T(25,2)

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Knot presentations

Planar diagram presentation	X_23,49,24,48 X_49,25,50,24 X_25,1,26,50 X_1,27,2,26 X_27,3,28,2 X_3,29,4,28 X_29,5,30,4 X_5,31,6,30 X_31,7,32,6 X_7,33,8,32 X_33,9,34,8 X_9,35,10,34 X_35,11,36,10 X_11,37,12,36 X_37,13,38,12 X_13,39,14,38 X_39,15,40,14 X_15,41,16,40 X_41,17,42,16 X_17,43,18,42 X_43,19,44,18 X_19,45,20,44 X_45,21,46,20 X_21,47,22,46 X_47,23,48,22
Gauss code	-4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15, -16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 24, -25, 1, -2, 3
Dowker-Thistlethwaite code	26 28 30 32 34 36 38 40 42 44 46 48 50 2 4 6 8 10 12 14 16 18 20 22 24
Conway Notation	Data:T(25,2)/Conway Notation

Polynomial invariants

Alexander polynomial	$t^{12}-t^{11}+t^{10}-t^{9}+t^{8}-t^{7}+t^{6}-t^{5}+t^{4}-t^{3}+t^{2}-t+1-t^{-1}+t^{-2}-t^{-3}+t^{-4}-t^{-5}+t^{-6}-t^{-7}+t^{-8}-t^{-9}+t^{-10}-t^{-11}+t^{-12}$
Conway polynomial	$z^{24}+23z^{22}+231z^{20}+1330z^{18}+4845z^{16}+11628z^{14}+18564z^{12}+19448z^{10}+12870z^{8}+5005z^{6}+1001z^{4}+78z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 25, 24 }
Jones polynomial	$-q^{37}+q^{36}-q^{35}+q^{34}-q^{33}+q^{32}-q^{31}+q^{30}-q^{29}+q^{28}-q^{27}+q^{26}-q^{25}+q^{24}-q^{23}+q^{22}-q^{21}+q^{20}-q^{19}+q^{18}-q^{17}+q^{16}-q^{15}+q^{14}+q^{12}$
HOMFLY-PT polynomial (db, data sources)	$z^{24}a^{-24}-24z^{22}a^{-24}-z^{22}a^{-26}+253z^{20}a^{-24}+22z^{20}a^{-26}-1540z^{18}a^{-24}-210z^{18}a^{-26}+5985z^{16}a^{-24}+1140z^{16}a^{-26}-15504z^{14}a^{-24}-3876z^{14}a^{-26}+27132z^{12}a^{-24}+8568z^{12}a^{-26}-31824z^{10}a^{-24}-12376z^{10}a^{-26}+24310z^{8}a^{-24}+11440z^{8}a^{-26}-11440z^{6}a^{-24}-6435z^{6}a^{-26}+3003z^{4}a^{-24}+2002z^{4}a^{-26}-364z^{2}a^{-24}-286z^{2}a^{-26}+13a^{-24}+12a^{-26}$
Kauffman polynomial (db, data sources)	$z^{24}a^{-24}+z^{24}a^{-26}+z^{23}a^{-25}+z^{23}a^{-27}-24z^{22}a^{-24}-23z^{22}a^{-26}+z^{22}a^{-28}-22z^{21}a^{-25}-21z^{21}a^{-27}+z^{21}a^{-29}+253z^{20}a^{-24}+232z^{20}a^{-26}-20z^{20}a^{-28}+z^{20}a^{-30}+210z^{19}a^{-25}+190z^{19}a^{-27}-19z^{19}a^{-29}+z^{19}a^{-31}-1540z^{18}a^{-24}-1350z^{18}a^{-26}+171z^{18}a^{-28}-18z^{18}a^{-30}+z^{18}a^{-32}-1140z^{17}a^{-25}-969z^{17}a^{-27}+153z^{17}a^{-29}-17z^{17}a^{-31}+z^{17}a^{-33}+5985z^{16}a^{-24}+5016z^{16}a^{-26}-816z^{16}a^{-28}+136z^{16}a^{-30}-16z^{16}a^{-32}+z^{16}a^{-34}+3876z^{15}a^{-25}+3060z^{15}a^{-27}-680z^{15}a^{-29}+120z^{15}a^{-31}-15z^{15}a^{-33}+z^{15}a^{-35}-15504z^{14}a^{-24}-12444z^{14}a^{-26}+2380z^{14}a^{-28}-560z^{14}a^{-30}+105z^{14}a^{-32}-14z^{14}a^{-34}+z^{14}a^{-36}-8568z^{13}a^{-25}-6188z^{13}a^{-27}+1820z^{13}a^{-29}-455z^{13}a^{-31}+91z^{13}a^{-33}-13z^{13}a^{-35}+z^{13}a^{-37}+27132z^{12}a^{-24}+20944z^{12}a^{-26}-4368z^{12}a^{-28}+1365z^{12}a^{-30}-364z^{12}a^{-32}+78z^{12}a^{-34}-12z^{12}a^{-36}+z^{12}a^{-38}+12376z^{11}a^{-25}+8008z^{11}a^{-27}-3003z^{11}a^{-29}+1001z^{11}a^{-31}-286z^{11}a^{-33}+66z^{11}a^{-35}-11z^{11}a^{-37}+z^{11}a^{-39}-31824z^{10}a^{-24}-23816z^{10}a^{-26}+5005z^{10}a^{-28}-2002z^{10}a^{-30}+715z^{10}a^{-32}-220z^{10}a^{-34}+55z^{10}a^{-36}-10z^{10}a^{-38}+z^{10}a^{-40}-11440z^{9}a^{-25}-6435z^{9}a^{-27}+3003z^{9}a^{-29}-1287z^{9}a^{-31}+495z^{9}a^{-33}-165z^{9}a^{-35}+45z^{9}a^{-37}-9z^{9}a^{-39}+z^{9}a^{-41}+24310z^{8}a^{-24}+17875z^{8}a^{-26}-3432z^{8}a^{-28}+1716z^{8}a^{-30}-792z^{8}a^{-32}+330z^{8}a^{-34}-120z^{8}a^{-36}+36z^{8}a^{-38}-8z^{8}a^{-40}+z^{8}a^{-42}+6435z^{7}a^{-25}+3003z^{7}a^{-27}-1716z^{7}a^{-29}+924z^{7}a^{-31}-462z^{7}a^{-33}+210z^{7}a^{-35}-84z^{7}a^{-37}+28z^{7}a^{-39}-7z^{7}a^{-41}+z^{7}a^{-43}-11440z^{6}a^{-24}-8437z^{6}a^{-26}+1287z^{6}a^{-28}-792z^{6}a^{-30}+462z^{6}a^{-32}-252z^{6}a^{-34}+126z^{6}a^{-36}-56z^{6}a^{-38}+21z^{6}a^{-40}-6z^{6}a^{-42}+z^{6}a^{-44}-2002z^{5}a^{-25}-715z^{5}a^{-27}+495z^{5}a^{-29}-330z^{5}a^{-31}+210z^{5}a^{-33}-126z^{5}a^{-35}+70z^{5}a^{-37}-35z^{5}a^{-39}+15z^{5}a^{-41}-5z^{5}a^{-43}+z^{5}a^{-45}+3003z^{4}a^{-24}+2288z^{4}a^{-26}-220z^{4}a^{-28}+165z^{4}a^{-30}-120z^{4}a^{-32}+84z^{4}a^{-34}-56z^{4}a^{-36}+35z^{4}a^{-38}-20z^{4}a^{-40}+10z^{4}a^{-42}-4z^{4}a^{-44}+z^{4}a^{-46}+286z^{3}a^{-25}+66z^{3}a^{-27}-55z^{3}a^{-29}+45z^{3}a^{-31}-36z^{3}a^{-33}+28z^{3}a^{-35}-21z^{3}a^{-37}+15z^{3}a^{-39}-10z^{3}a^{-41}+6z^{3}a^{-43}-3z^{3}a^{-45}+z^{3}a^{-47}-364z^{2}a^{-24}-298z^{2}a^{-26}+11z^{2}a^{-28}-10z^{2}a^{-30}+9z^{2}a^{-32}-8z^{2}a^{-34}+7z^{2}a^{-36}-6z^{2}a^{-38}+5z^{2}a^{-40}-4z^{2}a^{-42}+3z^{2}a^{-44}-2z^{2}a^{-46}+z^{2}a^{-48}-12za^{-25}-za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-za^{-39}+za^{-41}-za^{-43}+za^{-45}-za^{-47}+za^{-49}+13a^{-24}+12a^{-26}$
The A2 invariant	Data:T(25,2)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(25,2)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(25,2) 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(25,2)"];

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

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

Out[4]= $t^{12}-t^{11}+t^{10}-t^{9}+t^{8}-t^{7}+t^{6}-t^{5}+t^{4}-t^{3}+t^{2}-t+1-t^{-1}+t^{-2}-t^{-3}+t^{-4}-t^{-5}+t^{-6}-t^{-7}+t^{-8}-t^{-9}+t^{-10}-t^{-11}+t^{-12}$

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

Out[5]= $z^{24}+23z^{22}+231z^{20}+1330z^{18}+4845z^{16}+11628z^{14}+18564z^{12}+19448z^{10}+12870z^{8}+5005z^{6}+1001z^{4}+78z^{2}+1$

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]= { 25, 24 }

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

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

Out[8]= $-q^{37}+q^{36}-q^{35}+q^{34}-q^{33}+q^{32}-q^{31}+q^{30}-q^{29}+q^{28}-q^{27}+q^{26}-q^{25}+q^{24}-q^{23}+q^{22}-q^{21}+q^{20}-q^{19}+q^{18}-q^{17}+q^{16}-q^{15}+q^{14}+q^{12}$

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

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

Out[9]= $z^{24}a^{-24}-24z^{22}a^{-24}-z^{22}a^{-26}+253z^{20}a^{-24}+22z^{20}a^{-26}-1540z^{18}a^{-24}-210z^{18}a^{-26}+5985z^{16}a^{-24}+1140z^{16}a^{-26}-15504z^{14}a^{-24}-3876z^{14}a^{-26}+27132z^{12}a^{-24}+8568z^{12}a^{-26}-31824z^{10}a^{-24}-12376z^{10}a^{-26}+24310z^{8}a^{-24}+11440z^{8}a^{-26}-11440z^{6}a^{-24}-6435z^{6}a^{-26}+3003z^{4}a^{-24}+2002z^{4}a^{-26}-364z^{2}a^{-24}-286z^{2}a^{-26}+13a^{-24}+12a^{-26}$

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

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

Out[10]= $z^{24}a^{-24}+z^{24}a^{-26}+z^{23}a^{-25}+z^{23}a^{-27}-24z^{22}a^{-24}-23z^{22}a^{-26}+z^{22}a^{-28}-22z^{21}a^{-25}-21z^{21}a^{-27}+z^{21}a^{-29}+253z^{20}a^{-24}+232z^{20}a^{-26}-20z^{20}a^{-28}+z^{20}a^{-30}+210z^{19}a^{-25}+190z^{19}a^{-27}-19z^{19}a^{-29}+z^{19}a^{-31}-1540z^{18}a^{-24}-1350z^{18}a^{-26}+171z^{18}a^{-28}-18z^{18}a^{-30}+z^{18}a^{-32}-1140z^{17}a^{-25}-969z^{17}a^{-27}+153z^{17}a^{-29}-17z^{17}a^{-31}+z^{17}a^{-33}+5985z^{16}a^{-24}+5016z^{16}a^{-26}-816z^{16}a^{-28}+136z^{16}a^{-30}-16z^{16}a^{-32}+z^{16}a^{-34}+3876z^{15}a^{-25}+3060z^{15}a^{-27}-680z^{15}a^{-29}+120z^{15}a^{-31}-15z^{15}a^{-33}+z^{15}a^{-35}-15504z^{14}a^{-24}-12444z^{14}a^{-26}+2380z^{14}a^{-28}-560z^{14}a^{-30}+105z^{14}a^{-32}-14z^{14}a^{-34}+z^{14}a^{-36}-8568z^{13}a^{-25}-6188z^{13}a^{-27}+1820z^{13}a^{-29}-455z^{13}a^{-31}+91z^{13}a^{-33}-13z^{13}a^{-35}+z^{13}a^{-37}+27132z^{12}a^{-24}+20944z^{12}a^{-26}-4368z^{12}a^{-28}+1365z^{12}a^{-30}-364z^{12}a^{-32}+78z^{12}a^{-34}-12z^{12}a^{-36}+z^{12}a^{-38}+12376z^{11}a^{-25}+8008z^{11}a^{-27}-3003z^{11}a^{-29}+1001z^{11}a^{-31}-286z^{11}a^{-33}+66z^{11}a^{-35}-11z^{11}a^{-37}+z^{11}a^{-39}-31824z^{10}a^{-24}-23816z^{10}a^{-26}+5005z^{10}a^{-28}-2002z^{10}a^{-30}+715z^{10}a^{-32}-220z^{10}a^{-34}+55z^{10}a^{-36}-10z^{10}a^{-38}+z^{10}a^{-40}-11440z^{9}a^{-25}-6435z^{9}a^{-27}+3003z^{9}a^{-29}-1287z^{9}a^{-31}+495z^{9}a^{-33}-165z^{9}a^{-35}+45z^{9}a^{-37}-9z^{9}a^{-39}+z^{9}a^{-41}+24310z^{8}a^{-24}+17875z^{8}a^{-26}-3432z^{8}a^{-28}+1716z^{8}a^{-30}-792z^{8}a^{-32}+330z^{8}a^{-34}-120z^{8}a^{-36}+36z^{8}a^{-38}-8z^{8}a^{-40}+z^{8}a^{-42}+6435z^{7}a^{-25}+3003z^{7}a^{-27}-1716z^{7}a^{-29}+924z^{7}a^{-31}-462z^{7}a^{-33}+210z^{7}a^{-35}-84z^{7}a^{-37}+28z^{7}a^{-39}-7z^{7}a^{-41}+z^{7}a^{-43}-11440z^{6}a^{-24}-8437z^{6}a^{-26}+1287z^{6}a^{-28}-792z^{6}a^{-30}+462z^{6}a^{-32}-252z^{6}a^{-34}+126z^{6}a^{-36}-56z^{6}a^{-38}+21z^{6}a^{-40}-6z^{6}a^{-42}+z^{6}a^{-44}-2002z^{5}a^{-25}-715z^{5}a^{-27}+495z^{5}a^{-29}-330z^{5}a^{-31}+210z^{5}a^{-33}-126z^{5}a^{-35}+70z^{5}a^{-37}-35z^{5}a^{-39}+15z^{5}a^{-41}-5z^{5}a^{-43}+z^{5}a^{-45}+3003z^{4}a^{-24}+2288z^{4}a^{-26}-220z^{4}a^{-28}+165z^{4}a^{-30}-120z^{4}a^{-32}+84z^{4}a^{-34}-56z^{4}a^{-36}+35z^{4}a^{-38}-20z^{4}a^{-40}+10z^{4}a^{-42}-4z^{4}a^{-44}+z^{4}a^{-46}+286z^{3}a^{-25}+66z^{3}a^{-27}-55z^{3}a^{-29}+45z^{3}a^{-31}-36z^{3}a^{-33}+28z^{3}a^{-35}-21z^{3}a^{-37}+15z^{3}a^{-39}-10z^{3}a^{-41}+6z^{3}a^{-43}-3z^{3}a^{-45}+z^{3}a^{-47}-364z^{2}a^{-24}-298z^{2}a^{-26}+11z^{2}a^{-28}-10z^{2}a^{-30}+9z^{2}a^{-32}-8z^{2}a^{-34}+7z^{2}a^{-36}-6z^{2}a^{-38}+5z^{2}a^{-40}-4z^{2}a^{-42}+3z^{2}a^{-44}-2z^{2}a^{-46}+z^{2}a^{-48}-12za^{-25}-za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-za^{-39}+za^{-41}-za^{-43}+za^{-45}-za^{-47}+za^{-49}+13a^{-24}+12a^{-26}$

Vassiliev invariants

V₂ and V₃:

(78, 650)

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 $s=$ 24 is the signature of T(25,2). 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

24

25

χ

75

1

-1

73

0

71

1

0

69

0

67

1

0

65

0

63

1

0

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

25

1

23

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[25, 2]]
Out[2]=	25
In[3]:=	PD[TorusKnot[25, 2]]
Out[3]=	PD[X[23, 49, 24, 48], X[49, 25, 50, 24], X[25, 1, 26, 50], X[1, 27, 2, 26], X[27, 3, 28, 2], X[3, 29, 4, 28], X[29, 5, 30, 4], X[5, 31, 6, 30], X[31, 7, 32, 6], X[7, 33, 8, 32], X[33, 9, 34, 8], X[9, 35, 10, 34], X[35, 11, 36, 10], X[11, 37, 12, 36], X[37, 13, 38, 12], X[13, 39, 14, 38], X[39, 15, 40, 14], X[15, 41, 16, 40], X[41, 17, 42, 16], X[17, 43, 18, 42], X[43, 19, 44, 18], X[19, 45, 20, 44], X[45, 21, 46, 20], X[21, 47, 22, 46], X[47, 23, 48, 22]]
In[4]:=	GaussCode[TorusKnot[25, 2]]
Out[4]=	GaussCode[-4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15, -16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 24, -25, 1, -2, 3]
In[5]:=	BR[TorusKnot[25, 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, 1, 1}]
In[6]:=	alex = Alexander[TorusKnot[25, 2]][t]
Out[6]=	-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 1 + t - t + t - t + t - t + t - t + t - t + -2 1 2 3 4 5 6 7 8 9 10 11 12 t - - - t + t - t + t - t + t - t + t - t + t - t + t t
In[7]:=	Conway[TorusKnot[25, 2]][z]
Out[7]=	2 4 6 8 10 12 1 + 78 z + 1001 z + 5005 z + 12870 z + 19448 z + 18564 z + 14 16 18 20 22 24 11628 z + 4845 z + 1330 z + 231 z + 23 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[TorusKnot[25, 2]], KnotSignature[TorusKnot[25, 2]]}
Out[9]=	{25, 24}
In[10]:=	J=Jones[TorusKnot[25, 2]][q]
Out[10]=	12 14 15 16 17 18 19 20 21 22 23 24 q + q - q + q - q + q - q + q - q + q - q + q - 25 26 27 28 29 30 31 32 33 34 35 q + q - q + q - q + q - q + q - q + q - q + 36 37 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[TorusKnot[25, 2]][q]
Out[12]=	NotAvailable
In[13]:=	Kauffman[TorusKnot[25, 2]][a, z]
Out[13]=	NotAvailable
In[14]:=	{Vassiliev[2][TorusKnot[25, 2]], Vassiliev[3][TorusKnot[25, 2]]}
Out[14]=	{0, 650}
In[15]:=	Kh[TorusKnot[25, 2]][q, t]
Out[15]=	23 25 27 2 31 3 31 4 35 5 35 6 39 7 q + q + q t + q t + q t + q t + q t + q t + 39 8 43 9 43 10 47 11 47 12 51 13 51 14 q t + q t + q t + q t + q t + q t + q t + 55 15 55 16 59 17 59 18 63 19 63 20 67 21 q t + q t + q t + q t + q t + q t + q t + 67 22 71 23 71 24 75 25 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(25,2)

Contents

Knot presentations

Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

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