T(19,2)

T(17,2)

T(10,3)

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	T(19,2) Quick Notes

T(19,2) Further Notes and Views

Knot presentations

Planar diagram presentation	X13,33,14,32 X33,15,34,14 X15,35,16,34 X35,17,36,16 X17,37,18,36 X37,19,38,18 X19,1,20,38 X1,21,2,20 X21,3,22,2 X3,23,4,22 X23,5,24,4 X5,25,6,24 X25,7,26,6 X7,27,8,26 X27,9,28,8 X9,29,10,28 X29,11,30,10 X11,31,12,30 X31,13,32,12
Gauss code	-8, 9, -10, 11, -12, 13, -14, 15, -16, 17, -18, 19, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 1, -2, 3, -4, 5, -6, 7
Dowker-Thistlethwaite code	20 22 24 26 28 30 32 34 36 38 2 4 6 8 10 12 14 16 18
Conway Notation	Data:T(19,2)/Conway Notation

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 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} }[/math]
Conway polynomial	[math]\displaystyle{ z^{18}+17 z^{16}+120 z^{14}+455 z^{12}+1001 z^{10}+1287 z^8+924 z^6+330 z^4+45 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 19, 18 }
Jones polynomial	[math]\displaystyle{ -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^{11}+q^9 }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^{18} a^{-18} +18 z^{16} a^{-18} -z^{16} a^{-20} +136 z^{14} a^{-18} -16 z^{14} a^{-20} +560 z^{12} a^{-18} -105 z^{12} a^{-20} +1365 z^{10} a^{-18} -364 z^{10} a^{-20} +2002 z^8 a^{-18} -715 z^8 a^{-20} +1716 z^6 a^{-18} -792 z^6 a^{-20} +792 z^4 a^{-18} -462 z^4 a^{-20} +165 z^2 a^{-18} -120 z^2 a^{-20} +10 a^{-18} -9 a^{-20} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{18}a^{-18}+z^{18}a^{-20}+z^{17}a^{-19}+z^{17}a^{-21}-18z^{16}a^{-18}-17z^{16}a^{-20}+z^{16}a^{-22}-16z^{15}a^{-19}-15z^{15}a^{-21}+z^{15}a^{-23}+136z^{14}a^{-18}+121z^{14}a^{-20}-14z^{14}a^{-22}+z^{14}a^{-24}+105z^{13}a^{-19}+91z^{13}a^{-21}-13z^{13}a^{-23}+z^{13}a^{-25}-560z^{12}a^{-18}-469z^{12}a^{-20}+78z^{12}a^{-22}-12z^{12}a^{-24}+z^{12}a^{-26}-364z^{11}a^{-19}-286z^{11}a^{-21}+66z^{11}a^{-23}-11z^{11}a^{-25}+z^{11}a^{-27}+1365z^{10}a^{-18}+1079z^{10}a^{-20}-220z^{10}a^{-22}+55z^{10}a^{-24}-10z^{10}a^{-26}+z^{10}a^{-28}+715z^9a^{-19}+495z^9a^{-21}-165z^9a^{-23}+45z^9a^{-25}-9z^9a^{-27}+z^9a^{-29}-2002z^8a^{-18}-1507z^8a^{-20}+330z^8a^{-22}-120z^8a^{-24}+36z^8a^{-26}-8z^8a^{-28}+z^8a^{-30}-792z^7a^{-19}-462z^7a^{-21}+210z^7a^{-23}-84z^7a^{-25}+28z^7a^{-27}-7z^7a^{-29}+z^7a^{-31}+1716z^6a^{-18}+1254z^6a^{-20}-252z^6a^{-22}+126z^6a^{-24}-56z^6a^{-26}+21z^6a^{-28}-6z^6a^{-30}+z^6a^{-32}+462z^5a^{-19}+210z^5a^{-21}-126z^5a^{-23}+70z^5a^{-25}-35z^5a^{-27}+15z^5a^{-29}-5z^5a^{-31}+z^5a^{-33}-792z^4a^{-18}-582z^4a^{-20}+84z^4a^{-22}-56z^4a^{-24}+35z^4a^{-26}-20z^4a^{-28}+10z^4a^{-30}-4z^4a^{-32}+z^4a^{-34}-120z^3a^{-19}-36z^3a^{-21}+28z^3a^{-23}-21z^3a^{-25}+15z^3a^{-27}-10z^3a^{-29}+6z^3a^{-31}-3z^3a^{-33}+z^3a^{-35}+165z^2a^{-18}+129z^2a^{-20}-8z^2a^{-22}+7z^2a^{-24}-6z^2a^{-26}+5z^2a^{-28}-4z^2a^{-30}+3z^2a^{-32}-2z^2a^{-34}+z^2a^{-36}+9za^{-19}+za^{-21}-za^{-23}+za^{-25}-za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-10a^{-18}-9a^{-20} }[/math]
The A2 invariant	Data:T(19,2)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(19,2)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(19,2) 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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["T(19,2)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 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} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^{18}+17 z^{16}+120 z^{14}+455 z^{12}+1001 z^{10}+1287 z^8+924 z^6+330 z^4+45 z^2+1 }[/math]

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]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 19, 18 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -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^{11}+q^9 }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^{18} a^{-18} +18 z^{16} a^{-18} -z^{16} a^{-20} +136 z^{14} a^{-18} -16 z^{14} a^{-20} +560 z^{12} a^{-18} -105 z^{12} a^{-20} +1365 z^{10} a^{-18} -364 z^{10} a^{-20} +2002 z^8 a^{-18} -715 z^8 a^{-20} +1716 z^6 a^{-18} -792 z^6 a^{-20} +792 z^4 a^{-18} -462 z^4 a^{-20} +165 z^2 a^{-18} -120 z^2 a^{-20} +10 a^{-18} -9 a^{-20} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^{18}a^{-18}+z^{18}a^{-20}+z^{17}a^{-19}+z^{17}a^{-21}-18z^{16}a^{-18}-17z^{16}a^{-20}+z^{16}a^{-22}-16z^{15}a^{-19}-15z^{15}a^{-21}+z^{15}a^{-23}+136z^{14}a^{-18}+121z^{14}a^{-20}-14z^{14}a^{-22}+z^{14}a^{-24}+105z^{13}a^{-19}+91z^{13}a^{-21}-13z^{13}a^{-23}+z^{13}a^{-25}-560z^{12}a^{-18}-469z^{12}a^{-20}+78z^{12}a^{-22}-12z^{12}a^{-24}+z^{12}a^{-26}-364z^{11}a^{-19}-286z^{11}a^{-21}+66z^{11}a^{-23}-11z^{11}a^{-25}+z^{11}a^{-27}+1365z^{10}a^{-18}+1079z^{10}a^{-20}-220z^{10}a^{-22}+55z^{10}a^{-24}-10z^{10}a^{-26}+z^{10}a^{-28}+715z^9a^{-19}+495z^9a^{-21}-165z^9a^{-23}+45z^9a^{-25}-9z^9a^{-27}+z^9a^{-29}-2002z^8a^{-18}-1507z^8a^{-20}+330z^8a^{-22}-120z^8a^{-24}+36z^8a^{-26}-8z^8a^{-28}+z^8a^{-30}-792z^7a^{-19}-462z^7a^{-21}+210z^7a^{-23}-84z^7a^{-25}+28z^7a^{-27}-7z^7a^{-29}+z^7a^{-31}+1716z^6a^{-18}+1254z^6a^{-20}-252z^6a^{-22}+126z^6a^{-24}-56z^6a^{-26}+21z^6a^{-28}-6z^6a^{-30}+z^6a^{-32}+462z^5a^{-19}+210z^5a^{-21}-126z^5a^{-23}+70z^5a^{-25}-35z^5a^{-27}+15z^5a^{-29}-5z^5a^{-31}+z^5a^{-33}-792z^4a^{-18}-582z^4a^{-20}+84z^4a^{-22}-56z^4a^{-24}+35z^4a^{-26}-20z^4a^{-28}+10z^4a^{-30}-4z^4a^{-32}+z^4a^{-34}-120z^3a^{-19}-36z^3a^{-21}+28z^3a^{-23}-21z^3a^{-25}+15z^3a^{-27}-10z^3a^{-29}+6z^3a^{-31}-3z^3a^{-33}+z^3a^{-35}+165z^2a^{-18}+129z^2a^{-20}-8z^2a^{-22}+7z^2a^{-24}-6z^2a^{-26}+5z^2a^{-28}-4z^2a^{-30}+3z^2a^{-32}-2z^2a^{-34}+z^2a^{-36}+9za^{-19}+za^{-21}-za^{-23}+za^{-25}-za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-10a^{-18}-9a^{-20} }[/math]

Vassiliev invariants

V2 and V3:

(45, 285)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 Data:T(19,2)/V 2,1 Data:T(19,2)/V 3,1 Data:T(19,2)/V 4,1 Data:T(19,2)/V 4,2 Data:T(19,2)/V 4,3 Data:T(19,2)/V 5,1 Data:T(19,2)/V 5,2 Data:T(19,2)/V 5,3 Data:T(19,2)/V 5,4 Data:T(19,2)/V 6,1 Data:T(19,2)/V 6,2 Data:T(19,2)/V 6,3 Data:T(19,2)/V 6,4 Data:T(19,2)/V 6,5 Data:T(19,2)/V 6,6 Data:T(19,2)/V 6,7 Data:T(19,2)/V 6,8 Data:T(19,2)/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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]18 is the signature of T(19,2). 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 χ 57                                       1 -1 55                                         0 53                                   1 1   0 51                                         0 49                               1 1       0 47                                         0 45                           1 1           0 43                                         0 41                       1 1               0 39                                         0 37                   1 1                   0 35                                         0 33               1 1                       0 31                                         0 29           1 1                           0 27                                         0 25       1 1                               0 23                                         0 21     1                                   1 19 1                                       1 17 1                                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=17 }[/math] [math]\displaystyle{ i=19 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=9 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=10 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=11 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=12 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=13 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=14 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=15 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=16 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=17 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=18 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=19 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

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[19, 2]]
Out[2]=	19
In[3]:=	PD[TorusKnot[19, 2]]
Out[3]=	PD[X[13, 33, 14, 32], X[33, 15, 34, 14], X[15, 35, 16, 34], X[35, 17, 36, 16], X[17, 37, 18, 36], X[37, 19, 38, 18], X[19, 1, 20, 38], X[1, 21, 2, 20], X[21, 3, 22, 2], X[3, 23, 4, 22], X[23, 5, 24, 4], X[5, 25, 6, 24], X[25, 7, 26, 6], X[7, 27, 8, 26], X[27, 9, 28, 8], X[9, 29, 10, 28], X[29, 11, 30, 10], X[11, 31, 12, 30], X[31, 13, 32, 12]]
In[4]:=	GaussCode[TorusKnot[19, 2]]
Out[4]=	GaussCode[-8, 9, -10, 11, -12, 13, -14, 15, -16, 17, -18, 19, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 1, -2, 3, -4, 5, -6, 7]
In[5]:=	BR[TorusKnot[19, 2]]
Out[5]=	BR[2, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}]
In[6]:=	alex = Alexander[TorusKnot[19, 2]][t]
Out[6]=	-9 -8 -7 -6 -1 + Alternating - Alternating + Alternating - Alternating + -5 -4 -3 -2 Alternating - Alternating + Alternating - Alternating + 1 2 3 ----------- + Alternating - Alternating + Alternating - Alternating 4 5 6 7 Alternating + Alternating - Alternating + Alternating - 8 9 Alternating + Alternating
In[7]:=	Conway[TorusKnot[19, 2]][z]
Out[7]=	2 4 6 8 10 12 14 1 + 45 z + 330 z + 924 z + 1287 z + 1001 z + 455 z + 120 z + 16 18 17 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[TorusKnot[19, 2]], KnotSignature[TorusKnot[19, 2]]}
Out[9]=	{19, 18}
In[10]:=	J=Jones[TorusKnot[19, 2]][q]
Out[10]=	9 11 12 13 14 15 16 17 18 19 20 21 q + q - q + q - q + q - q + q - q + q - q + q - 22 23 24 25 26 27 28 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[19, 2]][q]
Out[12]=	NotAvailable
In[13]:=	Kauffman[TorusKnot[19, 2]][a, z]
Out[13]=	NotAvailable
In[14]:=	{Vassiliev[2][TorusKnot[19, 2]], Vassiliev[3][TorusKnot[19, 2]]}
Out[14]=	{0, 285}
In[15]:=	Kh[TorusKnot[19, 2]][q, t]
Out[15]=	17 19 2 21 3 25 4 25 q + q + Alternating q + Alternating q + Alternating q + 5 29 6 29 7 33 Alternating q + Alternating q + Alternating q + 8 33 9 37 10 37 Alternating q + Alternating q + Alternating q + 11 41 12 41 13 45 Alternating q + Alternating q + Alternating q + 14 45 15 49 16 49 Alternating q + Alternating q + Alternating q + 17 53 18 53 19 57 Alternating q + Alternating q + Alternating q