MATH3001
Project Report
Geometric Constructions
November 27, 2024
1 introduction
Knots and entanglements are common topological features observed not only in the macroscopic world, but also at the molecular level like Fig 1. So in our daily life, they make different appli-cations from tying shoes to tangling headphone’s cord. But in special circumstances, Knots can cause us a lot of trouble. For example, if a baby’s umbilical cord is tangled in a knot, it may affect the fetus’s respiratory tract and cause suffocation and death.
Figure 1: knots in different applications
At first, knot theory is a branch of mathematics that belongs to the field of topology. It studies knots in space, which are closed curves formed by connecting a piece of rope end to end. These knots can be curves in three-dimensional space or objects in higher-dimensional space.And a mathematical theory of knot was borned by Vandermonde (1771). Thomson (1867) hypothe- sized that atoms are composed of knotted vortices of the aether although this hypothesis is wrong after it has been certificated. As a result, this aspect has also aroused people’s enthusiasm for this theory. Importantly,the knot can be distorted but essentially it cannot altered the foundation.
Thus for the fundamental problem in knot theory is to determine whether a closed loop embedded in three dimensional space is knotted by the Reidemeister moves and the decision was made by Haken, and several other algorithms were created afterwards, one of which is Lakenby. Meanwhile, Knots are being found to play a role in more and more scientific contexts, so the knot play an important role in different areas, like chemistry, physics, anthropology, biology and etc.
2 literature review
2.1 classification and invention of knots
In traditional, the theory of knots are all applied with different branches in science like Fig 2. From mathematics point of view, a knot is a closed loop in a topology state and if they don’t connect with each other then this problem cannot be solved. Then from the technology part, this knot cannot be defined in open chains (open chains is an arrangement of atoms represented in a structural formula by a chain whose ends are not joined so as to form a ring). But in Biology, all of these knots are open chains. Therefore in a simple loop, if the two ends don’t untie themselves, then it’s a knot. Then we can get different types of knots in the systematization,such as: trefoil, torus, singular, hyperbolic and others. Fig 3 is the diagram which shows different types of knots from unknot to 77.
Figure 2: different fields for knots
Figure 3: Different types of knots
2.2 mathematical theorem of knots
We know that one of the theorem help knots is the Reidemeister moves. Reidemeister’s theo- rem states that two diagrams represent ambient isotopic knots (or links) if and only if there is a sequence of Reidemeister moves taking one diagram to the other and if two links are piecewise linearly equivalent (ambient isotopic), then there is a sequence of Reidemeister diagram moves taking a projection of one link to a projection of the other. Finally the Reidemeister move has been shown in Fig 4.
2.3 Applications of Knots
In the 21th century, knots can be used in different fields. First in biology, DNA is the most clas- sicial example in it. So for DNA topology, it shows the shape and path of the DNA helix in three dimensional space. Topoisomerases are enzymes that change the topology of DNA. Finally is ma- terial science, in polymers and high-molecular materials, the entanglement and disentanglement of chains have an important influence on the physical properties of the materials. Understanding the disentanglement characteristics of these chains can help develop new materials and optimize their properties. For example,the bulk of material is semital and their valence and conduction bands touch at near Fermi level in Fig 2.
P vs NP
Figure 4: Reidemeister moves
The algorithm of unknotted problem is a long-standing open question. In another way,given a knot K described by a knot diagram or by a triangulation of its complement, is there a fast algorithm to decide whether K is the unknot? Hence Hakan is the first person to prove there are any algorithms. Also any other algorithms are constructed by the survey of Lackenby (M.
2016). And Lackenby also change the bound to polynomial in order to show that the unknotted recognition problem is in NP.From others, like Welsh (D. J.1993) proposed the study of qualitative rather than quantitative bounds on the algorithmic complexity of problems in knot theory (and by extension, in low-dimensional topology). And then Hass, Lagarias give the response for the number of Reidemeister moves for the diagram of unknotted problem is in complexity class NP since P is the class of yes–no functions (or yes–no questions or decision problems) on input strings that can be computed in polynomial time and the best way to explain for NP is to give an issue about explain if the integer N (in binary) is a composite number.At the same time.
we know that co-NP is the subset of NP which NP is the branch of some decision problems for a polynomial time which return true and co-NP is the branch of all decision problems for a polynomial time which return true.
3 Research Question
The most curious for me is that how to use a reasonable solution to untie a knot,
4 Internal references and citations
Bibliography
Airas, U., & Heinonen, S. (2002). Clinical significance of true umbilical knots: a population-based analysis. American journal of perinatology, 19(03), 127-132.
Vandermonde, (1771) Acad´emie des sciences, et al. M´emoires de math´ematique et de physique, tirez des registres de l’Acad´emie royale des sciences. de l’Imprimerie royale, 1771.
Thomson, W. (1867). II. On vortex atoms. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 34(227), 15-24.
Lamport, L. (1994). LATEX: a document preparation system. User’s guide and reference manual. Addison Wesley, second edition.
Mittelbach, F., Goossens, M., Braams, J., Carlisle, D., and Rowley, C. (2004). The LATEX Companion. Addison Wesley, second edition.
Lackenby, M. (2016).Elementary knot theory. arXiv preprint arXiv:1604.03778.
Welsh, D. J. (1993). Complexity: knots, colourings and counting (Vol. 186). Cambridge university press.