매듭이론 (knot theory)

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개요

• 매듭(knot)
• 3차원 상에 놓인 원과 위상동형인 곡선, 또는 3차원 상에 놓인 자기자신과 만나지 않는 닫힌 곡선
• 동위(isotopy)
• 3차원 상에서 매듭을 끊지 않는 연속적인 변형
• 매듭 diagram
• 라이데마이스터 변형
• 20세기말에 통계역학, 양자군, 양자장론과의 관계가 발견되어 큰 발전

중요한 문제

• 주어진 두 매듭이 동위관계에 있는지를 판단하는 문제
• 매듭의 분류
• 중요 미해결 문제
• Does there exist a knot in R3, different from the unknot , whose Jones polynomial is equal to 1?”

매듭 diagram

• 3차원 공간에 놓인 매듭을 2차원 평면에 사영하여 얻어짐

라이데마이스터 변형

• 매듭 diagram 에 가하는 변형
• 매듭이 3차원 공간에서의 연속적인 변형을 통하여 다른 매듭으로 변하면, 매듭 diagram에 세가지 라이데마이스터 변형을 가하여 같은 결과를 얻을 수 있다
• 매듭으로부터 정의된 양이 불변량임을 증명하는데 흔히 사용
• 라이데마이스터 변형 1 - disapperanace of a little loop
• 라이데마이스터 변형 2 - twin crossing 의 제거
• 라이데마이스터 변형 3 - 크로싱 위로 thread의 이동

 라이데마이스터 변형 I 라이데마이스터 변형 II 라이데마이스터 변형 III

불변량

• 동위관계에 있는 두 매듭에 같은 값을 주는 양
• 동의관계에 있는 매듭에는 같은 다항식이 대응되나, 다항식이 같다고 매듭이 동위관계에 있다고는 말할수 없다
• 서로 다른 매듭을 구분할 수 있는 더 강력한 불변량을 찾는 것은 매듭이론의 중요한 주제이다
• 알렉산더-콘웨이 다항식
• HOMFLY 다항식
• 존스 다항식
• 바실리예프 다항식
• 실타래 관계를 이용하여 정의되는 경우가 많다

실타래 관계 (skein relation)

• 나머지 부분이 같고, 한 교차점에서만 다른 매듭의 oriented diagram을 실타래 diagram이라 한다
• 유향매듭 L이 있을때, 다음과 같이 $$L_{+},L_{-},L_{0}$$ 을 정의한다
• 다항식으로 정의되는 여러 불변량들은 이 세 실타래들이 만족시키는 관계를 가지며, 이를 실타래 관계라 한다
• 불변량을 재귀적으로 정의할 수 있게 된다

다항식 불변량의 예

알렉산더-콘웨이 다항식

• 각 매듭에 대해 정의되는 z를 변수로 가지는 정수계수다항식 $$\nabla(\cdot)$$
• 실타래 관계(skein relation)$\nabla(O) = 1$$\nabla(L_+) - \nabla(L_-) = z \nabla(L_0)$

존스 다항식

• 각 매듭에 대해 정의되는 $$t^{1/2}$$를 변수로 가지는 정수계수 로랑다항식 $$V(\cdot)$$
• 실타래 관계(skein relation)$V(O) = 1$$(t^{1/2} - t^{-1/2})V(L_0) = t^{-1}V(L_{+}) - tV(L_{-})$

홈플리 (HOMFLY) 다항식

• HOMFLY는 사람의 이름이 아니라, 발견자 여러 명의 머리글자이다
• 알렉산더-콘웨이 다항식과 존스 다항식의 일반화
• 매듭에 정의되는 이변수다항식 $$P(\cdot)$$
• 실타래 관계

역사

• 1984년 존스 다항식
• 1988년 위튼이 존스 다항식을 양자장론의 틀로 설명[Witten1989]
• 1990년 존스, 위튼 필즈메달 수상
• 수학사 연표

관련된 항목들

• 양자군 (quantum group)

노트

말뭉치

1. You may not have heard of knot theory.
2. Knot theory seeks to encode information about knots -- including what happens when they are altered in some way -- into algebraic expressions that can distinguish one knot from another.
3. Knot theory has uses in physics, biology and other fields, Menasco says.
4. Knot theory provides insight into how hard it is to unknot and reknot various types of DNA, shedding light on how much time it takes the enzymes to do their jobs.
5. Knot theory, in mathematics, the study of closed curves in three dimensions, and their possible deformations without one part cutting through another.
6. Although Kelvin’s theory was eventually rejected along with ether, knot theory continued to develop as a purely mathematical theory for about 100 years.
7. Other applications of knot theory have been made in biology, chemistry, and mathematical physics.
8. Therefore, a fundamental problem in knot theory is determining when two descriptions represent the same knot.
9. The original motivation for the founders of knot theory was to create a table of knots and links, which are knots of several components entangled with each other.
10. More than six billion knots and links have been tabulated since the beginnings of knot theory in the 19th century.
11. This record motivated the early knot theorists, but knot theory eventually became part of the emerging subject of topology.
12. Question: How can knot theory help us understand DNA packing?
13. It is perhaps not surprising then that connections between mathematical knot theory and biology have been discovered.
14. By thinking of DNA as a knot, we can use knot theory to estimate how hard DNA is to unknot.
15. Knot theory can help scientists discover the mechanisms by which these enzymes work.
16. Below, Edward Witten, Charles Simonyi Professor in the School of Natural Sciences, describes the history and development of the Jones polynomial and his interest in knot theory as a physicist.
17. But to understand knot theory, at least for the moment, we are going to imagine a world of only three spacetime dimensions—two space dimensions and one time dimension.
18. their properties is known as knot theory.
19. Knot theory is the mathematical branch of topology that studies mathematical knots, which are defined as embeddings of a circle in 3-dimensional Euclidean space, R3.
20. Research in knot theory began with the creation of knot tables and the systematic tabulation of knots.
21. In the last 30 years, knot theory has also become a tool in applied mathematics.
22. Chemists and biologists use knot theory to understand, for example, chirality of molecules and the actions of enzymes on DNA.
23. An introduction to knot theory which seems to be aimed at teachers of mathematics can be found at Los Alamos National Laboratory.
24. A huge page of links to pages on knots and knot theory of all kinds.
25. An on knot theory appears in the November 1997 issue of American Scientist.
26. A page at the Univ. of Liverpool for accessing preprints on knot theory.
27. There are various pages related to knot theory that are linked from the main articles.
28. Still, even without any other application in sight, the mathematical interest in knot theory continued at that point for its own sake.
29. One of the main goals of knot theory has always been to identify properties that truly distinguish knots—to find what are known as knot invariants.
30. When a better mathematical model (in the form of the Bohr atom) was discovered, mathematicians did not abandon knot theory.
31. Unexpectedly, the Jones polynomial and knot theory in general turned out to have wide-ranging applications in string theory.
32. Slice knots “provide a bridge between the three-dimensional and four-dimensional stories of knot theory,” Greene said.
33. Most people would see this as a frustrating test of dexterity, but for the mathematician it’s the real-life manifestation of a field that is still the focal point of research to this day: knot theory.
34. The fascination with knot theory kickstarted in the 1860’s, when the predominant ideology was that a mysterious substance called ‘ether’ permeated the entire universe.
35. Skip forward to the 1980’s and knot theory had found an application: biochemists discovered that DNA unknots and knots itself using tailor-made enzymes.
36. Cryptography, statistics and quantum computing all utilise aspects from knot theory, with developments in the field often occurring as a by-product from research into quantum physics.
37. In general it is very difficult problem to decide if two given knots are equivalent, and much of knot theory is devoted to developing techniques to aid in answering this question.
38. Most of knot theory concerns only tame knots, and these are the only knots examined here.
39. An introduction to knot theory.
40. Physical knot theory: an introduction to the study of the influence of knotting on the spatial characteristics of polymers.
41. Until molecular biologists discovered that knot theory could help them understand how DNA, the genetic material, is twisted and knotted inside cells, knot theory had no practical applications.
42. We will also explore the applications of knot theory to biology, chemistry, and physics.
43. “In knot theory, mathematicians spend most of their time developing tools to tell which knots are the same and which knots are different.
44. Why the public should care about knot theory is hard to convey – but care they should.
45. For Denne, the true challenge of Taping Shape was finding ways to make knot theory resonate with people of all ages and educational backgrounds.
46. In addition to knot theory, topology and geometry, the exhibit also touches on spatial relations, membranes, mathematics and tensile strength.
47. But from the mathematical viewpoint, a gold mine had been discovered: The branch of mathematics now known as "knot theory" has been burgeoning ever since.
48. The techniques of knot theory which are based on the study of knot diagrams are called combinatorial methods.
49. Significance Knot theory is a branch of topology that deals with study and classification of closed loops in 3D Euclidean space.
50. As a branch of topology, knot theory is a developing field, with many unresolved questions, including the ongoing search for an algorithm that will provide an exact identification of arbitrary knots.
51. Each of these cases is unique, as the rules of knot theory interact with the rules and restrictions of each underlying material and confinement.
52. The investigation of knotted fields is thus a specialized topic where certain theoretical aspects of knot theory emerge in a physical context.
53. Besides the relation of enveloping isotopy, in knot theory one studies other, coarser, equivalence relations between links.
54. Another equivalence relation studied in knot theory is concordance or cobordism.
55. The dream began in the 1860s with an ingenious knot theory of nature.
56. According to a press release, her master’s thesis focused on knot theory and her Ph.D. work focuses on braids and how to translate them into matrices, which are easier to understand and manipulate.
57. Knot theory is a broad field involving dimensional tangles and the work of untangling them.
58. Mathematicians use knot theory models to try to explain the stresses a knot is subject to.
59. Knot theory is a branch of pure mathematics, but it is increasingly being applied in a variety of sciences.
60. Discovering the Art of Knot Theory lets you, the explorer, investigate the mathematical concepts and ideas of knot theory using Tangles®.
61. This Journal is intended as a forum for new developments in knot theory, particularly developments that create connections between knot theory and other aspects of mathematics and natural science.

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• [{'LOWER': 'knot'}, {'LEMMA': 'theory'}]