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.
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by Sir Randolph Bacon III (cousin -‐in-‐law to Colin Adams). Abstract: Book", an elementary introduction to the mathematical theory of knots, "Why. Knot?. preckalohotchning.cf 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.
For the knot J K in Figure 1. There is no wa y t o tak e th e compositio n o f tw o nontrivia l knot s an d ge t th e un - knot. Maybe that part of the projec- tion corresponding t o the knot on the right somehow untangles that part of the projection correspondin g to the knot on the left.
We can thin k o f thi s re - sult a s analogou s t o th e fac t tha t th e intege r 1 is no t th e produc t o f tw o positive integers. I s th e unkno t composite? That is. Introduction 9 any two nontrivial knots. Both the trefoil knot an d the figure-eight kno t are prime knots. Since every kno t is the composition o f itself wit h the unknot. But how abou t th e knot i n Figur e 1. Wil l thi s choic e affec t th e outcome?
Althoug h al l th e positive integers aren't listed. W e have a choic e o f wher e w e remov e th e ar c from th e outsid e o f each projection. This direction is denoted b y placin g coherentl y directe d arrow s alon g th e projection o f th e knot in the direction of our choice.
It's lik e a table o f prim e numbers. One wa y tha t compositio n o f knot s doe s diffe r fro m multiplicatio n o f integers is that there is more than on e way t o take the composition o f tw o knots. The appendix table. A n orientatio n i s defined b y choosing a direction to travel around th e knot. We then say that the knot is oriented. All o f th e composition s o f th e tw o knot s wher e th e orientation s d o no t match u p wil l als o yiel d a singl e composit e knot.
Although thi s wil l no t b e th e cas e i n general. The first knot that is not invertible in the table at the end o f the book is the kno t 81 7. This occurs because on e of the factor knot s is in- vertible. Introduction1 1 tion on K in J K. In orde r. To convince ourselves that the first tw o compositions in Figure 1. A knot is invertible i f it can be deformed bac k to itself s o that a n orientation o n it is sent to the opposit e orientation.
I n the case that on e of the tw o knot s i s invertible. Al l o f th e composition s o f th e tw o knot s where th e orientation s d o matc h u p wil l yiel d th e sam e composit e knot. Composing i t wit h itsel f i n th e tw o differen t way s produce s two distinct composite knots that are not equivalent Figur e 1.
Kee p i n min d that this is highly deformable rubber. It's easiest to think of a kno t mad e o f string. A deformatio n o f a knot projectio n i s called a plana r isotopy i f i t de - forms th e projection plan e as if it were made o f rubber with the projectio n drawn upo n i t Figur e 1.
Th e wor d "isotopy " refer s t o th e deformatio n o f th e string. Knot the- orists call the rearranging o f the string. The word "planar " i s used her e because w e are onl y deformin g th e kno t withi n th e projectio n plane. Note that in a n ambient isotopy. If we made a knot out o f strin g that modeled th e first of the two projections. So far. Th e wor d "ambient " refer s t o th e fac t tha t th e strin g i s being de - formed throug h th e three-dimensional spac e that it sits in.
The first Rei- demeister mov e allow s u s t o put i n o r tak e ou t a twist i n th e knot. W e assume tha t th e projection remain s imchange d excep t fo r the chang e depicte d i n the figure. OR Figure 1. A Reidemeiste r mov e i s on e o f thre e way s t o chang e a projection o f the knot tha t will change the relation between th e crossings.
Th e secon d Reidemeiste r mov e allow s us t o eithe r ad d tw o crossing s o r remov e tw o crossing s a s i n Figure 1. The third Reidemeister move allow s us t o slide a strand o f the knot fro m one side of a crossing to the other side of the crossing. Introduction1 3 Figure 1. A s anothe r example. Figur e 1. For example. Eac h suc h move is an ambient isotopy.
Even if we could prov e that we cannot ge t from th e standard projectio n o f the trefoi l kno t t o it s mirro r imag e i n 1. I f th e two original projections hav e 1 0 crossings each. We just check whether o r not ther e is a sequence o f Reidemeiste r move s t o ge t u s fro m th e on e projectio n t o th e other.
The proo f tha t Reidemeiste r move s an d plana r isotop y suffic e t o ge t us from an y one projection o f a knot to any other projection o f that knot i s not particularly difficult. More on amphicheirality i n Chapter 7. Inci - dentally. A proo f appear s i n Burd e an d Zeischan g 1 Here is an interesting example. Introduction1 5 equivalent t o its mirror image. Believe it or not. It migh t now seem that the problem o f determining whether tw o projections repre - sent th e sam e knot would b e easy.
Although th e kno t tables do not list both a knot and its mirror image. It is highly unlikely such a constant exists. Not e tha t thi s proble m i s askin g a lo t more than just showing that this knot can be untangled.
O r perhap s you ca n fin d a sequenc e o f example s tha t prove s tha t th e increas e i n th e number o f crossing s i s sometime s greate r tha n an y functio n o f th e for m a. Exercise1 A2 Prov e tha t i n Exercis e 1. Or per - haps there is a sequence of examples that show s that the crossing increas e is sometimes greate r tha n any polynomia l i n n. In mathematical parlance.
But there was no reason to say that there could only be one loop that we knotted. I know of no set of ex- amples that demonstrate its nonexistence. Prett y much everythin g w e hav e sai d abou t knot s hold s tru e fo r links. This link is name d after th e Borromeas. Introduction1 7 A link i s a set o f knotte d loop s al l tangled u p together. Th e tabl e a t th e back o f th e book contain s projection s o f som e o f th e simple r links.
Since it is made up of two loops knotted with each other. Some - times it' s obviou s whe n a lin k i s splittable. Here are two projections o f on e of the simples t links. A lin k i s calle d splittabl e i f th e component s o f th e lin k ca n b e de - formed s o that the y lie on differen t side s of a plane i n three-space.
Figure Tw o projections o f the Whitehead link.
Adams, The Knot Book
Fo r in - stance. Her e is another well-know n lin k with thre e components. A kno t wil l b e considere d a lin k o f on e component. We will defin e what' s know n a s the link - ing number.
If the numbers ar e different. I n Figure 1. W e call th e firs t o f thes e th e unlin k o r trivia l link o f tw o components an d th e second th e Hop f link.
On e differenc e betwee n thes e two link s i s that th e unlin k i s splittable. Ther e is on e quic k wa y fo r tellin g certai n link s apart: If w e hav e tw o projection s o f links. Most o f the links that we will be interested i n are nonsplittable. We would lik e a method fo r measurin g numericall y ho w linked u p tw o component s are.
S o obviously. Then a t each crossin g between th e two components. The two components i n the oriented lin k pictured i n Figure 1. I f w e just loo k a t th e abso - lute value of the linking number. For th e Hop f link. Notice tha t i f w e revers e th e orienta - tion o n on e of th e tw o components.
W e d o no t coun t th e crossing s betwee n a componen t an d itself. Figure Linkin g number 2. Now revers e th e direction o n on e o f th e components an d recom - pute it. Sometimes i t is hard to determine from th e picture whether a crossing is of the first typ e or the second type. This will be the linking num - ber.
Note that if a crossing is of the first type. Fo r the unlink. Introduction1 9 the first type. Let's first loo k a t the effect o f the first Reidemeiste r mov e o n the link - ing number. I t ca n creat e o r eliminat e a self-crossin g i n on e o f th e tw o components. I s ar e assigne d t o eac h o f th e crossings. Sinc e we ca n get fro m an y on e projec - tion o f a link t o any othe r vi a a sequence o f Reidemeiste r moves. Even if we change the ori- entation o n on e of th e strands.
We show thi s by provin g tha t th e Reidemeiste r move s do no t chang e th e linking number. W e ar e assumin g tha t th e two strand s correspon d t o th e tw o differen t components. In Figur e 1. Notice tha t w e us e a particular projectio n o f th e link i n orde r t o com - pute th e linkin g number. I t ha s linkin g numbe r. Anothe r invarian t o f link s tha t w e hav e alread y men - tioned i s simply the number o f components in the link.
For instance. Tr y computin g th e link - ing numbe r fo r th e Whitehea d lin k i n Figur e 1. Introduction 2 1 Figure 1. Since we want t o dis- tinguish link s that d o no t alread y hav e orientation s o n them. It remains invarian t when th e projectio n o f th e lin k i s altered. Any tw o links with two compo - nents tha t hav e distinc t absolut e value s o f thei r linkin g number s hav e t o be different links. We can use linking number t o distinguish links. It is unchanged b y ambient isotopies of the link.
So no w you'r e thinking. We sa y tha t th e linkin g numbe r i s a n invarian t o f th e oriente d link. Maybe every one of those pro-. The n see if you can prove it. What about ellipses? Thin k up your own.
Th e fac t tha t thes e three ring s ar e locke d togethe r relie s o n th e presenc e o f al l thre e compo - nents. W e nee d som e othe r way s t o distinguis h variou s knot s an d links. These link s ar e name d afte r Herman n Brunn.
W e hav e not yet proved that there is any other knot besides the unknot. S o w e can' t eve n sho w that th e Whitehea d lin k i s differen t fro m th e trivia l lin k o f tw o compo - nents. In th e nex t section. A link is called Brunnian i f the link itself i s nontrivial. Bu t firs t let' s tak e anothe r look a t th e Borromea n ring s Figur e 1.
Bu t none o f th e crossing s i n eithe r pictur e hav e exactly two colors occurring. I n orde r tha t a projectio n b e tricol - orable. In th e firs t tricoloration. The point is that of course they can't be. Introduction 2 3 jections can be turned int o the projection o f the unknot throug h a series of Reidemeister moves.
W e will sa y tha t a projectio n o f a kno t o r lin k i s tricolorabl e i f each o f th e strand s i n th e projectio n ca n be colore d on e o f thre e differen t colors. S o we wil l prov e tha t ther e i s at least one othe r knot besides th e unknot. We will say tha t a strand in a projection o f a link i s a piece of th e lin k that goe s fro m on e undercrossin g t o anothe r wit h onl y overcrossing s i n between.
In order to do that. W e will prove tha t th e trefoil kno t i s not equiva - lent to the unknot. For ou r purposes.
Note tha t i n both thes e cases. If the tw o origina l strand s ar e the same color. Eithe r al l o f th e strand s tha t appea r in th e diagra m fo r th e Reidemeiste r mov e ar e th e sam e color. You needn't prov e that the knots that you describ e are actuall y different knots.
The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots
I f a knot ha s a crossing i n th e tricoloratio n tha t ha s only on e color. Introduction 2 5 ab Figure 1. Conclud e tha t th e figure-eigh t kno t an d th e trefoi l knot are distinct knots. Either every projection of a knot is tricolorable or no projec- tion of that knot is tricolorable.
We can. There are several cases to check. Sinc e th e usua l projectio n o f th e unkno t i s no t tri - colorable w e certainl y can' t us e a t leas t tw o color s o n i t sinc e i t doesn' t have distinc t strands. Remem - ber. By Exercise 1. Thi s i s the reverse of what happened fo r tricolorability for knots. Then show that such a labeling system on a knot projection i s preserved unde r Reidemeiste r move s Typ e III is the tricky one.
Notice tha t th e trivia l lin k o f tw o component s is tricolorable. Tricolorability fo r link s of two components i s slightly differen t Figur e 1. Although w e will se e a proof o f thi s fact later. Conclude tha t th e figure-eight kno t i s not th e trivial knot. Tha t is to say. An argument is needed. Thi s prove s tha t th e unkno t canno t b e th e composition of the trefoil knot with any other knot. S o i n th e projection. If we looked down at the plane.
How abou t fou r sticks? So it would hav e to be the unknot. The sticks can be an y lengt h tha t w e wan t Figur e 1. Q Knot s and Sticks Suppose we were given a bunch of straight sticks and we were told to glue them together en d to end in order to make a nontrivial knot. How man y stick s will it tak e t o make a nontrivial knot? Try playing with some sticks to see what happens. Introduction 2 7 Exercise 1.
If you can tangle up your arms and then clasp your hands together so that. In the projection o f the knot that we see. Although th e picture looks believable. The n it' s clea r tha t suc h a kno t coul d actuall y b e con - structed fro m sticks.
See Exercise 1. Let the vertices labeled P lie in the xy plane. But we ca n convince ourselve s tha t thi s constructio n work s withou t goin g t o tha t much trouble. We are only looking at a projection of the sticks in this picture. That's a total of five stick s that ar e attached en d t o end Figur e 1. One solutio n i s to actually build th e knot wit h rea l sticks. If w e wan t a hands-on demonstratio n tha t fiv e stick s won't suffic e t o make a knot.
How do we know tha t th e sticks needn't be bent o r warped t o fit together i n this way. How about fiv e sticks? Let's view the knot so that we are looking straight down one of th e sticks.
The vertices la- beled L li e low. For the same reason a s in the previou s paragraph. Th e vertice s labele d H lie high. So four stick s aren't enough to make a nontrivial knot. What happen s i f w e tr y t o mak e knot s usin g tw o peopl e holdin g hands and their "ten sticks"? What knots can we make? But supposedly.
Define th e stic k numbe r s K o f a kno t K t o b e th e leas t numbe r o f straight sticks necessary to make K. Introduction 2 9 the loo p forme d fro m thes e fiv e stick s i s knotted.
Don't hurt yourself. Adams et al. B y Exercise 1. The proo f i s elementary. Then invert the inequality. Seiya Negami. I t does no t chang e i n the cas e of the trefoi l knot. In- dependently.
So each new trefoi l onl y requires tw o more sticks. In Chapte r 7. Look straight down one edge and the n count crossings to obtai n a bound o n c K in terms of s K. But it seems unlikel y tha t th e sam e would b e tru e fo r all knots. Prove that if K is a nontrivial knot. I n a paper tha t appeared i n 1 I understand a reticulation of any number of meshes of two or more edges. The first work on tabulating knot projections wa s done in the 1 s by the Reverend Thoma s P.
To quote: By a knot of n crossings.
But it was Lord Kelvin' s theory tha t atom s were knotted vortice s i n the ether that sparked seriou s interest in determining the possible knots. These early explorations in knot theor y suffered fro m Kirkman' s opaque writing style. Mar y G. In spite of Kirkman's obfuscation. It wasn't unti l 1 97 4 that i t wa s discovere d tha t tw o o f th e knot s i n Little' s table were in fact the same knot and that there were only 42 distinct nonal - ternating knots o f 1 0 crossings.
I t remained th e only polynomial for knots until Figure 2. This was the firs t successfu l tabulatio n o f knots. I n During thi s early period i n the tabulation o f knots. There ensue d a lon g perio d o f inactivit y i n th e tabulation of knots.
Exercise 2. Their methods utilized th e first polynomial applie d t o knots. Th e two projection s tha t actuall y correspon d t o th e sam e kno t ar e no w calle d the Perko pair Figur e 2.
The duplication wa s discovered b y a part - time mathematicia n an d Ne w Yor k lawyer name d Kennet h A Perko. His tabl e wa s believe d t o be correc t fo r 7 5 years.
Kurt Reidemeiste r finishe d of f th e rigorou s classificatio n o f knot s u p to nin e crossing s i n 1 Little was th e first t o attac k th e proble m o f enumeratin g th e nonalternatin g knots. Little wen t o n t o publis h a censu s o f 1 1 -crossin g alternatin g knots. A professor a t the University o f Nebrask a name d C. Hasema n liste d al l amphicheira l knot s remember.
In the case that a knot is equivalent to its mirror image. Bu t because o f hi s wide-rangin g mathematical interests. When you get near to 1 4 cross- ings. Here is a list of the numbers of knots that have been determined s o far: Alain Caudro n o f th e Universit y o f Pari s produce d th e firs t correct lis t o f al l prime knot s throug h 1 1 crossings.
In the case that the knot is not equiv - alent t o it s mirro r image. Thi s compute r progra m resulte d i n a tabl e o f al l prime knot s throug h 1 2 crossings i n 1 Conwa y ha d firs t becom e intereste d i n knot s whil e i n hig h schoo l and formulate d man y o f hi s idea s then. An algorithm for generating knots that utilize d this notation was implemented o n the computer by an Englishman name d Morwen Thistlethwaite.
You won't always be able to close it u p after exactl y 1 4 crossings. What doe s a 1 4-crossin g knot loo k like? Let's draw one. To tackle the crossing knots. Conway invented a new nota - tion fo r knot s an d use d i t t o determin e al l o f th e prim e knot s o f 1 1 o r fewer crossing s an d al l o f th e prim e nonsplittabl e link s o f 1 0 o r fewe r crossings.
Thistlethwait e an d Week s deter - mine exactly which knots are amphicheiral. In this list of numbers. In the meantime. Morwe n Thistlethwaite began agai n to work o n tabulation. His tabulation wa s al l done by hand. The pape r the y co-authore d ha s th e wonderfu l title. Host e recruite d Jef f Weeks. Try to close up th e curve af - ter exactly 1 4 crossings Figure 2.
W e wil l tal k mor e abou t determinin g amphicheiralit y i n Chapter 6. Tabulating Knots 3 3 In Start drawin g a curv e on a piece of paper. No on e worke d o n extendin g th e lis t unti l ten year s late r whe n Ji m Host e bega n wor k wit h a grou p o f hig h schoo l students wh o ha d acces s t o a supercomputer. Using differen t meth - ods. It is remarkable that we cannot yet show this. Continu e i n thi s manne r unti l yo u hav e a 1 4-crossin g alternating knot. This i s probably har d an d requires som e ne w ideas.
Thi s give s yo u som e feelin g fo r ho w man y 1 4-crossin g knots there might be. Bu t man y o f th e scribble s an d choice s o f crossing s actuall y correspond t o the same knots. Claus Ernst and Dewit t Sumners. There ar e als o million s o f differen t scribble d curve s tha t w e coul d draw. Perhap s n o elemen - tary function give s this sequence. T o do this. Fin d al l o f th e 1 7-crossin g prim e knots. Determin e th e sequenc e o f integer s tha t begin s 1. Classif y th e alternatin g knot s o f 1 7 crossings.
Any 1 4-crossin g scribbled curv e corresponds to a crossing alternatin g knot. We do kno w tha t th e numbe r o f prim e knot s o f n crossing s grow s a t an exponential rate. Perhap s thi s se - quence o f number s givin g th e numbe r o f prim e knot s wit h a give n crossing number i s in fact a reasonable function.
Se e th e nex t sectio n fo r wh y thi s coul d b e difficult. In Whe n yo u restric t yourself t o alternating knots. This is a hard ope n question. I t seems lik e there ar e many mor e 1 4-crossin g knot s tha n w e coul d ever catalog. We talk more about this re- sult in Section 3. Leavin g that crossin g alon g the understan d i n the directio n o f th e orientation. Continue t o label th e crossing s wit h th e integer s i n sequenc e unti l yo u have gon e al l th e wa y aroun d th e kno t once.
Whe n yo u ar e done. Choose a n orientatio n o n th e knot. Tabulating Knots 3 5 Florida Stat e University. See Erns t an d Stunners. Not e tha t i n thi s lowe r bound. Notic e that. Suppose we have a pro- jection of an alternating knot that we want t o describe. Dominic Wels h o f Oxfor d Universit y ha s prove d tha t th e numbe r o f distinct prim e n-crossin g knot s i s bounded abov e by a n exponentia l i n n.
Continu e throug h tha t crossing on the same strand o f the knot. How abou t a second sequenc e of even integers that also represents the same projection o f 63? Sinc e 2 i s paired wit h 7. In this case. Star t b y drawin g jus t th e firs t cross - ing. Sa y the sequence is 8 10 12 2 14 6 4. Given a sequence o f eve n integers tha t represent s a projectio n o f a n alternatin g knot. 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 before you cut them completely, twist the middle strip again. I will get two entangled moebius strips, right!? Only one way to find out! I am not a skilled person. I attempted this a bunch of times and all I have are weird strips of paper. Can someone please share pics if you got something other than weirdly shaped strips of paper?
Clifford Ashley was a sailor who collected knots, and an accomplished painter and writer. If you find yourself in New Bedford, MA, you can see some of his work in the whaling museum which I highly recommend. I took a knot theory class as an undergrad, and I don't remember which book we used. It ended up being a pretty superficial introduction to the subject, which is both disappointing, and probably also how I got my first "A" in a math class since 11th grade which, I would argue, was wholly undeserved Several key takeaways: The figure 8 knot is the only 4 crossing knot.
If you climb, and use it as your tie-in, you can check that you've tied it correctly by checking that you have 5 pairs of strands in the knot. The figure 8 knot is amphichiral. There appear to be two variants like the left and right-handed trefoil knot , but they are transformable into each other via the "pretzel" configuration, which seems to be the canonical representation in math.
If you coil rope with only overhand or underhand loops and pull it out, you put a lot of twist into it. If you alternate overhand and underhand loops, it pulls out untwisted. This is most easily seen with ribbon, which has two distinct sides. This is a really nice introductory video imo on the topic by Up and Atom as a guest on Tom Scott's channel. Link to Video: For knot fans - there's also a code golf problem on Stack Exchange that only has one solution so far - Knot or Not?
The spectacular knot homology theories such as Khovanov Homology and Heegaard Floer Homology can detect the unknot 'on the blackboard' as well.
Actually it can detect the genus of the knot: 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. Knot theory is a mathematical field which studies knots in a three dimensional space. The books is a well written starter to the field. It talks about ideas, history, why it is important and applications. It is well written and does not require any special background in mathematics. Jonas rated it really liked it Jul 22, Qiaochu Yuan rated it really liked it Dec 27, Shanna-Mae Slight rated it really liked it Mar 07, Matthias rated it really liked it May 05, Alireza rated it it was amazing May 07, Chris rated it liked it Oct 14, Bryan Hallett rated it really liked it Apr 10, Michael Ponce rated it really liked it Jun 15, Mark rated it really liked it Dec 25, Kevin rated it really liked it Jul 25, Paul Vittay rated it really liked it Apr 08, Philip Kromer rated it really liked it Aug 05, James Peirce rated it it was amazing Feb 15, Miles Gould rated it liked it Jan 01, It's lik e a table o f prim e numbers.
We do kno w tha t th e numbe r o f prim e knot s o f n crossing s grow s a t an exponential rate. Cover Type: If we made a knot out o f strin g that modeled th e first of the two projections. Can a composite knot have unknottin g number 1 Figure 3. In general.
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