T(23,2): Difference between revisions

Revision as of 18:42, 28 August 2005

T(11,3)

T(6,5)

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T(23,2) Quick Notes

T(23,2) Further Notes and Views

Knot presentations

Planar diagram presentation	X_9,33,10,32 X_33,11,34,10 X_11,35,12,34 X_35,13,36,12 X_13,37,14,36 X_37,15,38,14 X_15,39,16,38 X_39,17,40,16 X_17,41,18,40 X_41,19,42,18 X_19,43,20,42 X_43,21,44,20 X_21,45,22,44 X_45,23,46,22 X_23,1,24,46 X_1,25,2,24 X_25,3,26,2 X_3,27,4,26 X_27,5,28,4 X_5,29,6,28 X_29,7,30,6 X_7,31,8,30 X_31,9,32,8
Gauss code	-16, 17, -18, 19, -20, 21, -22, 23, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15
Dowker-Thistlethwaite code	24 26 28 30 32 34 36 38 40 42 44 46 2 4 6 8 10 12 14 16 18 20 22
Conway Notation	Data:T(23,2)/Conway Notation

Polynomial invariants

Alexander polynomial	$t^{11}-t^{10}+t^{9}-t^{8}+t^{7}-t^{6}+t^{5}-t^{4}+t^{3}-t^{2}+t-1+t^{-1}-t^{-2}+t^{-3}-t^{-4}+t^{-5}-t^{-6}+t^{-7}-t^{-8}+t^{-9}-t^{-10}+t^{-11}$
Conway polynomial	$z^{22}+21z^{20}+190z^{18}+969z^{16}+3060z^{14}+6188z^{12}+8008z^{10}+6435z^{8}+3003z^{6}+715z^{4}+66z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 23, 22 }
Jones polynomial	$-q^{34}+q^{33}-q^{32}+q^{31}-q^{30}+q^{29}-q^{28}+q^{27}-q^{26}+q^{25}-q^{24}+q^{23}-q^{22}+q^{21}-q^{20}+q^{19}-q^{18}+q^{17}-q^{16}+q^{15}-q^{14}+q^{13}+q^{11}$
HOMFLY-PT polynomial (db, data sources)	$z^{22}a^{-22}+22z^{20}a^{-22}-z^{20}a^{-24}+210z^{18}a^{-22}-20z^{18}a^{-24}+1140z^{16}a^{-22}-171z^{16}a^{-24}+3876z^{14}a^{-22}-816z^{14}a^{-24}+8568z^{12}a^{-22}-2380z^{12}a^{-24}+12376z^{10}a^{-22}-4368z^{10}a^{-24}+11440z^{8}a^{-22}-5005z^{8}a^{-24}+6435z^{6}a^{-22}-3432z^{6}a^{-24}+2002z^{4}a^{-22}-1287z^{4}a^{-24}+286z^{2}a^{-22}-220z^{2}a^{-24}+12a^{-22}-11a^{-24}$
Kauffman polynomial (db, data sources)	$z^{22}a^{-22}+z^{22}a^{-24}+z^{21}a^{-23}+z^{21}a^{-25}-22z^{20}a^{-22}-21z^{20}a^{-24}+z^{20}a^{-26}-20z^{19}a^{-23}-19z^{19}a^{-25}+z^{19}a^{-27}+210z^{18}a^{-22}+191z^{18}a^{-24}-18z^{18}a^{-26}+z^{18}a^{-28}+171z^{17}a^{-23}+153z^{17}a^{-25}-17z^{17}a^{-27}+z^{17}a^{-29}-1140z^{16}a^{-22}-987z^{16}a^{-24}+136z^{16}a^{-26}-16z^{16}a^{-28}+z^{16}a^{-30}-816z^{15}a^{-23}-680z^{15}a^{-25}+120z^{15}a^{-27}-15z^{15}a^{-29}+z^{15}a^{-31}+3876z^{14}a^{-22}+3196z^{14}a^{-24}-560z^{14}a^{-26}+105z^{14}a^{-28}-14z^{14}a^{-30}+z^{14}a^{-32}+2380z^{13}a^{-23}+1820z^{13}a^{-25}-455z^{13}a^{-27}+91z^{13}a^{-29}-13z^{13}a^{-31}+z^{13}a^{-33}-8568z^{12}a^{-22}-6748z^{12}a^{-24}+1365z^{12}a^{-26}-364z^{12}a^{-28}+78z^{12}a^{-30}-12z^{12}a^{-32}+z^{12}a^{-34}-4368z^{11}a^{-23}-3003z^{11}a^{-25}+1001z^{11}a^{-27}-286z^{11}a^{-29}+66z^{11}a^{-31}-11z^{11}a^{-33}+z^{11}a^{-35}+12376z^{10}a^{-22}+9373z^{10}a^{-24}-2002z^{10}a^{-26}+715z^{10}a^{-28}-220z^{10}a^{-30}+55z^{10}a^{-32}-10z^{10}a^{-34}+z^{10}a^{-36}+5005z^{9}a^{-23}+3003z^{9}a^{-25}-1287z^{9}a^{-27}+495z^{9}a^{-29}-165z^{9}a^{-31}+45z^{9}a^{-33}-9z^{9}a^{-35}+z^{9}a^{-37}-11440z^{8}a^{-22}-8437z^{8}a^{-24}+1716z^{8}a^{-26}-792z^{8}a^{-28}+330z^{8}a^{-30}-120z^{8}a^{-32}+36z^{8}a^{-34}-8z^{8}a^{-36}+z^{8}a^{-38}-3432z^{7}a^{-23}-1716z^{7}a^{-25}+924z^{7}a^{-27}-462z^{7}a^{-29}+210z^{7}a^{-31}-84z^{7}a^{-33}+28z^{7}a^{-35}-7z^{7}a^{-37}+z^{7}a^{-39}+6435z^{6}a^{-22}+4719z^{6}a^{-24}-792z^{6}a^{-26}+462z^{6}a^{-28}-252z^{6}a^{-30}+126z^{6}a^{-32}-56z^{6}a^{-34}+21z^{6}a^{-36}-6z^{6}a^{-38}+z^{6}a^{-40}+1287z^{5}a^{-23}+495z^{5}a^{-25}-330z^{5}a^{-27}+210z^{5}a^{-29}-126z^{5}a^{-31}+70z^{5}a^{-33}-35z^{5}a^{-35}+15z^{5}a^{-37}-5z^{5}a^{-39}+z^{5}a^{-41}-2002z^{4}a^{-22}-1507z^{4}a^{-24}+165z^{4}a^{-26}-120z^{4}a^{-28}+84z^{4}a^{-30}-56z^{4}a^{-32}+35z^{4}a^{-34}-20z^{4}a^{-36}+10z^{4}a^{-38}-4z^{4}a^{-40}+z^{4}a^{-42}-220z^{3}a^{-23}-55z^{3}a^{-25}+45z^{3}a^{-27}-36z^{3}a^{-29}+28z^{3}a^{-31}-21z^{3}a^{-33}+15z^{3}a^{-35}-10z^{3}a^{-37}+6z^{3}a^{-39}-3z^{3}a^{-41}+z^{3}a^{-43}+286z^{2}a^{-22}+231z^{2}a^{-24}-10z^{2}a^{-26}+9z^{2}a^{-28}-8z^{2}a^{-30}+7z^{2}a^{-32}-6z^{2}a^{-34}+5z^{2}a^{-36}-4z^{2}a^{-38}+3z^{2}a^{-40}-2z^{2}a^{-42}+z^{2}a^{-44}+11za^{-23}+za^{-25}-za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-za^{-39}+za^{-41}-za^{-43}+za^{-45}-12a^{-22}-11a^{-24}$
The A2 invariant	Data:T(23,2)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(23,2)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(23,2) Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

In[3]:= K = Knot["T(23,2)"];

In[4]:= Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= $t^{11}-t^{10}+t^{9}-t^{8}+t^{7}-t^{6}+t^{5}-t^{4}+t^{3}-t^{2}+t-1+t^{-1}-t^{-2}+t^{-3}-t^{-4}+t^{-5}-t^{-6}+t^{-7}-t^{-8}+t^{-9}-t^{-10}+t^{-11}$

In[5]:= Conway[K][z]

Out[5]= $z^{22}+21z^{20}+190z^{18}+969z^{16}+3060z^{14}+6188z^{12}+8008z^{10}+6435z^{8}+3003z^{6}+715z^{4}+66z^{2}+1$

In[6]:= Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]= $\{1\}$

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 23, 22 }

In[8]:= Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]= $-q^{34}+q^{33}-q^{32}+q^{31}-q^{30}+q^{29}-q^{28}+q^{27}-q^{26}+q^{25}-q^{24}+q^{23}-q^{22}+q^{21}-q^{20}+q^{19}-q^{18}+q^{17}-q^{16}+q^{15}-q^{14}+q^{13}+q^{11}$

In[9]:= HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]= $z^{22}a^{-22}+22z^{20}a^{-22}-z^{20}a^{-24}+210z^{18}a^{-22}-20z^{18}a^{-24}+1140z^{16}a^{-22}-171z^{16}a^{-24}+3876z^{14}a^{-22}-816z^{14}a^{-24}+8568z^{12}a^{-22}-2380z^{12}a^{-24}+12376z^{10}a^{-22}-4368z^{10}a^{-24}+11440z^{8}a^{-22}-5005z^{8}a^{-24}+6435z^{6}a^{-22}-3432z^{6}a^{-24}+2002z^{4}a^{-22}-1287z^{4}a^{-24}+286z^{2}a^{-22}-220z^{2}a^{-24}+12a^{-22}-11a^{-24}$

In[10]:= Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]= $z^{22}a^{-22}+z^{22}a^{-24}+z^{21}a^{-23}+z^{21}a^{-25}-22z^{20}a^{-22}-21z^{20}a^{-24}+z^{20}a^{-26}-20z^{19}a^{-23}-19z^{19}a^{-25}+z^{19}a^{-27}+210z^{18}a^{-22}+191z^{18}a^{-24}-18z^{18}a^{-26}+z^{18}a^{-28}+171z^{17}a^{-23}+153z^{17}a^{-25}-17z^{17}a^{-27}+z^{17}a^{-29}-1140z^{16}a^{-22}-987z^{16}a^{-24}+136z^{16}a^{-26}-16z^{16}a^{-28}+z^{16}a^{-30}-816z^{15}a^{-23}-680z^{15}a^{-25}+120z^{15}a^{-27}-15z^{15}a^{-29}+z^{15}a^{-31}+3876z^{14}a^{-22}+3196z^{14}a^{-24}-560z^{14}a^{-26}+105z^{14}a^{-28}-14z^{14}a^{-30}+z^{14}a^{-32}+2380z^{13}a^{-23}+1820z^{13}a^{-25}-455z^{13}a^{-27}+91z^{13}a^{-29}-13z^{13}a^{-31}+z^{13}a^{-33}-8568z^{12}a^{-22}-6748z^{12}a^{-24}+1365z^{12}a^{-26}-364z^{12}a^{-28}+78z^{12}a^{-30}-12z^{12}a^{-32}+z^{12}a^{-34}-4368z^{11}a^{-23}-3003z^{11}a^{-25}+1001z^{11}a^{-27}-286z^{11}a^{-29}+66z^{11}a^{-31}-11z^{11}a^{-33}+z^{11}a^{-35}+12376z^{10}a^{-22}+9373z^{10}a^{-24}-2002z^{10}a^{-26}+715z^{10}a^{-28}-220z^{10}a^{-30}+55z^{10}a^{-32}-10z^{10}a^{-34}+z^{10}a^{-36}+5005z^{9}a^{-23}+3003z^{9}a^{-25}-1287z^{9}a^{-27}+495z^{9}a^{-29}-165z^{9}a^{-31}+45z^{9}a^{-33}-9z^{9}a^{-35}+z^{9}a^{-37}-11440z^{8}a^{-22}-8437z^{8}a^{-24}+1716z^{8}a^{-26}-792z^{8}a^{-28}+330z^{8}a^{-30}-120z^{8}a^{-32}+36z^{8}a^{-34}-8z^{8}a^{-36}+z^{8}a^{-38}-3432z^{7}a^{-23}-1716z^{7}a^{-25}+924z^{7}a^{-27}-462z^{7}a^{-29}+210z^{7}a^{-31}-84z^{7}a^{-33}+28z^{7}a^{-35}-7z^{7}a^{-37}+z^{7}a^{-39}+6435z^{6}a^{-22}+4719z^{6}a^{-24}-792z^{6}a^{-26}+462z^{6}a^{-28}-252z^{6}a^{-30}+126z^{6}a^{-32}-56z^{6}a^{-34}+21z^{6}a^{-36}-6z^{6}a^{-38}+z^{6}a^{-40}+1287z^{5}a^{-23}+495z^{5}a^{-25}-330z^{5}a^{-27}+210z^{5}a^{-29}-126z^{5}a^{-31}+70z^{5}a^{-33}-35z^{5}a^{-35}+15z^{5}a^{-37}-5z^{5}a^{-39}+z^{5}a^{-41}-2002z^{4}a^{-22}-1507z^{4}a^{-24}+165z^{4}a^{-26}-120z^{4}a^{-28}+84z^{4}a^{-30}-56z^{4}a^{-32}+35z^{4}a^{-34}-20z^{4}a^{-36}+10z^{4}a^{-38}-4z^{4}a^{-40}+z^{4}a^{-42}-220z^{3}a^{-23}-55z^{3}a^{-25}+45z^{3}a^{-27}-36z^{3}a^{-29}+28z^{3}a^{-31}-21z^{3}a^{-33}+15z^{3}a^{-35}-10z^{3}a^{-37}+6z^{3}a^{-39}-3z^{3}a^{-41}+z^{3}a^{-43}+286z^{2}a^{-22}+231z^{2}a^{-24}-10z^{2}a^{-26}+9z^{2}a^{-28}-8z^{2}a^{-30}+7z^{2}a^{-32}-6z^{2}a^{-34}+5z^{2}a^{-36}-4z^{2}a^{-38}+3z^{2}a^{-40}-2z^{2}a^{-42}+z^{2}a^{-44}+11za^{-23}+za^{-25}-za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-za^{-39}+za^{-41}-za^{-43}+za^{-45}-12a^{-22}-11a^{-24}$

Vassiliev invariants

V₂ and V₃:

(66, 506)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$ , alternation over $r$ ). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s+1$ , where $s=$ 22 is the signature of T(23,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

χ

69

1

-1

67

0

65

1

0

63

0

61

1

0

59

0

57

1

0

55

0

53

1

0

51

0

49

1

0

47

0

45

1

0

43

0

41

1

0

39

0

37

1

0

35

0

33

1

0

31

0

29

1

0

27

0

25

1

23

1

21

1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[TorusKnot[23, 2]]
Out[2]=	23
In[3]:=	PD[TorusKnot[23, 2]]
Out[3]=	PD[X[9, 33, 10, 32], X[33, 11, 34, 10], X[11, 35, 12, 34], X[35, 13, 36, 12], X[13, 37, 14, 36], X[37, 15, 38, 14], X[15, 39, 16, 38], X[39, 17, 40, 16], X[17, 41, 18, 40], X[41, 19, 42, 18], X[19, 43, 20, 42], X[43, 21, 44, 20], X[21, 45, 22, 44], X[45, 23, 46, 22], X[23, 1, 24, 46], X[1, 25, 2, 24], X[25, 3, 26, 2], X[3, 27, 4, 26], X[27, 5, 28, 4], X[5, 29, 6, 28], X[29, 7, 30, 6], X[7, 31, 8, 30], X[31, 9, 32, 8]]
In[4]:=	GaussCode[TorusKnot[23, 2]]
Out[4]=	GaussCode[-16, 17, -18, 19, -20, 21, -22, 23, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15]
In[5]:=	BR[TorusKnot[23, 2]]
Out[5]=	BR[2, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}]
In[6]:=	alex = Alexander[TorusKnot[23, 2]][t]
Out[6]=	-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 1 -1 + t - t + t - t + t - t + t - t + t - t + - + t 2 3 4 5 6 7 8 9 10 11 t - t + t - t + t - t + t - t + t - t + t
In[7]:=	Conway[TorusKnot[23, 2]][z]
Out[7]=	2 4 6 8 10 12 1 + 66 z + 715 z + 3003 z + 6435 z + 8008 z + 6188 z + 14 16 18 20 22 3060 z + 969 z + 190 z + 21 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[TorusKnot[23, 2]], KnotSignature[TorusKnot[23, 2]]}
Out[9]=	{23, 22}
In[10]:=	J=Jones[TorusKnot[23, 2]][q]
Out[10]=	11 13 14 15 16 17 18 19 20 21 22 23 q + q - q + q - q + q - q + q - q + q - q + q - 24 25 26 27 28 29 30 31 32 33 34 q + q - q + q - q + q - q + q - q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[TorusKnot[23, 2]][q]
Out[12]=	NotAvailable
In[13]:=	Kauffman[TorusKnot[23, 2]][a, z]
Out[13]=	NotAvailable
In[14]:=	{Vassiliev[2][TorusKnot[23, 2]], Vassiliev[3][TorusKnot[23, 2]]}
Out[14]=	{0, 506}
In[15]:=	Kh[TorusKnot[23, 2]][q, t]
Out[15]=	21 23 25 2 29 3 29 4 33 5 33 6 37 7 q + q + q t + q t + q t + q t + q t + q t + 37 8 41 9 41 10 45 11 45 12 49 13 49 14 q t + q t + q t + q t + q t + q t + q t + 53 15 53 16 57 17 57 18 61 19 61 20 65 21 q t + q t + q t + q t + q t + q t + q t + 65 22 69 23 q t + q t

This category should contain all the individual knots pages, like 7_5, K11n67, L8a2 and T(5,3)

@@ Line 1: / Line 1: @@
-<!-- Script generated - do not edit! -->
+<!--  -->
 <!--  -->
@@ Line 10: / Line 10: @@
 |- valign=top
 |[[Image:{{PAGENAME}}.jpg]]
-|Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-16,17,-18,19,-20,21,-22,23,-1,2,-3,4,-5,6,-7,8,-9,10,-11,12,-13,14,-15,16,-17,18,-19,20,-21,22,-23,1,-2,3,-4,5,-6,7,-8,9,-10,11,-12,13,-14,15/goTop.html {{PAGENAME}}'s page] at [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html Knotilus]!
+|{{Torus Knot Site Links|m=23|n=2|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-16,17,-18,19,-20,21,-22,23,-1,2,-3,4,-5,6,-7,8,-9,10,-11,12,-13,14,-15,16,-17,18,-19,20,-21,22,-23,1,-2,3,-4,5,-6,7,-8,9,-10,11,-12,13,-14,15/goTop.html}}
-Visit [http://www.math.toronto.edu/~drorbn/KAtlas/TorusKnots/23.2.html {{PAGENAME}}'s page] at the original [http://www.math.toronto.edu/~drorbn/KAtlas/index.html Knot Atlas]!
 {{:{{PAGENAME}} Quick Notes}}
@@ Line 22: / Line 20: @@
 {{Knot Presentations}}
-===Knot presentations===
-{|
-|'''[[Planar Diagrams|Planar diagram presentation]]'''
-|style="padding-left: 1em;" | X<sub>9,33,10,32</sub> X<sub>33,11,34,10</sub> X<sub>11,35,12,34</sub> X<sub>35,13,36,12</sub> X<sub>13,37,14,36</sub> X<sub>37,15,38,14</sub> X<sub>15,39,16,38</sub> X<sub>39,17,40,16</sub> X<sub>17,41,18,40</sub> X<sub>41,19,42,18</sub> X<sub>19,43,20,42</sub> X<sub>43,21,44,20</sub> X<sub>21,45,22,44</sub> X<sub>45,23,46,22</sub> X<sub>23,1,24,46</sub> X<sub>1,25,2,24</sub> X<sub>25,3,26,2</sub> X<sub>3,27,4,26</sub> X<sub>27,5,28,4</sub> X<sub>5,29,6,28</sub> X<sub>29,7,30,6</sub> X<sub>7,31,8,30</sub> X<sub>31,9,32,8</sub>
-|-
-|'''[[Gauss Codes|Gauss code]]'''
-|style="padding-left: 1em;" | <math>\{-16,17,-18,19,-20,21,-22,23,-1,2,-3,4,-5,6,-7,8,-9,10,-11,12,-13,14,-15,16,-17,18,-19,20,-21,22,-23,1,-2,3,-4,5,-6,7,-8,9,-10,11,-12,13,-14,15\}</math>
-|-
-|'''[[DT (Dowker-Thistlethwaite) Codes|Dowker-Thistlethwaite code]]'''
-|style="padding-left: 1em;" | 24 26 28 30 32 34 36 38 40 42 44 46 2 4 6 8 10 12 14 16 18 20 22
-|}
 {{Polynomial Invariants}}
 {{Vassiliev Invariants}}
@@ Line 157: / Line 141: @@
   q   t   + q   t</nowiki></pre></td></tr>
 </table>
+ {{Category:Knot Page}}

T(23,2): Difference between revisions

Revision as of 18:42, 28 August 2005

Contents

Knot presentations

Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

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