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Knot Module Lecture Notes (As of June 2005) Day 1 - Crossing and Linking Number

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Knot Module Lecture Notes (as of June 2005) Day 1 - Crossing and Linking number 1. Definition and crossing number. (Reference: §§1.1 & 3.3 of [A]. References are listed at the end of these notes.) • Introduce idea of knot using physical representations (e.g., extension cord, rope, wire, chain, tangle toys). • Need to connect ends or else can undo knot; we’re allowed to do anything but cut. • The simplest knot is the unknot, but even this can look quite complicated. (Before class, scrunch up a big elastic band to make it look complicated. Then, in class, unravel it to illustrate that it’s the same as the unknot). • The next is the trefoil. There are two fairly simple projections which look quite different (demonstrate with extension cord). Figure 1: Two projections of the trefoil. Figure 2: How to get from one projection to the other. • We say the trefoil has crossing number three (we will write C(trefoil) = 3), since it has no projection with fewer than three crossings. We will prove this by showing projections of 0, 1, or 2 crossings are the unknot. Show that projections of 1 crossing are all the unknot. You may wish to identify projections that are the same up to rotation. In this case, there are four types of 1 Figure 3: Some 1-crossing projections. 1-crossing projection. If you also allow reflections, then there are only two types, one represented by the left column of Figure 3 and the other by any of the eight other projections. The crossing number two case is homework. • There are also two different flavours of trefoil, left and right. They are mirror reflections of one another. Figure 4: Left and Right trefoils. 2 ? = Figure 5: Is the figure 8 the same as its mirror reflection? This doesn’t happen for next most complicated knot, the figure eight, since it’s its own mirror reflection. Allow students to explore going between different projections of unknot, trefoil and figure eight. Figure 6: How to get from the figure 8 to its reflection. 3 Figure 7: Famous knots in history. • Write up some definitions (or elicit them from students): A knot is a closed curve in space which doesn’t intersect itself. A knot projection is a picture of the knot with crossings shown by breaks in the curve. • The knots up to five crossings are famous enough to have their own names. Aside from the three we’ve already discussed, there are two 5-crossing knots: the star knot and the tweeny knot. To distinguish between these two, label the crossings a, b, c, d, and e. When you trace along the knot, you will pass through each crossing twice before returning to the starting point. For a star knot, the sequence of crossings is repeated twice, i.e., if you visit crossings a, c, d, e, and b in that order to start with, then you will again visit a, c, d, e, and b before returning to the start point. • It will take some time for students to feel comfortable copying knots into their notes, so consider providing them with handouts of important diagrams. We encourage you to use the names of the knots whenever convenient to assist note takers. 2. Orientation, crossing sign, and linking number. (Reference: §1.4 of [A].) • We can distinguish between the two trefoils using crossing sign. • To determine crossing sign, first orient the knot by choosing a direction to travel around the knot (indicated by adding arrows). As in Figure 8, right- and left- handed crossings will be labelled with the “sign” +1 and -1, respectively. There are several, equivalent, ways to determine if a crossing is left or right handed; use whichever works best for you and your students: 4 Figure 8: Left- and Right-handed crossings. Methods for determining crossing sign For (a) and (b) below, you will need to use your left hand for a left-handed crossing and your right hand for a right-handed crossing. If the knot is drawn on paper, it’s often more convenient to move the paper around rather than trying to contort your hand into the correct position. (a) Place thumb in direction of the overstrand and imagine curling fingers around the overstrand so that fingertips point in the direction of the understrand. (b) Lay your hand palm down on the paper with fingertips pointing in the direc- tion of the overstrand. Then your thumb will point in the direction of the understrand. (c) Rotate the understrand so as to make the arrows line up. For a right-handed crossing you will rotate clockwise and for a left-handed crossing you will rotate counter-clockwise. (Compare to rotating the understrand to sweep out water – see notes for Day 2 below – when forming the Lake and Island polynomial). • A three crossing projection of the right trefoil has all right-hand crossings while a three crossing left trefoil has only left-handed crossings. • A link is a collection of closed curves. The closed curves are called components of the link. • We can often distinguish links using linking number. • To evaluate the linking number, add the signs of crossings between two compo- nents of the link, take the absolute value, and divide by two. • Some famous links (see Figure 9) are the unlink, Hopf link, King Solomon’s link, Whitehead link, and the Borromean rings. Hopf was a German mathematician and J.H.C. Whitehead came up with his link as a counterexample to a project proposed by John Milnor as a way to prove the Poincar´e Conjecture. The Bor- romeo’s were an Italian family prominent during the Renaissance and these rings appeared in their family crest. The rings symbolise strength in unity. If any one of the three is removed, the other two fall apart. • Calculate the linking number of the unlink, Hopf link, and Solomon’s link. (In Homework students will learn that the Whitehead link, like the unlink, has linking number zero. So linking number is not a complete invariant, i.e., it’s not one to one.) 5 Figure 9: Famous links in history. 6 Homework: 1. Complete the proof that the trefoil knot has crossing number three by examining projections with two crossings. Try to draw all possible knot projections with two crossings. You should have more than ten projections. Do you think you’ve drawn them all? Explain why or why not. Show that each of the knot projections you’ve drawn is a projection of the unknot. 2. For each of the following projections (see Figure 10), determine if it represents an unknot, trefoil, figure 8, star knot, or the tweeny knot. Figure 10: Which knots are these?. 7 3. Give 5 different projections of the unknot (all different from those in the previous exercise). Give three different projections of the trefoil. Find some “nice” projections of the figure eight knot. 4. Explain why any knot has a projection with over 1000 crossings. 5. An alternating projection is one where the over- and undercrossings alternate as we trace along the curve. Identify the alternating projections in Figure 10. Also list the number of crossings in each of the projections as well as the crossing number of the knot it represents. Knot a b ... Alt/Non N A ... # Crossings 4 4 ... C(K) 0 4 ... Can you see a connection between crossing number and whether or not a projection is alternating? When does an alternating knot have C(K) crossings? How about a non-alternating knot? 6. Just as there are two trefoils, there are right and left versions of the 5-crossing knots. Draw projections of the left star knot and the right star knot. Which of the two tweeny knots would you call “left”, and which “right”? 7. What happens to crossing signs in a knot projection when the orientation is reversed. How about in a link projection? Why do we take the absolute value in computing linking number? 8. Why is the linking number an integer (rather than a half integer)? 9. Calculate the linking number of the links in Figure 11. Figure 11: Calculate the linking number. 8 Extension Problems 1. Show that by changing some undercrossings to overcrossings any projection can be made into a projection of the unknot. (See §3.1 of [A].) (Of course changing crossings will change the knot.) 2. Show that by changing crossings any projection can be made into an alternating pro- jection. 3. A link is Brunnian if the link is not an unlink, but removing any one component results in the unlink. For example, the Borromean rings are Brunnian. Find a Brunnian link of four components. Find Brunnian links with an arbitrary number of components. 4. Find examples of more complicated knots, e.g., knots with 6 or 7 crossings. Which are the same as their mirror reflections? 5. To keep track of knots, mathematicians have organised them into tables. In 1974, an amateur mathematician, K.A. Perko, noticed an error in the knot tables which had gone undetected for almost 100 years. He noticed that two of the following four projections actually represent the same knot. (Knot theorists had thought they were different). Which two are the same knot? (Note: This is a very difficult problem.) Figure 12: Which two of these four are the same knot? 6. Make up a conjecture and see if you can prove it. 9 Day 2 — Lake & Island Polynomial and Unknotting number 1. Homework review (Reference: §6.2 of [A].) • Revisit Homework including idea of alternating knot and connection with cross- ing number.
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