Knots and their projections I

Transcription

1 Knots and their projections I Uwe Kaiser Boise State University REU Lecture series on Topological Quantum Computing, Talk 1 June 7, 2011

2 A knot is formed by glueing the two ends of a tangled rope in 3-space.

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4 A link is a disjoint union of knots in 3-space.

5 Definition A (regular) link projection, also called link diagram is a graph with only 4-valent vertices and distinguished overcrossing/undercrossing information at each vertex. resolves to or

6 A mathematical definition of knots and links is given by considering smooth mappings of circles in 3-space with nowhere vanishing tangent vectors. Two links are considered isotopic or equivalent if they can be deformed smoothly into each other through links. In particular the number of circles does not change under such a deformation. We will not need to work with this definition directly. It is called the number of components of the link. It is a theorem of Reidemeister (1927) that each link admits a regular diagram. Moreover two diagrams represent equivalent links if and only if they differ by finite sequences of deformations of planar graphs and the following three famous moves:

7 Reidemeister moves RI-move R II-move R III-move

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9 Central questions of knot theory Given a knot diagram when does it represent the unknot? Given two knot diagrams, when do they represent the same knot? Definition A knot or link invariant is just a map {link diagrams}/(ri III ) algebraic or well known object Algebraic objects could be Z or other nice groups, or polynomials etc., just something where it is easy to decide when two objects are different or the same. When we say knot we often mean equivalence class of knot, and the same for links.

10 Definition A connected sum of two knots (or links) is formed deleting from each knot an arc and joining the two knots by a band. The connected sum of two knots K 1 and K 2 is denoted K 1 K 2.

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12 The connected sum of knots is well-defined. For links it is only well-defined up to the choice of components joined. Note that the connected sum of an r-component link and an s-somponent link is an r + s 1-component link. Definition A knot is prime if it is not the connected sum two nontrivial knots. Theorem (Schubert) Each knot can be decomposed uniquely, up to ordering the factors, as a connected sum of prime knots. Thus prime knots are the building blocks of knot theory just like prime numbers are the multiplicative building blocks of the integers.

13 crossing number Definition The crossing number c(l) of a link L is the minimal crossing number over all possible diagrams of the link L. The crossing number is difficult to determine. Knots are tabulated using the crossing number. It is trivial that c(k 1 K 2 ) c(k 1 ) + c(k 2 ). It is a long standing conjecture that that c(k 1 K 2 ) = c(k 1 ) + c(k 2 ). Lackenby proved the following in 2009: c(k 1 ) c(k n ) 152 c(k 1... K n ) c(k 1 ) c(k n )

14 unknotting numbers Definition The unknotting number of a knot is the minimal number of crossing changes necessary to untie the knot. Some facts. If a knot has unknotting number n then there exists a diagram of the knot such that n crossing changes in that diagram will give a diagram of the unknot (even though usually a difficult one). There exist knots with arbitrarily high unknotting number. It is very difficult to determine the unknotting number of a knot. Theorem (Scharlemann): Unknotting number 1 knots are prime.

15 Definition An oriented link is a link with a direction assigned to each component This is an oriented version of the 5 1 torus knot. It is a difficult question whether an oriented knot is equivalent to its inverse, i. e. the same knot but with reversed orientation.

16 For an oriented link diagram to each crossing we can assign a sign in the following way: Definition The writhe w(d) of a link diagram is the sum over all signs for all crossings. The writhe is an invariant of diagrams up to Reidemeister II and III moves but is changed under the Reidemeister I move. The equivalence of diagrams uder planar isotopy and Reidemeister II and III moves is called regular isotopy

17 Gauss code of an oriented knot Gauss code: (over-crossing positive, under-crossing negative) extended Gauss code: (the second time record positive/negative crossing)

18 Here are some movies to watch concerning knot theory. Traditionally KNOT THEORY is considered part of a mathematicial field called TOPOLOGY. Topology studies more generally the placement problem of one space into another space. More generally it studies rough geometric features like holes in spaces like subsets of R 3. This in particular applies to the complements R 3 \ K of knots K R 3. It has been proved in 1986 by Gordon and Luecke that knots are determined by their complements. Not Knots

19 Literature 1. Colin Adams: The Knot Book, Freeman Co Charles Livingston: Knot Theory, MAA Kunio Murasugi: Introduction to Knot Theory

20 Exercises 1. Untie the bottom knot on page 7 and record the Reidemeister moves you used. 2. How does the writhe of a diagram change if all components get the reverse orientation? 3. Find the Gauss codes of the knots in the table on page Draw the oriented knot diagram with extended Gauss code A knot diagram is alternating if when traversing the diagram overcrossings and undercrossings alternate. What is special about the extended Gauss code of an alternating diagram? 6. What sequences occur as extended Gauss-codes of a knot?