T(15,2)

T(5,4)

T(8,3)

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	T(15,2) Quick Notes

T(15,2) Further Notes and Views

Knot presentations

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

Polynomial invariants

Alexander polynomial	$t^{7}-t^{6}+t^{5}-t^{4}+t^{3}-t^{2}+t-1+t^{-1}-t^{-2}+t^{-3}-t^{-4}+t^{-5}-t^{-6}+t^{-7}$
Conway polynomial	$z^{14}+13z^{12}+66z^{10}+165z^{8}+210z^{6}+126z^{4}+28z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 15, 14 }
Jones polynomial	$-q^{22}+q^{21}-q^{20}+q^{19}-q^{18}+q^{17}-q^{16}+q^{15}-q^{14}+q^{13}-q^{12}+q^{11}-q^{10}+q^{9}+q^{7}$
HOMFLY-PT polynomial (db, data sources)	$z^{14}a^{-14}+14z^{12}a^{-14}-z^{12}a^{-16}+78z^{10}a^{-14}-12z^{10}a^{-16}+220z^{8}a^{-14}-55z^{8}a^{-16}+330z^{6}a^{-14}-120z^{6}a^{-16}+252z^{4}a^{-14}-126z^{4}a^{-16}+84z^{2}a^{-14}-56z^{2}a^{-16}+8a^{-14}-7a^{-16}$
Kauffman polynomial (db, data sources)	$z^{14}a^{-14}+z^{14}a^{-16}+z^{13}a^{-15}+z^{13}a^{-17}-14z^{12}a^{-14}-13z^{12}a^{-16}+z^{12}a^{-18}-12z^{11}a^{-15}-11z^{11}a^{-17}+z^{11}a^{-19}+78z^{10}a^{-14}+67z^{10}a^{-16}-10z^{10}a^{-18}+z^{10}a^{-20}+55z^{9}a^{-15}+45z^{9}a^{-17}-9z^{9}a^{-19}+z^{9}a^{-21}-220z^{8}a^{-14}-175z^{8}a^{-16}+36z^{8}a^{-18}-8z^{8}a^{-20}+z^{8}a^{-22}-120z^{7}a^{-15}-84z^{7}a^{-17}+28z^{7}a^{-19}-7z^{7}a^{-21}+z^{7}a^{-23}+330z^{6}a^{-14}+246z^{6}a^{-16}-56z^{6}a^{-18}+21z^{6}a^{-20}-6z^{6}a^{-22}+z^{6}a^{-24}+126z^{5}a^{-15}+70z^{5}a^{-17}-35z^{5}a^{-19}+15z^{5}a^{-21}-5z^{5}a^{-23}+z^{5}a^{-25}-252z^{4}a^{-14}-182z^{4}a^{-16}+35z^{4}a^{-18}-20z^{4}a^{-20}+10z^{4}a^{-22}-4z^{4}a^{-24}+z^{4}a^{-26}-56z^{3}a^{-15}-21z^{3}a^{-17}+15z^{3}a^{-19}-10z^{3}a^{-21}+6z^{3}a^{-23}-3z^{3}a^{-25}+z^{3}a^{-27}+84z^{2}a^{-14}+63z^{2}a^{-16}-6z^{2}a^{-18}+5z^{2}a^{-20}-4z^{2}a^{-22}+3z^{2}a^{-24}-2z^{2}a^{-26}+z^{2}a^{-28}+7za^{-15}+za^{-17}-za^{-19}+za^{-21}-za^{-23}+za^{-25}-za^{-27}+za^{-29}-8a^{-14}-7a^{-16}$
The A2 invariant	Data:T(15,2)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(15,2)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(15,2) 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(15,2)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{7}-t^{6}+t^{5}-t^{4}+t^{3}-t^{2}+t-1+t^{-1}-t^{-2}+t^{-3}-t^{-4}+t^{-5}-t^{-6}+t^{-7}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{14}+13z^{12}+66z^{10}+165z^{8}+210z^{6}+126z^{4}+28z^{2}+1$

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]=

{ 15, 14 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{22}+q^{21}-q^{20}+q^{19}-q^{18}+q^{17}-q^{16}+q^{15}-q^{14}+q^{13}-q^{12}+q^{11}-q^{10}+q^{9}+q^{7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{14}a^{-14}+14z^{12}a^{-14}-z^{12}a^{-16}+78z^{10}a^{-14}-12z^{10}a^{-16}+220z^{8}a^{-14}-55z^{8}a^{-16}+330z^{6}a^{-14}-120z^{6}a^{-16}+252z^{4}a^{-14}-126z^{4}a^{-16}+84z^{2}a^{-14}-56z^{2}a^{-16}+8a^{-14}-7a^{-16}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{14}a^{-14}+z^{14}a^{-16}+z^{13}a^{-15}+z^{13}a^{-17}-14z^{12}a^{-14}-13z^{12}a^{-16}+z^{12}a^{-18}-12z^{11}a^{-15}-11z^{11}a^{-17}+z^{11}a^{-19}+78z^{10}a^{-14}+67z^{10}a^{-16}-10z^{10}a^{-18}+z^{10}a^{-20}+55z^{9}a^{-15}+45z^{9}a^{-17}-9z^{9}a^{-19}+z^{9}a^{-21}-220z^{8}a^{-14}-175z^{8}a^{-16}+36z^{8}a^{-18}-8z^{8}a^{-20}+z^{8}a^{-22}-120z^{7}a^{-15}-84z^{7}a^{-17}+28z^{7}a^{-19}-7z^{7}a^{-21}+z^{7}a^{-23}+330z^{6}a^{-14}+246z^{6}a^{-16}-56z^{6}a^{-18}+21z^{6}a^{-20}-6z^{6}a^{-22}+z^{6}a^{-24}+126z^{5}a^{-15}+70z^{5}a^{-17}-35z^{5}a^{-19}+15z^{5}a^{-21}-5z^{5}a^{-23}+z^{5}a^{-25}-252z^{4}a^{-14}-182z^{4}a^{-16}+35z^{4}a^{-18}-20z^{4}a^{-20}+10z^{4}a^{-22}-4z^{4}a^{-24}+z^{4}a^{-26}-56z^{3}a^{-15}-21z^{3}a^{-17}+15z^{3}a^{-19}-10z^{3}a^{-21}+6z^{3}a^{-23}-3z^{3}a^{-25}+z^{3}a^{-27}+84z^{2}a^{-14}+63z^{2}a^{-16}-6z^{2}a^{-18}+5z^{2}a^{-20}-4z^{2}a^{-22}+3z^{2}a^{-24}-2z^{2}a^{-26}+z^{2}a^{-28}+7za^{-15}+za^{-17}-za^{-19}+za^{-21}-za^{-23}+za^{-25}-za^{-27}+za^{-29}-8a^{-14}-7a^{-16}$

Vassiliev invariants

V2 and V3:

(28, 140)

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(15,2)/V 2,1 Data:T(15,2)/V 3,1 Data:T(15,2)/V 4,1 Data:T(15,2)/V 4,2 Data:T(15,2)/V 4,3 Data:T(15,2)/V 5,1 Data:T(15,2)/V 5,2 Data:T(15,2)/V 5,3 Data:T(15,2)/V 5,4 Data:T(15,2)/V 6,1 Data:T(15,2)/V 6,2 Data:T(15,2)/V 6,3 Data:T(15,2)/V 6,4 Data:T(15,2)/V 6,5 Data:T(15,2)/V 6,6 Data:T(15,2)/V 6,7 Data:T(15,2)/V 6,8 Data:T(15,2)/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 $t^{r}q^{j}$ 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=$14 is the signature of T(15,2). 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 χ 45                               1 -1 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                       0 19                                 0 17     1                           1 15 1                               1 13 1                               1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=13$ $i=15$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=1$ $r=2$ ${\mathbb {Z} }$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=4$ ${\mathbb {Z} }$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=6$ ${\mathbb {Z} }$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=8$ ${\mathbb {Z} }$ $r=9$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=10$ ${\mathbb {Z} }$ $r=11$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=12$ ${\mathbb {Z} }$ $r=13$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=14$ ${\mathbb {Z} }$ $r=15$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

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

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[TorusKnot[15, 2]]
Out[2]=	15
In[3]:=	PD[TorusKnot[15, 2]]
Out[3]=	PD[X[9, 25, 10, 24], X[25, 11, 26, 10], X[11, 27, 12, 26], X[27, 13, 28, 12], X[13, 29, 14, 28], X[29, 15, 30, 14], X[15, 1, 16, 30], X[1, 17, 2, 16], X[17, 3, 18, 2], X[3, 19, 4, 18], X[19, 5, 20, 4], X[5, 21, 6, 20], X[21, 7, 22, 6], X[7, 23, 8, 22], X[23, 9, 24, 8]]
In[4]:=	GaussCode[TorusKnot[15, 2]]
Out[4]=	GaussCode[-8, 9, -10, 11, -12, 13, -14, 15, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 1, -2, 3, -4, 5, -6, 7]
In[5]:=	BR[TorusKnot[15, 2]]
Out[5]=	BR[2, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}]
In[6]:=	alex = Alexander[TorusKnot[15, 2]][t]
Out[6]=	-7 -6 -5 -4 -1 + Alternating - Alternating + Alternating - Alternating + -3 -2 1 Alternating - Alternating + ----------- + Alternating - Alternating 2 3 4 5 Alternating + Alternating - Alternating + Alternating - 6 7 Alternating + Alternating
In[7]:=	Conway[TorusKnot[15, 2]][z]
Out[7]=	2 4 6 8 10 12 14 1 + 28 z + 126 z + 210 z + 165 z + 66 z + 13 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[TorusKnot[15, 2]], KnotSignature[TorusKnot[15, 2]]}
Out[9]=	{15, 14}
In[10]:=	J=Jones[TorusKnot[15, 2]][q]
Out[10]=	7 9 10 11 12 13 14 15 16 17 18 19 q + q - q + q - q + q - q + q - q + q - q + q - 20 21 22 q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[TorusKnot[15, 2]][q]
Out[12]=	26 28 30 32 34 58 60 62 q + q + 2 q + q + q - q - q - q
In[13]:=	Kauffman[TorusKnot[15, 2]][a, z]
Out[13]=	2 -7 8 z z z z z z z 7 z z --- - --- + --- - --- + --- - --- + --- - --- + --- + --- + --- - 16 14 29 27 25 23 21 19 17 15 28 a a a a a a a a a a a 2 2 2 2 2 2 2 3 3 2 z 3 z 4 z 5 z 6 z 63 z 84 z z 3 z ---- + ---- - ---- + ---- - ---- + ----- + ----- + --- - ---- + 26 24 22 20 18 16 14 27 25 a a a a a a a a a 3 3 3 3 3 4 4 4 4 6 z 10 z 15 z 21 z 56 z z 4 z 10 z 20 z ---- - ----- + ----- - ----- - ----- + --- - ---- + ----- - ----- + 23 21 19 17 15 26 24 22 20 a a a a a a a a a 4 4 4 5 5 5 5 5 35 z 182 z 252 z z 5 z 15 z 35 z 70 z ----- - ------ - ------ + --- - ---- + ----- - ----- + ----- + 18 16 14 25 23 21 19 17 a a a a a a a a 5 6 6 6 6 6 6 7 7 126 z z 6 z 21 z 56 z 246 z 330 z z 7 z ------ + --- - ---- + ----- - ----- + ------ + ------ + --- - ---- + 15 24 22 20 18 16 14 23 21 a a a a a a a a a 7 7 7 8 8 8 8 8 9 28 z 84 z 120 z z 8 z 36 z 175 z 220 z z ----- - ----- - ------ + --- - ---- + ----- - ------ - ------ + --- - 19 17 15 22 20 18 16 14 21 a a a a a a a a a 9 9 9 10 10 10 10 11 9 z 45 z 55 z z 10 z 67 z 78 z z ---- + ----- + ----- + --- - ------ + ------ + ------ + --- - 19 17 15 20 18 16 14 19 a a a a a a a a 11 11 12 12 12 13 13 14 14 11 z 12 z z 13 z 14 z z z z z ------ - ------ + --- - ------ - ------ + --- + --- + --- + --- 17 15 18 16 14 17 15 16 14 a a a a a a a a a
In[14]:=	{Vassiliev[2][TorusKnot[15, 2]], Vassiliev[3][TorusKnot[15, 2]]}
Out[14]=	{0, 140}
In[15]:=	Kh[TorusKnot[15, 2]][q, t]
Out[15]=	13 15 2 17 3 21 4 21 q + q + Alternating q + Alternating q + Alternating q + 5 25 6 25 7 29 Alternating q + Alternating q + Alternating q + 8 29 9 33 10 33 Alternating q + Alternating q + Alternating q + 11 37 12 37 13 41 Alternating q + Alternating q + Alternating q + 14 41 15 45 Alternating q + Alternating q