personal communication, 2017.
- Humphrey, W.; Dalke, A.; Schulten, K. VMD—Visual Molecular Dynamics. J. Mol. Graph.
**1996**, 14, 33–38. [Google Scholar] [CrossRef] - Shi, D.; Yu, X.; Roth, L.; Morizono, H.; Tuchman, M.; Allewell, N.M. Structures of N-acetylornithine transcarbamoylase from Xanthomonas campestris complexed with substrates and substrate analogs imply mechanisms for substrate binding and catalysis. Proteins
**2006**, 64, 532–542. [Google Scholar] [CrossRef] [PubMed] - Berman, H.M.; Westbrook, J.; Feng, Z.; Gilliland, G.; Bhat, T.N.; Weissig, H.; Shindyalov, I.N.; Bourne, P.E. The Protein Data Bank. Nucleic Acids Res.
**2000**, 28, 235–242. [Google Scholar] [CrossRef] [PubMed]

**Figure 2.**Examples of knotoid diagrams: (

**a**) trivial knotoid; (

**b**) a non-trivial planar knotoid that is trivial as spherical; (

**c**,

**d**) two proper knotoids; (

**e**) a knot-type knotoid.

**Figure 5.**The overpass and the underpass closures of a knotoid diagram resulting in different knots.

**Figure 7.**Some examples of braidoid diagrams: (

**a**) a braidoid diagram with one free strand; (

**b**–

**e**) braidoid diagrams with two free strands.

**Figure 19.**A sliding triangle violating the endpoint condition and further subdivision of the up-arc.

**Figure 26.**The choice of the arc connecting the endpoints does not affect the resulting braid up to L-equivalence.

**Figure 30.**The configuration of the backbone of the protein 3KZN in 3D and its simplified configuration.

**Figure 31.**The knotoid of the protein 3KZN and a corresponding braidoid diagram with algebraic expression ${l}_{2}{\sigma}_{1}^{3}{h}_{2}$.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gügümcü, N.; Lambropoulou, S.
Knotoids, Braidoids and Applications. *Symmetry* **2017**, *9*, 315.
https://doi.org/10.3390/sym9120315

**AMA Style**

Gügümcü N, Lambropoulou S.
Knotoids, Braidoids and Applications. *Symmetry*. 2017; 9(12):315.
https://doi.org/10.3390/sym9120315

**Chicago/Turabian Style**

Gügümcü, Neslihan, and Sofia Lambropoulou.
2017. "Knotoids, Braidoids and Applications" *Symmetry* 9, no. 12: 315.
https://doi.org/10.3390/sym9120315