T(7,3)

T(13,2)

T(5,4)

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	T(7,3) Quick Notes

T(7,3) Further Notes and Views

Knot presentations

Planar diagram presentation	X1,11,2,10 X20,12,21,11 X21,3,22,2 X12,4,13,3 X13,23,14,22 X4,24,5,23 X5,15,6,14 X24,16,25,15 X25,7,26,6 X16,8,17,7 X17,27,18,26 X8,28,9,27 X9,19,10,18 X28,20,1,19
Gauss code	-1, 3, 4, -6, -7, 9, 10, -12, -13, 1, 2, -4, -5, 7, 8, -10, -11, 13, 14, -2, -3, 5, 6, -8, -9, 11, 12, -14
Dowker-Thistlethwaite code	10 -12 14 -16 18 -20 22 -24 26 -28 2 -4 6 -8
Conway Notation	Data:T(7,3)/Conway Notation

Polynomial invariants

Alexander polynomial	$t^{6}-t^{5}+t^{3}-t^{2}+1-t^{-2}+t^{-3}-t^{-5}+t^{-6}$
Conway polynomial	$z^{12}+11z^{10}+44z^{8}+78z^{6}+60z^{4}+16z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 1, 8 }
Jones polynomial	$-q^{14}+q^{8}+q^{6}$
HOMFLY-PT polynomial (db, data sources)	$z^{12}a^{-12}+12z^{10}a^{-12}-z^{10}a^{-14}+55z^{8}a^{-12}-11z^{8}a^{-14}+121z^{6}a^{-12}-44z^{6}a^{-14}+z^{6}a^{-16}+132z^{4}a^{-12}-78z^{4}a^{-14}+6z^{4}a^{-16}+66z^{2}a^{-12}-60z^{2}a^{-14}+10z^{2}a^{-16}+12a^{-12}-16a^{-14}+5a^{-16}$
Kauffman polynomial (db, data sources)	$z^{12}a^{-12}+z^{12}a^{-14}+z^{11}a^{-13}+z^{11}a^{-15}-12z^{10}a^{-12}-12z^{10}a^{-14}-11z^{9}a^{-13}-11z^{9}a^{-15}+55z^{8}a^{-12}+55z^{8}a^{-14}+44z^{7}a^{-13}+44z^{7}a^{-15}-121z^{6}a^{-12}-122z^{6}a^{-14}-z^{6}a^{-16}-78z^{5}a^{-13}-78z^{5}a^{-15}+132z^{4}a^{-12}+138z^{4}a^{-14}+6z^{4}a^{-16}+60z^{3}a^{-13}+60z^{3}a^{-15}-66z^{2}a^{-12}-76z^{2}a^{-14}-10z^{2}a^{-16}-16za^{-13}-16za^{-15}+12a^{-12}+16a^{-14}+5a^{-16}$
The A2 invariant	Data:T(7,3)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(7,3)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(7,3) 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(7,3)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

$z^{12}+11z^{10}+44z^{8}+78z^{6}+60z^{4}+16z^{2}+1$

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]=

{ 1, 8 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{14}+q^{8}+q^{6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{12}a^{-12}+12z^{10}a^{-12}-z^{10}a^{-14}+55z^{8}a^{-12}-11z^{8}a^{-14}+121z^{6}a^{-12}-44z^{6}a^{-14}+z^{6}a^{-16}+132z^{4}a^{-12}-78z^{4}a^{-14}+6z^{4}a^{-16}+66z^{2}a^{-12}-60z^{2}a^{-14}+10z^{2}a^{-16}+12a^{-12}-16a^{-14}+5a^{-16}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{12}a^{-12}+z^{12}a^{-14}+z^{11}a^{-13}+z^{11}a^{-15}-12z^{10}a^{-12}-12z^{10}a^{-14}-11z^{9}a^{-13}-11z^{9}a^{-15}+55z^{8}a^{-12}+55z^{8}a^{-14}+44z^{7}a^{-13}+44z^{7}a^{-15}-121z^{6}a^{-12}-122z^{6}a^{-14}-z^{6}a^{-16}-78z^{5}a^{-13}-78z^{5}a^{-15}+132z^{4}a^{-12}+138z^{4}a^{-14}+6z^{4}a^{-16}+60z^{3}a^{-13}+60z^{3}a^{-15}-66z^{2}a^{-12}-76z^{2}a^{-14}-10z^{2}a^{-16}-16za^{-13}-16za^{-15}+12a^{-12}+16a^{-14}+5a^{-16}$

Vassiliev invariants

V2 and V3:

(16, 56)

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(7,3)/V 2,1 Data:T(7,3)/V 3,1 Data:T(7,3)/V 4,1 Data:T(7,3)/V 4,2 Data:T(7,3)/V 4,3 Data:T(7,3)/V 5,1 Data:T(7,3)/V 5,2 Data:T(7,3)/V 5,3 Data:T(7,3)/V 5,4 Data:T(7,3)/V 6,1 Data:T(7,3)/V 6,2 Data:T(7,3)/V 6,3 Data:T(7,3)/V 6,4 Data:T(7,3)/V 6,5 Data:T(7,3)/V 6,6 Data:T(7,3)/V 6,7 Data:T(7,3)/V 6,8 Data:T(7,3)/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=$8 is the signature of T(7,3). 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 χ 29                   1 -1 27                   1 -1 25               1 1   0 23           1     1   0 21           1 1       0 19       1 1           0 17         1           1 15     1               1 13 1                   1 11 1                   1

Integral Khovanov Homology (db, data source)

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

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[7, 3]]
Out[2]=	14
In[3]:=	PD[TorusKnot[7, 3]]
Out[3]=	PD[X[1, 11, 2, 10], X[20, 12, 21, 11], X[21, 3, 22, 2], X[12, 4, 13, 3], X[13, 23, 14, 22], X[4, 24, 5, 23], X[5, 15, 6, 14], X[24, 16, 25, 15], X[25, 7, 26, 6], X[16, 8, 17, 7], X[17, 27, 18, 26], X[8, 28, 9, 27], X[9, 19, 10, 18], X[28, 20, 1, 19]]
In[4]:=	GaussCode[TorusKnot[7, 3]]
Out[4]=	GaussCode[-1, 3, 4, -6, -7, 9, 10, -12, -13, 1, 2, -4, -5, 7, 8, -10, -11, 13, 14, -2, -3, 5, 6, -8, -9, 11, 12, -14]
In[5]:=	BR[TorusKnot[7, 3]]
Out[5]=	BR[3, {1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2}]
In[6]:=	alex = Alexander[TorusKnot[7, 3]][t]
Out[6]=	-6 -5 -3 -2 1 + Alternating - Alternating + Alternating - Alternating - 2 3 5 6 Alternating + Alternating - Alternating + Alternating
In[7]:=	Conway[TorusKnot[7, 3]][z]
Out[7]=	2 4 6 8 10 12 1 + 16 z + 60 z + 78 z + 44 z + 11 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[TorusKnot[7, 3]], KnotSignature[TorusKnot[7, 3]]}
Out[9]=	{1, 8}
In[10]:=	J=Jones[TorusKnot[7, 3]][q]
Out[10]=	6 8 14 q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[TorusKnot[7, 3]][q]
Out[12]=	22 24 26 28 30 32 34 38 40 42 q + q + 2 q + 2 q + 2 q + q + q - q - 2 q - 2 q - 44 46 56 2 q - q + q
In[13]:=	Kauffman[TorusKnot[7, 3]][a, z]
Out[13]=	2 2 2 3 3 5 16 12 16 z 16 z 10 z 76 z 66 z 60 z 60 z --- + --- + --- - ---- - ---- - ----- - ----- - ----- + ----- + ----- + 16 14 12 15 13 16 14 12 15 13 a a a a a a a a a a 4 4 4 5 5 6 6 6 6 z 138 z 132 z 78 z 78 z z 122 z 121 z ---- + ------ + ------ - ----- - ----- - --- - ------ - ------ + 16 14 12 15 13 16 14 12 a a a a a a a a 7 7 8 8 9 9 10 10 44 z 44 z 55 z 55 z 11 z 11 z 12 z 12 z ----- + ----- + ----- + ----- - ----- - ----- - ------ - ------ + 15 13 14 12 15 13 14 12 a a a a a a a a 11 11 12 12 z z z z --- + --- + --- + --- 15 13 14 12 a a a a
In[14]:=	{Vassiliev[2][TorusKnot[7, 3]], Vassiliev[3][TorusKnot[7, 3]]}
Out[14]=	{0, 56}
In[15]:=	Kh[TorusKnot[7, 3]][q, t]
Out[15]=	11 13 2 15 4 17 3 19 q + q + Alternating q + Alternating q + Alternating q + 4 19 5 21 6 21 Alternating q + Alternating q + Alternating q + 5 23 8 23 7 25 Alternating q + Alternating q + Alternating q + 8 25 9 27 9 29 Alternating q + Alternating q + Alternating q