1

Mathematical theory of knots, quantum physics, string theory (connections with the Fibonaccis numbers, Lies numbers and partition numbers)

Michele Nardelli, Francesco Di Noto, Pier Francesco Roggero

Abstract

In this paper we show some possible connections between knots theory and string theory, based on Fibonaccis numbers, Lie numbers and partition of numbers.

Here we show a few possible relationship between string theory and the mathematical theory of knots, using the common connection with the Fibonaccis numbers, Lies numbers and partition numbers. Predictions and estimates on the values of N(n) from N(17) to N(20)

On the site of the newspaper "Il Sole 24 Ore" an article of knot theory in physics, and also on the issue of the paper of 20/10/2013 there is an article of Umberto Bottazzini, "Knots and strings of physics," with a lesson in Milan of the physicist Edward Witten, well-known studious of the strings, and available on the web at the same name on Google. Intrigued by the news for our interest about string theories with which the theory of knots seems interconnected, and with the suspicion that they might be involved in some way the Fibonaccis numbers (and then also the Lies numbers

2

and the partitions of number), we found a document suitable for our purpose, and which we report in full:

Knot in the Enciclopedia Treccani www.treccani.it Enciclopedia

2. Knot theory

In topology , studying the geometrical properties , in particular groups of homotopy of the complementary set in R3 , or of a knotted circuit , ie of a simple closed curve not reducible with continuous deformation to a circumference (k. trivial) . All the k are homeomorphic between them, however, because of the different manner in which they are immersed in R3 they are classified into types of k. equivalent or of the same type, i.e. that can correspond via a homeomorphism. The orthogonal projection on a plane makes it possible to classify the different types of k. : There are only one type for the k. with 3 or 4 self-intersections (on the plane), 2 and 3 types for those with 5 or 6 self-intersections , 7 and 21 for those with 7 or 8 self-intersections respectively (in Fig. 2 are given some examples). However, also if it was possible to distinguish the various types with a maximum of 10 self-intersections, a complete classification of all the possible cases is still unknown ...

The correlation between the theory of k. and modern physics was established in the early 1980s , when L.H. Kauffman was able to find a way to describe the Alexanders polynomial (and later the Joness polynomial ) in the form of a partition function of the statistical mechanics, and V. Jones discovered entirely new invariants of k. and links ( a link is the union of a finite number of k. that does not have in common sections of string), directly related to problems of statistical mechanics. A few years later, E. Witten showed how all these buildings could be understood in terms of quantum field theory, thereby giving rise to the new field of study concerning the quantum topology and the topological theory of quantized fields."

There is a little connection with the Fibonaccis numbers (and the partitions of a number n), already found also in string theories. We have marked in red this possible connection (and in blue a reference to the complete classification), that we will see after in the appropriate table.

As we can see easily, all pairs of numbers marked in red relate to one of the Fibonaccis numbers 2,3,5,8,21 (not included, however, the number 13). But in the voice of Wikipedia "Prime Node", we find the following table:

3

"The more simple prime knots The prime knots are generally described by diagrams, with increasing order of crossings. The simplest prime knot in this description is the trefoil knot with 3 crossings, followed by the eight knot with four crossings. The following table shows the number of prime nodes with n crossings.

n 1 2 3 4 5 6 7 8 9 10

Number of prime nodes with n crossings 0 0 1 1 2 3 7 21 49 165

Where we can see the connection with our summary table, in fact 8 is the number of intersections (self-intersections) and is connected to 21, the number of prime knots, with 8 and 21 Fibonaccis numbers, for 9 crossings there are 49 knots, and with 10 crossings we have 165 nodes. The connection with Fibonacci that we have hypothesized loosens more and more (49 and 165 arent Fibonaccis numbers). We remember also that the number 8 is connected with the modes that correspond to the physical vibrations of a superstring by the following Ramanujan function:

( )

++

+

=

42710

421110log

'

142

'

cosh'cos

log4

318

2

'

'

4

0'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

pi

pi

pi

pi

.

4

TABLE 2

Types

Number of knots

Self-intersections (crossings)

Fibonaccis numbers in number of knots or immediately preceding

Fibonaccis numbers in number of knots or differences

1 3 or 4 1 3 2 and 3 5 or 6 2 e 3 5 7 and 21 7 or 8 7 8 e 21 8 49 9 34 49-34 = 15

13 165 10 144 165-144= 21

Now we can create a new table, Table 3, with the subsequent ratios between the number of knots and the number of crossings, to see the medium progress of their increase:

TABLE 3

Number of knots N(n)

Number of crossings n

Subsequent ratios

r = N(n) /n 0 1 - 0 2 - 1 3 0,33 1 4 0,25 2 5 0,4 3 6 0,5 7 7 1

5

21 8 2,62 49 9 5,44 165 10 16,5 435 ? 11 39,6 ? see below 1188 ? 12 99 ?

We note that since Nn = n = 7, the ratio reaches and then exceeds the unit, and then increase proportionately to about 2.4*r

val. estimated val. real

Indeed 1*2,4 = 2,4 2,65 = r

2,65 * 2,4 = 6,36 5,44 = r

5,44*2,4 = 13,05 16,5 = r

16,5*2,4 = 39,6 ? real value of r

41,25 *2,4 = 99 ? real value of r

From which it goes back to N(n) for n = 11 and n = 12, multiplying by 11 and 12, obtaining the estimated values of N(n) 435 and 1188 respectively for n = 11 and n = 12.

These estimates are approximated by default, as we shall see in the second part, with more precise estimates and therefore more reliable.

6

But let us for a moment the Fibonaccis series, and move to the partitions, since the series of numbers Nn, number of knots, strangely resembles to the numerical series of partition numbers p(N), ie, N(n) p(n) . We compare the two columns up to n = 10

1 2 3 4 5 6 7 8 9 10

Number of prime knots with n crossings 0 0 1 1 2 3 7 21 49 165

Number of partitions up to n = 10 1 1 2 3 5 7 11 15 22 30

This similarity is limited up to n = 8, then the two series stray for n = 9 and n = 10. For successive values, 11 and 12, we have the estimated values 435 and 1188 that, as regards the values of partition, we have 435 as about the arithmetic mean between the partition numbers 385 and 490, in fact: (385 +490) / 2 = 875 / 2 = 437.5 435 estimated value for n = 11. Same thing for 1188 as the mean between 1002 and 1255 in fact (1002 + 1255) / 2 = 2257/2 = 1128 1188 estimated value for n = 12.

But also the previous values of N(n) can approach to the arithmetic means of numbers of partitions:

49 = (42 + 56)/2 = 98/2 = 49 165 = (135 + 176)/2 = 311/2 = 155,5 165

7

and also the previous smaller numbers N(n) can be mean of partition numbers

1 2 3 4 5 6 7 8 9 10

Number of prime knots with n crossings 0 0 1 1 2 3 7 21 49 165

Number of partitions up to n = 10 1 1 2 3 5 7 11 15 22 30

See the following list of numbers with the pairs of partition numbers in blue interested to the arithmetic means, although 1, 2, 3 and 7 are themselves partition numbers

1 = (1+1)/2 = 1 2 = (2+3)/2 = 5/2 = 2,5 2 3 =(2+3)/2 = 5/2 = 2,5 3 7=(5+7)/2 = 12/2= 6 7 21 = (15+30)/2 = 45/2 = 22,5 21 for 49 and 165 see above

Practically, a couple yes and a couple no gives rise to a mean which is very near to a number N(n)

8

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 31185, 37338, 44583, 53174, 63261, 75175, 89134, 105558, 124754, 147273, 173525

The next pair, for n = 13 crossings should be 2436 and 3010 with mean (2436 + 3010) / 2 = 5446 = 2723, instead of 99 * 2.4 * 13 = 3088, which is very near to 3010 partition number: for n = 14, we have 99 * 2.4 * 2.4 * 14 = 7983.36 (6842 +8349) / 2 = 15191/2 = 7595.5, we return to an arithmetic mean, but with a couple not more alternate

Although these estimates are approximated by default, as we shall see in the second part, with more precise estimates and therefore more reliable.

From this we can deduce that the numbers N(n) have a logarithmic curve similar to that of the partition, only that its values, except the initial 1, 2, 3, 7, are very often about a mean of two consecutive numbers of partitions, but the values of n crossings of N(n) and n of p(n) partitions do not coincide perfectly, and there is the problem of connect them.

Here the comparison, interesting, in confirmation of the above, with the Fibonaccis numbers, Lies numbers and principally, of the numbers of partitions: Number of knots N(n) with the equation preferred of the Nature n2 + n + 1 which gives the numbers of Lie, very near to the Fibonaccis numbers and to the partition numbers. Number of nodes in red

9

1 2 3 4 5 6 7 8 9 10

Number of prime knots with n crossings 0 0 1 1 2 3 7 21 49 165

TABLE 4

Fibonacci Lies numbers Number of partitions 1 1 1 1 1 2 2 3 3 3 5 5 8 7 7 7 13 13 11, 15 21 21 22 21 34 31 30 55 57 42, 56 mean

(42+56)/2= 49 89 91 77, 101 144 133 135, 176 mean

(135+176)/2 = 155,5 165

233 240 231

As we see, all the known prime numbers of N(n), number of knots for n crossings coincide or are very near, or are means of numbers of partitions (and, to a lesser extent, also to the Fibonaccis numbers and to the Lies numbers). Expanding on the table yet, so probably will be also for subsequent numbers of knots with n crossings that will be discovered and counted in the future.

10

This is our basic hypothesis, which connects the theory of knots to the partitions and, more generally, to the string theories, as is said in the interview to Eward Witten mentioned at the beginning. The way in which increase the number of knots N(n) to the increase of the n crossings, it might be useful to deepen the relationship between the mathematical theory of knots and the string theories.

Conclusions first part Having noted the above clear relationship between knots types and the Fibonaccis series F(n), of the Lies numbers and of partition of numbers p(n), using various formulas and tables, especially the last Table 4, we leave to the experts in physics and in mathematics of the strings and knot theory the complete classification of all possible cases, which could be facilitated in some way by our Tables.

We want to remember here that we have discovered how to increases the number of partitions p(n) as n increases: the ratio between a number of partitions p(n) and the number of previous partitions p(n-1) is always smaller, and tends to 1 as n increases, which means that p(n) increases more and more slowly as n increases:

p(n) / (pn-1) 1 for n

Will deepen more about this topic (subsequent reports) in the second part

11

Theory of knots.

Second part.

Michele Nardelli, Pierfrancesco Roggero, Francesco Di Noto

Abstract

In this second part we show other observations about Knots Theory

By unifying the various lists of numbers potentially related with N(n), numbers of knots with n crossings, we have the following general table, followed by our new observations with respect to the first part, Ref 1)

TABLE 1

1 2 3 4 5 6 7 8 9 10

Number N(n) of prime knots with n crossings 0 0 1 1 2 3 7 21 49 165 Fibonaccis numbers 0 1 1 2 3 5 8 13 21 34

Partitions numbers 1 1 2 3 5 7 11 15 22 30

Lies numbers 1 3 7 13 21 31 43 57 73 91

As we see, initially the first three series are very near, and connected to the formula of the projective geometry n2 + n + 1, which gives the numbers of Lie 1, 3, 7, 13, 21, 31, 43, 57, where 3, 7, 21 are also

12

numbers N(n). See previous work (Ref. 1)

Here is the final table of the Rif. 1, slightly modified, to remember this possible relationship emerged already with only the first ten already known values of N(n):

TABLE 4

Fibonacci Lies numbers Partitions numbers 1 1 1 1 1 2 2 3 3 3 5 5 8 7 7 7 13 13 11, 15 21 21 22 21 34 31 30 55 57 42, 56 mean

(42+56)/2= 49 > 42 89 91 77, 101 144 133 135, 176 mean

(135+176)/2 = 155,5 165 > 135

233 240 231

13

As we see, all the first known numbers of N(n), number of knots for n crossings coincide or are very near, or are means of numbers of partitions (and, to a lesser extent, also to the Fibonaccis numbers and the Lies numbers). Expanding yet the table, it will be so also for the next number of knots with n crossings that will be discovered and counted in future. This is our basic hypothesis, which connects the theory of knots to the partitions and, more generally, to the string theories, as is hinted in an interview with Edward Witten mentioned at the beginning. The way in which increases the number of knots N(n) with increasing of the n crossings, it might be useful to deepen the relationship between the mathematical theory of knots and the string theories.

Conclusions second part.

We can conclude by saying that the news reported in this second part confirm the conclusions of the previous part (Ref.1) on the possible relationship between knots n, crossings, and number N(n) of knots with n crossings and particularly with the partitions of numbers.

Note 1

Resume the Table 1:

TABLE 1

1 2 3 4 5 6 7 8 9 10

Number N(n) of prime nodes with n crossings 0 0 1 1 2 3 7 21 49 165 Fibonaccis numbers 0 1 1 2 3 5 8 13 21 34

Partitions numbers 1 1 2 3 5 7 11 15 22 30

Lies numbers 1 3 7 13 21 31 43 57 73 91

14

Regarding the number N(n) of knots, the numbers 2 and 3 are also Fibonaccis numbers and numbers of partitions and 3 is also a Lies numbers. The number 7 is a number of partitions and Lies number and sum of the two Fibonaccis numbers 2 and 5. The number 21 is a Fibonaccis number and Lies number, and is also the sum of the three partition numbers 1, 5 and 15. The 49 is given from the mean of the two Lies numbers 91 and 7 ((91 + 7) / 2 = 98/2 = 49) and is also the mean between the 10th and 11th partition number, ie between 42 and 56 (42 +56 = 98; 98/2 = 49). The number 49, in addition, it is also the sum of the three Fibonaccis numbers 2, 13 and 34. Finally, 165 is given from the sum of the two Fibonaccis numbers 144 + 21 = 165 and from the sum of the three Lies numbers 91, 43 and 31 (91 +43 +31 = 165). The 165 is also the sum of the 9th partition number, 30, and of the 14th partition number 135 (135 +30 = 165). Note 2

We now analyze a more comprehensive list up to n = 16, and of the numbers of knots and numbers of partitions.

A002863 as a simple table

More complete list of the number of knots n

a(n) 1

0

2

0

3

1

4

1

5

2

6

3

7

7

8

21

9

49

10

165

11

552

12

2176

13

9988

15

14

46972

15

253293

16

1388705

[0,0,1,1,2,3,7,21,49,165,552,2176,9988,46972, 253293,1388705]

A000041 as a simple table

We note that the 11th number of knots is 552 very near to that we have estimated (435, difference 117). While for the 12th , the 13th and the 14th number of knots we have: 2176 1188 * 2 (2376, diff. 200), thence about twice of the value that we have estimated, 9988 3088 * 3 (9264, diff. 724) , thence about three times the value that we have estimated and finally 46972 7983 * 6 (47898, diff. 926), and thence about six times the value that we have estimated. The 15th number of knots is 253 293, the value that we have estimated was 19807 that is about 13 * 19807 = 257 491. The 16th number of knots is 1388705, the value that we have estimated was 48878 which is about 28 * 48878 = 1368584.

More complete list of the number of partitions

n

a(n) 0

1

1

1

2

2

3

3

4

5

5

7

6

11

7

15

8

22

9

30

10

42

16

11

56

12

77

13

101

14

135

15

176

16

231

17

297

18

385

19

490

20

627

21

792

22

1002

23

1255

24

1575

25

1958

26

2436

27

3010

28

3718

29

4565

30

5604

31

6842

32

8349

33

10143

34

12310

35

14883

36

17977

37

21637

38

26015

39

31185

40

37338

41

44583

42

53174

43

63261

44

75175

45

89134

46

105558

47

124754

48

147273

49

173525

[1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231, 297,385,490,627,792,1002,1255,1575,1958,2436,3010, 3718,4565,5604,6842,8349,10143,12310,14883,17977, 21637,26015,31185,37338,44583,53174,63261,75175,

17

89134,105558,124754,147273,173525]

We see that 552 is the sum of the partition numbers 176, 135, 101, 77, 56 and 7. The number 2176 is given from the sum of the partition numbers 1958, 176 and 42. The number 9988 is given from the sum of the partition numbers 8349, 1575, 42 and 22. The number 46972 is given from the sum of the partition numbers 37338, 8349, 1255 and 30. The number 253293 is the sum of the partition numbers 147273, 105558, 385 and 77.

So, from the 11th to the 15th number of knots we have numbers corresponding "exactly" to sums of numbers of partitions. While up to 10th number of knots, we have numbers corresponding to the Fibonaccis numbers, Lies numbers and partition numbers (and / or their sums and means).

Now we look more closely at the text on the divergence between the our estimates and the real values from the 11th to the 15th number of knots:

We note that the 11th number of knots is 552 very near to that we have estimated (435, difference 117). While for the 12th , the 13th and the 14th number of knots we have: 2176 1188 * 2 (2376, diff. 200), thence about twice of the value that we have estimated, 9988 3088 * 3 (9264, diff. 724) , thence about three times the value that we have estimated and finally 46972 7983 * 6 (47898, diff. 926), and thence about six times the value that we have estimated. The 15th number of knots is 253 293, the value that we have estimated was 19807 that is about 13 * 19807 = 257 491. The 16th number of knots is 1388705, the value that we have estimated was 48878 which is about 28 * 48878 = 1368584.

18

We note that this divergence could be linked, once again, to the Fibonaccis numbers, as from the following table:

our previous estimate multiplied by the first Fibonaccis numbers

Fibonaccis numbers

new best estimates

11th number

estimate 435 real value 552

Estimate real ratio

435/552 = 0,78 1

(1,26 1,618=1,27)

435* 1 = 435

435/1 = 435

435/0,78= 557,69

Integer difference 5 12th number

estimate 1188

real value 2176

2176/1188 = 1,83 2

2176/2 = 1088 1188

Difference 100

13th number

estimate 3088

real value

9988

3,23 3

3088*3 = 9264 diff.724

9988/3088 = 3,23

9988/3 = 3329

Integer difference 241

14th number

estimate 7983

real value 46972

46972/7983 = 5,88 5 46972/5 = 9394,4

Integer difference 1411

46972/5,88= 7988

19

15th number

estimate 19807

real value 253293

253293/19807=

= 12,78 13

253293/13 = 19484

Integer difference 323

16th number

estimate 48878

real value 1388705

1388705/48878 =

= 28,41 27 arit.mean

bet. 21 and 34 = 27,5

1388705/27 = 51433,51

Integer difference

51433- 48878 = 2555

From the 11th to the 15th number, thence, the estimated value of "v" still more reliable is: "v" = estimate * successive Fibonaccis numbers 1, 2, 3, 5,13, or their means, for example, 27 for the 16th number. Predicted estimate of the number of knots for the 17th number: our previous estimate * 34 next Fibonaccis number after the mean 27, and so on.

Estimates, however, now superseded by this further addition.

Further addition

Now we return to the previous table, and now complete it with the exact numbers N(n), from 11th to the 16th

N(n) real values estimates values (in brown)

N(n-1) r =N(n) / N(n-1) from 1 onwards. Real values

r n/3 estimates values, near to the real ones

1 n = 4 1 1 4/3= 1,3 2 5 1 2 5/3= 1,6 3 6 2 1,5 6/3=0 2 7 7 3 2,33 7/3= 2,3 21 8 7 3 2,66

20

49 9 21 2,33 3 165 (10)th number) 49 3,36 3.33 552 11 165 3,34 3,6 2176 12 552 3,94 4 9 988 13 2 176 4,59 4,33 46 972 14 9 988 4,70 4,66 253 293 15 46 972 5,39 5 1 388 705 16 253 293 5,48 5,33 1 388 705*5,66 =

7 860 070 n =17

1 388 705

Real value

17/3=5,66

7 860 070*6 = 47160420

n=18

7 860 070 estimate value 18/3=6

47160420 *6,33 =

298 525 458

n=19

47160420 19/3=6,33

298 525 458*6,66=

1 988 179 550

n=20

298 525 458 20/3=6,66

In this way, we can estimate with good approximation of the values of the 17th , 18th , 19th and 20th number of knots, ie N(17), N(18), N(19) and N(20), estimating with r n/3 the relationship between one of them and the previous one. Further calculations for the real values will confirm whether or not these our actual provisional estimates, the 20th number of knots N(20), for example, should not must be very far from the two billion of knots with 20 crossings.

An estimate for defect would be the product between a real value and the last known relationship, for example 1388705 * 5,48 = 7610103 integer. The real

21

value would then be between the two estimates 7610103 and 7860070, and possibly around their mean (7 610 103 + 7 860 070) = 7 735 086. Further calculations could confirm this our prediction of estimate from the 17th to the 20th number of knots

Note 3

On the Jones Polynomial concerning a knot

With regard the Jones polynomial concerning a knot, Edward Witten has developed the following expression:

( )( ) ( ) 0 ,exp KAHolTrikCSD R (1)

We define the Chern-Simons function ( )CS , for any connection , possibly complex-valued by the following expression:

( )

+=V

TrxdCS pi 32

41 3

. (2)

Writing with h the dual Coxeter number of the gauge group G , we can write a formula equivalent to (2) in terms of a trace adTr in the adjoint representation of G

( )

+=V ad

Trxdh

CS

pi 32

81 3

. (3)

Furthermore, in the (1) the term ( )KAHolTrR , can be written also as follows:

( ) ( ) == KRRR APTrKAHolTrKW exp, . (4)

Thence, the (1) can be rewritten as follows:

[ 0 exp ikD

+ ]32

81 3

V adTrxd

h

pi KRAPTr exp . (5)

The Chern-Simons action for a gauge theory with gauge group G (where G is a compact Lie group) and gauge field A on an oriented three-manifold W , can be written:

22

+=

WAAAdAATrkI

32

4pi. (6)

Remember that in the gauge theory YMg is the gauge coupling constant and the gauge theory theta-angle. The action I of 4=N super Yang-Mills theory on a four-manifold V is the sum of a term proportional to 2/1 YMg , which contains the kinetic energy for all fields, and a term proportional to :

+= V VkinYM

FTrFxdigxdg

I

pi

42

42 321

L . (7)

The part of kinL that involves ,A only is (in Euclidean signature):

[ ]

+++=

2,,

21

21

RDDFFTrAkinL . (8)

We can thence to connect the eq. (6) and (7), obtaining:

+= W AAAdAATr

kI32

4pi +

V VkinYM

FTrFxdigxdg

pi

42

42 321

L . (9)

Thence, the eq. (1) can be written also as follows:

+

0exp

321

exp 424

2 KRVkinVYM

APTrFTrFxdigxdg

ikDA

pi

L , (10)

that is connected with the eq. (5), thence, in conclusion:

[ 0 exp ikD

+ ]32

81 3

V adTrxd

h

piKR APTr exp

+

0exp

321

exp 424

2 KRVkinVYM

APTrFTrFxdigxdg

ikDA

pi

L . (11)

We note as in the eq. (11) there are the Fibonaccis numbers 2, 3 and 8 and that this number is connected with the modes corresponding to the physical vibrations of the superstrings by the following Ramanujan modular equation:

23

( )

++

+

=

42710

421110log

'

142

'

cosh'cos

log4

318

2

'

'

4

0'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

pi

pi

pi

pi

.

Furthermore, we note as 32 = 1 + 2 + 3 + 5 + 8 + 13 that are all Fibonaccis numbers. We have also 32 = 24 + 8 where 24 is also connected with the modes corresponding to the physical vibrations of the bosonic strings by the other Ramanujans modular equation:

( )

++

+

=

42710

421110log

'

142

'

cosh'cos

log4

24

2

'

'

4

0'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

pi

pi

pi

pi

.

Always present in the eq. (5) is, also, the that is connected with the aurea section by the following relations:

5cos2 pi = or

103

sin2 pi =

The link between and , i.e. between 3.14 and 0.618 , can be obtained also from the simple relation: arccos = 0.2879 ; namely

arccos 0.618 = 0.2879

that in radians with the use of the calculator is done so:

rad inv(cos) 0,61803398 = 0,9045569 and 0,2879 3,14159265 = 0,9044645.

From which we see as 0,90455 is very near to another value, i.e. 0,90446 with a small difference 0,00009.

24

References

All the recent papers on our site regarding the string theory and the Fibonaccis sequence as "The equation preferred from the Nature," etc.

1) Wikipedia Nodo primo I nodi primi pi semplici

I nodi primi sono generalmente descritti tramite diagrammi, con ordine crescente di incroci. Il nodo primo pi semplice in questa descrizione il nodo a trifoglio con 3 incroci, seguito dal nodo a otto con quattro incroci. La tabella seguente mostra il numero di nodi primi con incroci.

1 2 3 4 5 6 7 8 9 10

Numero di nodi primi con incroci 0 0 1 1 2 3 7 21 49 165

2) Nodi e stringhe della fisica sul sito de Il Sole 24 Ore, dal quale riportiamo brevemente

Ancora nelle pi avanzate regioni di confine tra matematica e fisica, nella interazione tra teoria dei quanti, teoria delle stringhe e teoria matematica dei nodi si collocano le sue ricerche pi recenti. Un nodo un oggetto familiare e a prima vista non sembra offrire argomento di interesse matematico. Umberto Bottazzini - Il Sole 24 Ore - leggi su http://24o.it/hfOqI

25

3) Teoria dei nodi e i numeri di Fibonacci Sul nostro sito http://nardelli.xoom.it/virgiliowizard/

4) Teoria dei nodi Da Wikipedia, l'enciclopedia libera.

La teoria dei nodi una branca della topologia, a sua volta branca della matematica, che si occupa di nodi, ovvero di curve chiuse intrecciate nello spazio. La teoria ha applicazioni in fisica subatomica, chimica supramolecolare e biologia.

Fregio del cancello del dipartimento di matematica di Cambridge.

Per i suoi stretti legami con lo studio delle variet di basse dimensioni (1, 2, 3 e 4), la teoria dei nodi spesso considerata una branca della topologia della dimensione bassa.

5) sito http://www.aromatic.com/rudi/026.pdf :

6) http://www.unimib.it/open/news/I-polinomi-che-governano-la-complessita/6413442428496323404

I polinomi che governano la complessit

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Grazie ai formidabili progressi fatti in questi ultimi anni, in particolare dalla teoria dei nodi, ora possibile identificare e seguire nel tempo lannodamento e lo snodamento di filamenti fluidi nello spazio. Semplici polinomi come x+x3x-4 identificano in modo univoco ognuno degli infiniti ed evanescenti nodi e legami che si formano nel fluido, a cui si associano poi propriet dinamiche ed energetiche

7) The Jones Polynomial

http://math.berkeley.edu/~vfr/jones.pdf

8) Intervista a Vaughan Jones di Francesco Vaccarino - Politecnico di Torino

da "La Stampa - Tuttoscienze" del 18 marzo 2009

http://areeweb.polito.it/didattica/polymath/htmlS/Interventi/Articoli/VaccarinoNodi/VaccarinoNod...

9)Nodi e fisica Enciclopedia della Scienza e della Tecnica (2007)

di Louis H. Kauffman

http://www.treccani.it/enciclopedia/nodi-e-fisica_(Enciclopedia-della-Scienza-e-della-Tecnica)/

10)Edward Witten A new look at the Jones Polynomial of a Knot Clay Conference, Oxford, October 1, 2013

11)Edward Witten Fivebranes and Knots arXiv:1101.3216v2 [hep-th] - 11/08/2011