T(21,2)

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Knot presentations

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

Polynomial invariants

Alexander polynomial	$t^{10}-t^{9}+t^{8}-t^{7}+t^{6}-t^{5}+t^{4}-t^{3}+t^{2}-t+1-t^{-1}+t^{-2}-t^{-3}+t^{-4}-t^{-5}+t^{-6}-t^{-7}+t^{-8}-t^{-9}+t^{-10}$
Conway polynomial	$z^{20}+19z^{18}+153z^{16}+680z^{14}+1820z^{12}+3003z^{10}+3003z^{8}+1716z^{6}+495z^{4}+55z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 21, 20 }
Jones polynomial	$-q^{31}+q^{30}-q^{29}+q^{28}-q^{27}+q^{26}-q^{25}+q^{24}-q^{23}+q^{22}-q^{21}+q^{20}-q^{19}+q^{18}-q^{17}+q^{16}-q^{15}+q^{14}-q^{13}+q^{12}+q^{10}$
HOMFLY-PT polynomial (db, data sources)	$z^{20}a^{-20}+20z^{18}a^{-20}-z^{18}a^{-22}+171z^{16}a^{-20}-18z^{16}a^{-22}+816z^{14}a^{-20}-136z^{14}a^{-22}+2380z^{12}a^{-20}-560z^{12}a^{-22}+4368z^{10}a^{-20}-1365z^{10}a^{-22}+5005z^{8}a^{-20}-2002z^{8}a^{-22}+3432z^{6}a^{-20}-1716z^{6}a^{-22}+1287z^{4}a^{-20}-792z^{4}a^{-22}+220z^{2}a^{-20}-165z^{2}a^{-22}+11a^{-20}-10a^{-22}$
Kauffman polynomial (db, data sources)	$z^{20}a^{-20}+z^{20}a^{-22}+z^{19}a^{-21}+z^{19}a^{-23}-20z^{18}a^{-20}-19z^{18}a^{-22}+z^{18}a^{-24}-18z^{17}a^{-21}-17z^{17}a^{-23}+z^{17}a^{-25}+171z^{16}a^{-20}+154z^{16}a^{-22}-16z^{16}a^{-24}+z^{16}a^{-26}+136z^{15}a^{-21}+120z^{15}a^{-23}-15z^{15}a^{-25}+z^{15}a^{-27}-816z^{14}a^{-20}-696z^{14}a^{-22}+105z^{14}a^{-24}-14z^{14}a^{-26}+z^{14}a^{-28}-560z^{13}a^{-21}-455z^{13}a^{-23}+91z^{13}a^{-25}-13z^{13}a^{-27}+z^{13}a^{-29}+2380z^{12}a^{-20}+1925z^{12}a^{-22}-364z^{12}a^{-24}+78z^{12}a^{-26}-12z^{12}a^{-28}+z^{12}a^{-30}+1365z^{11}a^{-21}+1001z^{11}a^{-23}-286z^{11}a^{-25}+66z^{11}a^{-27}-11z^{11}a^{-29}+z^{11}a^{-31}-4368z^{10}a^{-20}-3367z^{10}a^{-22}+715z^{10}a^{-24}-220z^{10}a^{-26}+55z^{10}a^{-28}-10z^{10}a^{-30}+z^{10}a^{-32}-2002z^{9}a^{-21}-1287z^{9}a^{-23}+495z^{9}a^{-25}-165z^{9}a^{-27}+45z^{9}a^{-29}-9z^{9}a^{-31}+z^{9}a^{-33}+5005z^{8}a^{-20}+3718z^{8}a^{-22}-792z^{8}a^{-24}+330z^{8}a^{-26}-120z^{8}a^{-28}+36z^{8}a^{-30}-8z^{8}a^{-32}+z^{8}a^{-34}+1716z^{7}a^{-21}+924z^{7}a^{-23}-462z^{7}a^{-25}+210z^{7}a^{-27}-84z^{7}a^{-29}+28z^{7}a^{-31}-7z^{7}a^{-33}+z^{7}a^{-35}-3432z^{6}a^{-20}-2508z^{6}a^{-22}+462z^{6}a^{-24}-252z^{6}a^{-26}+126z^{6}a^{-28}-56z^{6}a^{-30}+21z^{6}a^{-32}-6z^{6}a^{-34}+z^{6}a^{-36}-792z^{5}a^{-21}-330z^{5}a^{-23}+210z^{5}a^{-25}-126z^{5}a^{-27}+70z^{5}a^{-29}-35z^{5}a^{-31}+15z^{5}a^{-33}-5z^{5}a^{-35}+z^{5}a^{-37}+1287z^{4}a^{-20}+957z^{4}a^{-22}-120z^{4}a^{-24}+84z^{4}a^{-26}-56z^{4}a^{-28}+35z^{4}a^{-30}-20z^{4}a^{-32}+10z^{4}a^{-34}-4z^{4}a^{-36}+z^{4}a^{-38}+165z^{3}a^{-21}+45z^{3}a^{-23}-36z^{3}a^{-25}+28z^{3}a^{-27}-21z^{3}a^{-29}+15z^{3}a^{-31}-10z^{3}a^{-33}+6z^{3}a^{-35}-3z^{3}a^{-37}+z^{3}a^{-39}-220z^{2}a^{-20}-175z^{2}a^{-22}+9z^{2}a^{-24}-8z^{2}a^{-26}+7z^{2}a^{-28}-6z^{2}a^{-30}+5z^{2}a^{-32}-4z^{2}a^{-34}+3z^{2}a^{-36}-2z^{2}a^{-38}+z^{2}a^{-40}-10za^{-21}-za^{-23}+za^{-25}-za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-za^{-39}+za^{-41}+11a^{-20}+10a^{-22}$
The A2 invariant	Data:T(21,2)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(21,2)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(21,2) 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(21,2)"];

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

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

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

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

Out[5]= $z^{20}+19z^{18}+153z^{16}+680z^{14}+1820z^{12}+3003z^{10}+3003z^{8}+1716z^{6}+495z^{4}+55z^{2}+1$

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]= { 21, 20 }

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

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

Out[8]= $-q^{31}+q^{30}-q^{29}+q^{28}-q^{27}+q^{26}-q^{25}+q^{24}-q^{23}+q^{22}-q^{21}+q^{20}-q^{19}+q^{18}-q^{17}+q^{16}-q^{15}+q^{14}-q^{13}+q^{12}+q^{10}$

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

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

Out[9]= $z^{20}a^{-20}+20z^{18}a^{-20}-z^{18}a^{-22}+171z^{16}a^{-20}-18z^{16}a^{-22}+816z^{14}a^{-20}-136z^{14}a^{-22}+2380z^{12}a^{-20}-560z^{12}a^{-22}+4368z^{10}a^{-20}-1365z^{10}a^{-22}+5005z^{8}a^{-20}-2002z^{8}a^{-22}+3432z^{6}a^{-20}-1716z^{6}a^{-22}+1287z^{4}a^{-20}-792z^{4}a^{-22}+220z^{2}a^{-20}-165z^{2}a^{-22}+11a^{-20}-10a^{-22}$

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

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

Out[10]= $z^{20}a^{-20}+z^{20}a^{-22}+z^{19}a^{-21}+z^{19}a^{-23}-20z^{18}a^{-20}-19z^{18}a^{-22}+z^{18}a^{-24}-18z^{17}a^{-21}-17z^{17}a^{-23}+z^{17}a^{-25}+171z^{16}a^{-20}+154z^{16}a^{-22}-16z^{16}a^{-24}+z^{16}a^{-26}+136z^{15}a^{-21}+120z^{15}a^{-23}-15z^{15}a^{-25}+z^{15}a^{-27}-816z^{14}a^{-20}-696z^{14}a^{-22}+105z^{14}a^{-24}-14z^{14}a^{-26}+z^{14}a^{-28}-560z^{13}a^{-21}-455z^{13}a^{-23}+91z^{13}a^{-25}-13z^{13}a^{-27}+z^{13}a^{-29}+2380z^{12}a^{-20}+1925z^{12}a^{-22}-364z^{12}a^{-24}+78z^{12}a^{-26}-12z^{12}a^{-28}+z^{12}a^{-30}+1365z^{11}a^{-21}+1001z^{11}a^{-23}-286z^{11}a^{-25}+66z^{11}a^{-27}-11z^{11}a^{-29}+z^{11}a^{-31}-4368z^{10}a^{-20}-3367z^{10}a^{-22}+715z^{10}a^{-24}-220z^{10}a^{-26}+55z^{10}a^{-28}-10z^{10}a^{-30}+z^{10}a^{-32}-2002z^{9}a^{-21}-1287z^{9}a^{-23}+495z^{9}a^{-25}-165z^{9}a^{-27}+45z^{9}a^{-29}-9z^{9}a^{-31}+z^{9}a^{-33}+5005z^{8}a^{-20}+3718z^{8}a^{-22}-792z^{8}a^{-24}+330z^{8}a^{-26}-120z^{8}a^{-28}+36z^{8}a^{-30}-8z^{8}a^{-32}+z^{8}a^{-34}+1716z^{7}a^{-21}+924z^{7}a^{-23}-462z^{7}a^{-25}+210z^{7}a^{-27}-84z^{7}a^{-29}+28z^{7}a^{-31}-7z^{7}a^{-33}+z^{7}a^{-35}-3432z^{6}a^{-20}-2508z^{6}a^{-22}+462z^{6}a^{-24}-252z^{6}a^{-26}+126z^{6}a^{-28}-56z^{6}a^{-30}+21z^{6}a^{-32}-6z^{6}a^{-34}+z^{6}a^{-36}-792z^{5}a^{-21}-330z^{5}a^{-23}+210z^{5}a^{-25}-126z^{5}a^{-27}+70z^{5}a^{-29}-35z^{5}a^{-31}+15z^{5}a^{-33}-5z^{5}a^{-35}+z^{5}a^{-37}+1287z^{4}a^{-20}+957z^{4}a^{-22}-120z^{4}a^{-24}+84z^{4}a^{-26}-56z^{4}a^{-28}+35z^{4}a^{-30}-20z^{4}a^{-32}+10z^{4}a^{-34}-4z^{4}a^{-36}+z^{4}a^{-38}+165z^{3}a^{-21}+45z^{3}a^{-23}-36z^{3}a^{-25}+28z^{3}a^{-27}-21z^{3}a^{-29}+15z^{3}a^{-31}-10z^{3}a^{-33}+6z^{3}a^{-35}-3z^{3}a^{-37}+z^{3}a^{-39}-220z^{2}a^{-20}-175z^{2}a^{-22}+9z^{2}a^{-24}-8z^{2}a^{-26}+7z^{2}a^{-28}-6z^{2}a^{-30}+5z^{2}a^{-32}-4z^{2}a^{-34}+3z^{2}a^{-36}-2z^{2}a^{-38}+z^{2}a^{-40}-10za^{-21}-za^{-23}+za^{-25}-za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-za^{-39}+za^{-41}+11a^{-20}+10a^{-22}$

Vassiliev invariants

V₂ and V₃:

(55, 385)

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 $s=$ 20 is the signature of T(21,2). 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

χ

63

1

-1

61

0

59

1

0

57

0

55

1

0

53

0

51

1

0

49

0

47

1

0

45

0

43

1

0

41

0

39

1

0

37

0

35

1

0

33

0

31

1

0

29

0

27

1

0

25

0

23

1

21

1

19

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[21, 2]]
Out[2]=	21
In[3]:=	PD[TorusKnot[21, 2]]
Out[3]=	PD[X[11, 33, 12, 32], X[33, 13, 34, 12], X[13, 35, 14, 34], X[35, 15, 36, 14], X[15, 37, 16, 36], X[37, 17, 38, 16], X[17, 39, 18, 38], X[39, 19, 40, 18], X[19, 41, 20, 40], X[41, 21, 42, 20], X[21, 1, 22, 42], X[1, 23, 2, 22], X[23, 3, 24, 2], X[3, 25, 4, 24], X[25, 5, 26, 4], X[5, 27, 6, 26], X[27, 7, 28, 6], X[7, 29, 8, 28], X[29, 9, 30, 8], X[9, 31, 10, 30], X[31, 11, 32, 10]]
In[4]:=	GaussCode[TorusKnot[21, 2]]
Out[4]=	GaussCode[-12, 13, -14, 15, -16, 17, -18, 19, -20, 21, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11]
In[5]:=	BR[TorusKnot[21, 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}]
In[6]:=	alex = Alexander[TorusKnot[21, 2]][t]
Out[6]=	-10 -9 -8 -7 -6 -5 -4 -3 -2 1 2 1 + t - t + t - t + t - t + t - t + t - - - t + t - t 3 4 5 6 7 8 9 10 t + t - t + t - t + t - t + t
In[7]:=	Conway[TorusKnot[21, 2]][z]
Out[7]=	2 4 6 8 10 12 1 + 55 z + 495 z + 1716 z + 3003 z + 3003 z + 1820 z + 14 16 18 20 680 z + 153 z + 19 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[TorusKnot[21, 2]], KnotSignature[TorusKnot[21, 2]]}
Out[9]=	{21, 20}
In[10]:=	J=Jones[TorusKnot[21, 2]][q]
Out[10]=	10 12 13 14 15 16 17 18 19 20 21 22 q + q - q + q - q + q - q + q - q + q - q + q - 23 24 25 26 27 28 29 30 31 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[21, 2]][q]
Out[12]=	NotAvailable
In[13]:=	Kauffman[TorusKnot[21, 2]][a, z]
Out[13]=	NotAvailable
In[14]:=	{Vassiliev[2][TorusKnot[21, 2]], Vassiliev[3][TorusKnot[21, 2]]}
Out[14]=	{0, 385}
In[15]:=	Kh[TorusKnot[21, 2]][q, t]
Out[15]=	19 21 23 2 27 3 27 4 31 5 31 6 35 7 q + q + q t + q t + q t + q t + q t + q t + 35 8 39 9 39 10 43 11 43 12 47 13 47 14 q t + q t + q t + q t + q t + q t + q t + 51 15 51 16 55 17 55 18 59 19 59 20 63 21 q t + q t + q t + q t + q t + q t + q t

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

T(21,2)

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Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

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