- Contents
- The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots
- Colin Adams - Google Scholar Citations
- Adams, The Knot Book
- No Mathematical Work is Ever Wasted

The Knot Book. An Elementary Introduction to the Mathematical Theory of Knots. Colin C Adams. American Mathematical Society. Providence, Rhod e Islan d. in a high school English class, for example, could work with the book over If you come across such a word Bhagavad-Gita As It Is. 1, Pages·· The Knot Book is also about the excitement of doing mathematics. Colin Adams engages the reader with fascinating examples, superb figures.

Author: | CONCEPCION SCHEHL |

Language: | English, Spanish, Hindi |

Country: | Belgium |

Genre: | Personal Growth |

Pages: | 739 |

Published (Last): | 16.08.2016 |

ISBN: | 197-7-49529-328-1 |

Distribution: | Free* [*Sign up for free] |

Uploaded by: | MARCELLA |

by Sir Randolph Bacon III (cousin -‐in-‐law to Colin Adams). Abstract: Book", an elementary introduction to the mathematical theory of knots, "Why. Knot?. domarepthestten.ga I had a class with Adams while at Williams (multivariable Calculus, never got to take his Knot Theory class). Wow - I took introductory topology with Colin just a few years ago. EDIT: also forgot to mention the knot book and his topology book are. Meeting Location - LCB Textbooks -. The Knot Book by Colin C. Adams. Knots and Links by Peter Cromwell. Knot Theory by W.B. Raymond Lickorish.

Students of all mathematical backgrounds are welcome. Week 1 Tuesday: What is a knot? When are two knots the same? Projections and Reidemeister moves. Connected sums and prime knots. Assign homework groups. Homework due Thursday Review the syllabus.

Rating details. Sort order. Aug 03, Barb rated it liked it Shelves: The book provides a general overview that is fairly accessible to a lay person like myself. I had some difficulty with explanations of rational knots. The accompanying illustration seemed different from the text explanation. The explanation of hyperbolic knots requires much more mathematical knowledge than I currently have.

Fortunately, the annotated bibliography is very thorough, so I'll try a couple of those books. I had to return this book to the library before I could finish it. I expect to r The book provides a general overview that is fairly accessible to a lay person like myself. I expect to reread it at some future time. Aug 29, Jon rated it really liked it. Covers a large number of topics e. Provides lots of interesting things to think about, and will often gloss over the technical details, which I think makes this book work very well actually.

My favorite parts were probably supercoiling, knot and graph polynomials, and the discussions of three-manifolds. Apr 25, Philip Rideout rated it really liked it. This book is a rarity because it sits exactly at the right level of math for me: I admit that I just never "got" some of the more advanced stuff like hyperbolic knots, but Jones Polynomials etc are fairly easy to understand and fascinating.

The most readable math text about what I find to be the most interesting field in math. Feb 26, kye rated it really liked it. They were great; I wish I could've read the rest of the book! Great diagrams and explanations. Wished there were solutions to the exercises.

Jan 08, Bob Woodley rated it it was amazing. A great introduction to Knot Theory. So accessible and approachable yet the book takes us to far-out mathematical realms such as topological embeddings, Dehn surgery, the Poincare conjecture. It has a well-annotated bibliography as well. Why don't more writers annotate their bibliographies? This book is a wonderful introduction to knot theory. If you're curious about what knot theory is, give this a read.

It's not only one of the few math books that you can simply read cover-to-cover, but it's also enjoyable for both mathematicians and non-mathematicians. Dec 13, Saman rated it liked it.

We extend th e understran d o f th e kno t and the n dra w i n th e nex t crossing. Because th e kno t i s. Note that we have two choices as to how t o circl e around: All th e way along.

Tabulating Knots 3 7 alternating. But if one of the la- bels has occurre d before. We continue thi s process unti l th e nex t intege r tha t shoul d b e place d o n a crossing alread y labels a n existin g crossing. I f neithe r o f th e label s o n th e nex t cross - ing has occurred before.

For the time being. We continue i n thi s manner. W e then kno w tha t th e kno t mus t no w circl e around t o pass through tha t crossing. We continue the overstrand throug h thi s crossing to the next crossin g where it becomes the understrand. Whe n th e kno t i s am - phicheiral.

Th e advantag e t o projectin g ont o a sphere i s that ther e i s n o special oute r regio n wit h infinit e are a a s ther e i s i n a projectio n ont o th e plane. The best way to see this is to think of pro- jecting th e knot onto a sphere Figur e 2. Ou r choic e can change th e resultin g knot. Although th e possibl e projection s loo k different.

Note tha t th e tw o knot s ar e composit e knots. When th e permutation o f the even number s ca n be broken int o two sepa - rate subpermutations. Tabulating Knots 3 9 Figure 2. If the even integer i s assigned t o the crossing whil e we ar e o n th e overstran d a t that crossing.

The system that we have explained work s very well for describing th e projection o f a n alternatin g knot. Thi s exercise gives us an upper bound o n the number of possible alternatin g kno t projection s wit h seve n crossings. Our rul e i s as follows: Whe n traversin g th e knot usin g th e labelin g system tha t w e have described.

But if the even integer is assigned t o the crossing while we are on th e understrand o f tha t crossing.

How abou t recognizin g a Typ e II Reidemeister mov e that will reduce th e number o f crossing s by two? See Figure 2. Dowker's notation allow s us to feed projection s o f knots into the com- puter simpl y a s a sequence o f numbers. In particular, suppos e w e wante d to attemp t a classificatio n o f 1 4-crossin g knots. The numbe r o f sequence s of the 1 4 numbers 2,4, 6, 8,10,12,14,16,18,20, 22 , 24, 26, 28 is 14!

Of course, there aren't thi s man y different knot s with 1 4 crossings. Lots of the sequences represent the sam e knot. I n fact , lot s o f th e sequence s represen t th e sam e projectio n o f th e same knot. Morwen Thistlethwait e use d th e Dowke r notatio n t o lis t al l o f th e prime knots of 1 3 or fewer crossings.

Perhaps it will turn out to be the best way to list knots of 1 4 or fewer crossings. This was the notation he used in order to tabulate the prime knots throug h 11 crossings an d prim e link s throug h 1 0 crossings i n 1 Although h e did no t use a computer, he missed onl y four knots. The Conway notatio n has been utilized in order to prove numerous results and recently has been applied t o knotting i n DNA se e Section 7. It is par - ticularly suited to calculations involving what are called tangles.

A tangle in a knot or link projection i s a region in the projection plan e surrounded b y a circle such tha t th e knot o r link crosse s the circle exactl y four time s Figure 2. We will always think of the four point s where th e knot o r lin k crosse s th e circl e as occurrin g i n th e fou r compas s direction s NW, NE, SW, and SE. We can use tangle s as the building blocks of knot an d lin k projection s Figure 2. Therefore, understanding tangle s will be very helpful i n un - derstanding knots.

W e will sa y tw o tangle s ar e equivalen t i f w e ca n ge t from on e to the other by a sequence of Reidemeister move s while the fou r endpoints o f the strings in the tangle remain fixed an d whil e the strings of the tangl e neve r journe y outsid e th e circl e definin g th e tangle. So, for in - stance, th e tw o tangle s i n Figur e 2. Notice tha t i f we for m a knot fro m a single tangl e by gluin g togethe r the ends in pairs as we did i n Figure 2.

Let' s loo k a t some particula r tangle s tha t ar e eas y to form. On e o f th e simples t tangle s is tw o vertica l strings , a s i n Figur e 2.

We denote th e tangl e consistin g o f tw o horizonta l string s a s th e 0 tangle. W e coul d win d tw o horizonta l string s aroun d eac h othe r t o ge t Figure 2. W e denot e thi s tangl e b y th e numbe r o f left-hande d twist s we pu t in. If w e ha d twiste d th e othe r wa y around, w e would hav e denote d th e resulting tangl e by —3. Note tha t fo r a positive-intege r twist , th e overstran d alway s ha s a positive slope , if w e think of it as a small segment of a line.

We are goin g t o for m a mor e complicate d tangle , startin g fro m th e 3 tangle. Thin k o f thi s reflectio n a s i f w e reflected i n a mirror that was perpendicular t o the projection plan e and in - tersected th e projectio n plan e alon g th e N W an d S E diagona l line. Not e that the two end s of th e tangle along the diagonal ar e fixed whe n w e per - form th e reflection, whil e the two ends of the tangle that are not on the di- agonal ar e switched. I t i s sometime s difficul t t o pictur e wha t happen s t o the crossing s unde r th e reflection.

Usually , w e ca n figure ou t wha t hap - pens to one crossing and the n we can infer wha t must happen t o the othe r crossings. Not e tha t fo r a positive-intege r twist , i t i s stil l tru e tha t afte r reflection th e overstrand ha s positive slope. Now w e win d th e tw o right-han d end s o f th e tangl e aroun d eac h other t o get Figure 2. We denote thi s tangle by 3 2, as the original tan - gle had thre e twist s o f th e horizonta l string s followe d b y a reflection an d then two twists of the horizontal strings.

Let's complicate this tangle still further. First , we take the tangle 3 2, as in Figur e 2. W e denot e this tangle 3 2 — 4. We call an y tangl e tha t w e coul d construc t i n thi s manne r a rationa l tangle.

The n w e coul d twis t th e tw o right-han d endpoint s around eac h other whil e keeping th e left-hand endpoint s fixed. W e could then alternatel y twis t th e botto m tw o endpoint s an d th e righ t tw o end - points t o create the tangle. A positive-integer twis t alway s give s the over - strand a positive slope, regardless of whether th e twist is occurring in tw o vertical strands or two horizontal strands Figure 2.

Similarly, if the rational tangle is represented b y an odd numbe r o f in- tegers, w e ca n construc t i t by startin g wit h tw o horizonta l string s th e 0 tangle an d alternatel y twistin g th e tw o right-han d endpoint s appropri - ately, followed b y twisting the two bottom endpoints appropriately.

Amazingly enough , ther e is an extremely simpl e way t o tell if two ra - tional tangles are equivalent. Suppose the two tangles are given by the se- quences o f integer s — 2 3 2 and 3 — 2 3. We compute th e so-calle d contin - ued fraction s correspondin g t o thes e integers. Th e continue d fractio n corresponding to - 2 3 2 is. We put the first — 2 in the denominator o f a fraction wit h numerator 1.

We then added th e last numbe r to the result. Notice that we can clean up this fraction:. I t i s i n fac t th e cas e that , sinc e thei r continue d fractions ar e equal, these two rationa l tangle s are equivalent Figur e 2. Thus , this tangle i s distinct fro m th e two equivalen t tangles - 2 3 2 and 3 -2 3. In general , suppos e w e hav e tw o rationa l tangle s give n b y th e se - quences o f integer s ijk. Im and npq. We can comput e th e cor - responding continued fraction s.

So for instance. We can us e ou r nota - tion fo r rationa l tangle s t o denote th e correspondin g rationa l knot. We call this notation Conway's notation. I n th e table at the end of the book. The two tangle s ar e equivalen t if and only if these two rational numbers are the same.

I f you ar e interested. The proof tha t tw o rationa l tangle s ar e equivalen t i f an d onl y i f thei r continued fraction s yiel d th e sam e rationa l numbe r i s difficult. As an exam - ple. Not e that this definition o f multiplication fit s i n nicely with ou r definitio n o f a rational tangle.

We ca n us e th e rationa l tangle s t o construc t mor e complicate d tan - gles. Note also that multiplying a rational tangle by a n intege r tangle will always generate a rational tangle. We reflect th e first tangl e across its NW t o S E diagonal line. Ti Figure 2. Tabulating Knots 4 7 Exercise 2. You do not need t o use the given projections.

We can also "add" together tw o tangles. We can think o f the rationa l tangle 3 2 as comin g fro m multiplyin g togethe r th e tw o tangle s 3 and 2. W e denot e th e lin k th e sam e wa y w e denot e th e correspondin g tangle. S o w e have th e operation s o f additio n o f tangle s an d multiplicatio n o f tangles.

Such a link is also sometimes called an arborescent link. An algebrai c lin k i s simpl y a lin k forme d whe n w e connec t th e N W string t o the NE strin g an d th e SW string t o the S E string o n a n algebrai c tangle.

So we would denot e the tangle for 8 5 by 3. We will call any tangl e obtaine d b y th e operation s o f additio n an d multi - plication on rational tangles an algebraic tangle.

Numerous additiona l example s appea r i n th e appendi x table. A knot obtaine d from a tangl e represente d b y a finite numbe r o f integer s separate d b y commas is often calle d a pretzel knot. We form a new kno t b y cutting the knot open along four point s on each of the four string s comin g out o f Ti. While w e ar e discussin g tangles. It's not true that ab -ba fo r al l tangles. Any of these thre e operation s i s calle d a mutation. More- over.

If we did bot h operations in turn. Suppose we have a knot K that we think of a s being formed fro m tw o tangles. Tabulating Knots 4 9 These algebrai c tangle s ar e behavin g a lo t lik e th e rea l numbers.

Th e resulting kno t look s lik e Figur e 2. We call 1 a multiplicative identity. Multiplication o n tangles is also not associative.

In the real numbers. Although man y tangle s are algebraic. The rea l number s als o hav e th e numbe r 1 s o tha t multiplyin g an y number by 1 doesn't change it. I s it the sam e if you multiply a tangle by it on the right side or the left side? There ar e difference s betwee n th e structur e o f th e rea l number s an d the structure of algebraic tangles.

A Figure 2. But for a tangle T. Bu t the real numbers hav e an element 0 so that adding 0 to a number doesn' t chang e the number. We call 0 an additive identity for the real numbers. Figure Th e Kinoshita-Terasaka mutants. Figure Mutan t knots. Note that w e can then permute n tangles in a row. In Chapter 7. From a projectio n o f a kno t o r link.

W e discuss the m agai n i n Chapter 6. Tabulating Knots 5 1 ab Figure Sho w tha t thes e knot s ar e related throug h a sequence o f mu - tations. Figure Two nasty mutants. A t least. What i s a graph? I t consists o f a set o f point s calle d vertice s an d a se t of edge s tha t connec t them. First shade every othe r regio n of the link pro- jection so that the infinite outermos t region is not shaded Figur e 2. Her e w e ar e interested i n planar graphs. Although mutatio n ca n tur n on e kno t int o another.

I t provide s a bridg e betwee n kno t theory an d grap h theory. We call the result a signed planar graph Figure 2. Figure 2 36 Shade d link projections. W e now hav e a wa y t o turn any link projection into a signed planar graph. It doesn't depen d i n any way on whether a crossing is an over- crossing o r a n undercrossing. This is the graph corresponding to our projection. Put a vertex at the center of each shaded region and then connect with an edge any two vertices that are in regions that share a crossing Figur e S o w e defin e crossing s t o b e positiv e o r negative depending o n which wa y the y cross as in Figure 2.

There is only one problem. Tabulating Knots 5 3 Figure Sign s on crossings. Connect th e edges inside each region of the graph as in Figure 2. Figure A signed planar graph from a knot projection. Shade those areas that contai n a vertex. The result is a link Fig- ure 2. What if we want to go in the other direction? Can we turn any signed planar grap h int o a kno t projection? Certainl y Startin g wit h th e signe d planar graph. I n particular. This is equivalent t o askin g whether o r no t ther e i s a sequenc e o f Reidemeiste r move s tha t take s u s from th e given projection t o the projection o f the unknot.

Fo r example. Mak e sur e yo u conside r what happens when different region s are shaded. We will com e back t o signe d plana r graph s whe n w e loo k a t the rela - tionship between knot theory and statistical mechanics in Section 7. The question o f whether kno t projections ar e equivalent unde r Reidemeis - ter move s become s on e o f whethe r signe d plana r graph s ar e equivalen t under operation s that the Reidemeister moves become.

We can turn Reidemeiste r move s into operations o n signed plana r graphs. Tabulating Knots 5 5 But w e ca n tur n kno t an d lin k projection s int o signe d plana r graphs. W e denot e th e unknottin g numbe r o f a knot by u K. W e say tha t 7 2 has unknottin g numbe r 1. Notic e first o f al l that if we changed th e crossing circled i n Figure 3. More generally. Invariants of Knots 3.

We begi n with a very intuitiv e invariant. Figure 31 Th e knot 7 2 becomes the unknot. Th e on e chang e o f crossin g completel y un - knots th e knot. Given a projection o f a knot. Aside fo r peopl e wh o kno w th e traditiona l definitio n o f unknottin g number: I n ou r definitio n o f th e unknottin g number. The fact tha t ever y kno t ha s a finite unknottin g numbe r follow s fro m the fact tha t ever y projection o f a knot ca n be changed int o a projection o f the unknot by changing som e subset o f the crossings in the projection.

W e can the n d o anothe r ambien t isotop y t o a new projectio n befor e w e chang e ou r thir d crossing. B y the tim e w e ar e finishe d with ou r n crossin g changes.

A s w e d o our ambien t isotop y t o anothe r projection. The n w e continu e throug h tha t crossin g o n ou r merr y way alon g th e knot. That thes e tw o defini - tions ar e equivalen t follow s fro m th e fac t tha t w e ca n kee p trac k o f eac h crossing change in the second definitio n wit h an arc that runs to and fro m the tw o point s o n th e kno t wher e th e crossin g chang e occurs.

Al - though thi s fac t appeare d a s Exercis e 1. Th e firs t tim e tha t w e arriv e a t a particula r crossing. Exercise 3. But whe n w e loo k a t ou r projectio n from th e side. Th e altered projection is the trivial knot. Thin k o f the z axis a s coming straigh t ou t o f th e projectio n plan e towar d us.

Start - ing a t th e initia l poin t again. Hence thi s kno t i s a trivial knot.

To see that thi s i s the trivia l knot. Invariants of Knots 5 9 returned t o ou r initia l startin g point. Note then that when w e look straigh t dow n th e z axi s a t ou r knot.

But how d o w e kno w tha t ther e isn' t som e othe r projectio n o f 7 4 that ca n b e unknotted by only one crossing change? In order to prove that the unknot - ting number i s 2. We might expect the answer to be no.

Here's a n interesting question: Can a composite knot have unknottin g number 1 Figure 3. It's not hard t o find two crossin g change s tha t mak e thi s projectio n int o th e unknot. Fin d them. CO O Figure 3. Figure 3. Bu t how d o we know ther e isn't som e othe r projectio n o f this kno t that ca n b e mad e int o th e unkno t wit h on e crossin g change? Kanenob u and Murakam i applie d th e powerfu l Cycli c Surger y Theorem. Martin Scharleman n a t th e Universit y o f California-Sant a.

For exam- ple. Fin d a simple proof tha t a knot with unknotting number 1 is prime. Ol d conjecture: Exercise 3 A Sho w that a knot like the one in Figure 3. I f K is a knot with unknotting numbe r 1. Invariants of Knots 6 1 Barbara Scharlemann.

I f K i s a n alternatin g kno t wit h unknottin g numbe r 1. His proof i s very technical. Ca n the unknotting number u K of any knot be realized by chang - ing a single crossing in a minimal crossin g projection. The n sho w tha t i t ha s unknottin g number 1 by showing that there is a crossing in this projection tha t can be changed to yield the trivial knot. Unsolve d Questio n 3 i n fac t implie s Unsolve d Question 4.

Sho w this. I s i t tru e tha t a kno t wit h unknottin g numbe r 2 cannot b e a com - posite kno t mad e fro m thre e facto r knots. Nalsanishi independently discovere d th e followin g example. I t i s know n tha t thi s kno t canno t b e drawn wit h fewe r crossings. It i s also known tha t this is the only projection u p to planar isotop y an d mir - ror reflection o f this knot with 1 0 crossings. It has Conway no- tation 2 —2 2 —2 2 4. That's surprising. Her e i s a kno t wit h Conway notatio n 51 4 Figur e 3.

Amazingl y enough. These results were generalized by a Princeton University undergradu - ate named Jame s Bernhard. Bernhard Steve Bleiler and Y. Check for yoursel f tha t th e two continue d fraction s give the same rational number. Here is another projection of the same knot Figure 3. We say tha t tw o knot s o r link s ar e fc-equivalent i f w e ca n ge t fro m a projection o f on e t o a projection o f th e othe r throug h a serie s o f fc-moves and — it-moves.

Nakanishi o f Kob e University. Invariants of Knots 6 3 While w e ar e a t it. We allow ourselve s t o change th e projections vi a ambien t isotopies between the various moves that we perform.

It' s surprisin g tha t n o one has succeeded in proving or disproving the conjecture yet. A — k- move will be the same. Thi s i s th e leas t numbe r o f suc h unknotte d arc s i n an y projection o f thes e knots. The links without tricoloratio n would b e exactl y thos e link s three-equivalen t t o th e trivia l lin k o f on e component. Note that if the first unsolve d conjectur e i s proved t o be true. A maximal overpas s i s an overpas s tha t.

In general. I n both o f th e pictures. Think of the darkene d portions o f th e knots as lying above the plane an d th e rest of th e knots a s lying belo w th e plane. Eac h kno t intersect s th e plan e i n fou r vertices.

Thi s i s known t o b e tru e fo r rationa l knots. I n thes e pictures. Se e Kirby. Not e that the y ar e unknotte d an d untangled. I t ma y hel p t o thin k abou t th e case s where the projection is alternating or nonalternating separately. Invariants of Knots 6 5 could no t b e mad e an y longe r Figur e 3.

Al l o f th e crossing s cam e fro m a maxima l overpas s an d one of these arcs. The bridge number of K. The bridge number o f the projection i s then th e number o f maxima l overpasse s i n th e projectio n thos e maxima l overpasses formin g th e bridges ove r th e res t o f th e knot.

Suppose we cut a two-bridge kno t ope n alon g the pro- jection plane. Both o f it s endpoint s occu r just before w e go under a crossing. So from th e side view. We would b e lef t wit h tw o unknotte d untangle d arc s fro m the kno t abov e th e plane.

Knots that have bridge number 2 are a special class of knots. The tricky part i s that although the strings to each side of the plan e are individuall y unknotted. Note tha t eac h crossing i n the projection mus t hav e som e maximal overpas s tha t crosse s over it. Given a pictur e o f a two-bridge kno t a s i n Figur e 3. V TO ab Figure 3. Now we can see that this two- bridge knot is in fact simply a rational knot. How d o we determin e th e crossing number o f a knot K?

Invariants of Knots 6 7 The two-bridge knots are a very well understood clas s of knots. The two-bridge knot s ar e well under - stood. No on e has yet un - derstood all of the three-bridge knots. Since two-bridge knot s ar e known t o be prime. The n w e know th e crossin g numbe r i s n o r smaller. Note that we are counting a knot and its mirror image as dis- tinct knots if they are not equivalent. The crossing number of a knot K. The first three-bridge knot in the appendix table is 8i0 Figure 3.

Kuni o Murasug i from th e Universit y o f Toronto. Here is an al- ternating kno t i n a reduce d alternatin g projectio n o f 2 3 crossings Figur e. No - body yet knows what all the knots of 1 4 crossings are. K5oO Figure 3. This last fact is very difficult an d get s at the essence of knot the- ory. Cal l a projectio n o f a kno t reduce d i f ther e ar e no easil y remove d crossings. The answer will have to wait until Chapter 7. They utilize d th e Jones polyno - mial fo r knot s i n orde r t o prove this.

We discuss th e Jones polynomia l i n Chapter 6. There cannot be a projec- tion o f suc h a kno t wit h fewe r crossings.

Since we ca n tel l by just lookin g a t a n alternatin g projectio n whethe r or not it is reduced. Note that Kauffman. Invariants of Knots 6 9 3. This problem has been open for 1 0 0 years. The question of determining th e crossing number fo r a nonalternatin g knot i s stil l very muc h open.

Hence its crossing number i s There cannot be a projection o f thi s knot with fewer tha n 23 crossings. Mura - sugi. We will come back to crossing number when we discuss particular cat - egories of knots. They are just the surface of the object. But first of all. Surfaces and Knots 4. Figure 41 Som e surfaces.

Note that these are not solid objects. Kee p in mind tha t we think of the glaze as being infinitely thin. We call the sur- face of a doughnut a torus. Figure 4. I n each o f th e thre e examples. The y fail t o be surfaces becaus e each of the m ha s at leas t one poin t suc h tha t th e regio n o n th e objec t surroundin g tha t poin t doe s not for m a dis k o n th e object.

At any point on a surface. The dis k doesn' t hav e t o b e flat. We wil l eventuall y generaliz e t o three-manifold s i n Chapte r 9. In what follows. Fin d tw o differen t one-mani - folds. This idea of treating objects a s if they were mad e o f rubbe r i s the fundamenta l concep t behind topology. Surfaces and Knots 7 3 Another nam e fo r a surface i s a two-manifold.

Exercise 4. Tw o surface s i n spac e that ar e equivalen t unde r a rubbe r deformatio n ar e calle d isotopi c sur - faces. Isotop y i s a generi c nam e fo r a rubbe r deformation. Mathematician s cal l suc h a rubbe r defor - mation a n isotopy. In order to apply surfaces t o the study of knots. An y thoughts o n wha t th e definitio n o f a three-manifol d shoul d be?

Ou r spa - tial universe appears to be a three-manifold. A two-manifold i s de- fined t o be an y objec t suc h tha t ever y poin t i n tha t objec t ha s a neighbor - hood in the object that is a possibly nonflat disk. Th e triangles have to fit together nicely along their edges so that they cover th e entire surface.

The tw o surface s i n Figur e 4. Figure 4 J Thes e three surfaces are all isotopic. A proo f tha t the y ar e no t isotopi c woul d requir e a lo t more work. The triangle s needn' t b e flat wit h straigh t edges.

In order t o better wor k wit h surfaces. We call such a division of a surface int o triangles a triangulation. Given a surface wit h a triangulation. Surfaces and Knots 7 5 like all the other object s in topology. Examples of triangulations o f the sphere and th e torus are given in Figure 4. W e can think of them a s disk s wit h a boundary mad e u p o f thre e edge s connectin g thre e vertices. We could alway s fin d a triangulatio n that contained thi s set of circles and edge s as part of the union of the set of edges.

We simpl y cut alon g a subset o f th e edge s o f a triangulatio n tha t for m a circle. Notice that we didn't even draw the rest of the triangulation. We can see the chain of cutting an d glu - ing tha t take s u s fro m th e on e surface t o th e other.

The fact tha t these two surfaces ar e not isotopic is not obvious an d would tak e some work to prove. Product details Paperback: American Mathematical Society August 11, Language: English ISBN Tell the Publisher! I'd like to read this book on Kindle Don't have a Kindle? Share your thoughts with other customers. Write a customer review. Read reviews that mention knot theory good introduction book is written knots mathematics subject concepts exercises topology basic ideas major questions words.

Top Reviews Most recent Top Reviews. There was a problem filtering reviews right now. Please try again later. Paperback Verified download. I downloadd this book as the textbook for a senior-level mathematics course in my undergrad.

However, this book does not "read like a textbook. The pictures are clear, the words are concise, the ideas are organized logically and in proper order such that ideas are clearly described and explained in a manner that you don't have to be a mathematics major to understand what is being talked about. Highly recommended. Very well-done book, especially for a topic as "obscure" as knot theory. Readers of the first edition of this book published by W. Freeman and Company , or its "Second printing" also dated , should check page to see if, in their copy, hyperbolic knots were discovered in or The author makes no mention of this and other corrections.

In fact it's quite good. I read this as an undergraduate. It was sufficiently interesting that I'm reading it again just because. One person found this helpful.

This book is written very well. I use this book in conjunction with more formal Knot Theory books. The writing is simpler and easier to understand than the more technical books. It has been very useful in helping me understand simple concepts needed to do my research.

Hardcover Verified download. Awesome book, very understandable. People get a kick out of it. This a really great book. It has everything needed for doing a Course in Knot Theory just by your own. It contains all level exercises from the bare beginner to the open questions in the field.

If you love Math, I pretty sure you'll enjoy this book pretty much. Wonderful mathematics book.