T(8,3)

T(15,2)

T(17,2)

	Visit [[[:Template:KnotilusURL]] T(8,3)'s page] at Knotilus! Visit T(8,3)'s page at the original Knot Atlas!
	T(8,3) Quick Notes

Banco Internacional do Funchal [1]

Knot presentations

Planar diagram presentation	X11,1,12,32 X22,2,23,1 X23,13,24,12 X2,14,3,13 X3,25,4,24 X14,26,15,25 X15,5,16,4 X26,6,27,5 X27,17,28,16 X6,18,7,17 X7,29,8,28 X18,30,19,29 X19,9,20,8 X30,10,31,9 X31,21,32,20 X10,22,11,21
Gauss code	2, -4, -5, 7, 8, -10, -11, 13, 14, -16, -1, 3, 4, -6, -7, 9, 10, -12, -13, 15, 16, -2, -3, 5, 6, -8, -9, 11, 12, -14, -15, 1
Dowker-Thistlethwaite code	22 -24 26 -28 30 -32 2 -4 6 -8 10 -12 14 -16 18 -20
Conway Notation	Data:T(8,3)/Conway Notation

Polynomial invariants

Alexander polynomial	$t^{7}-t^{6}+t^{4}-t^{3}+t-1+t^{-1}-t^{-3}+t^{-4}-t^{-6}+t^{-7}$
Conway polynomial	$z^{14}+13z^{12}+65z^{10}+157z^{8}+189z^{6}+105z^{4}+21z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 3, 10 }
Jones polynomial	$-q^{16}+q^{9}+q^{7}$
HOMFLY-PT polynomial (db, data sources)	$z^{14}a^{-14}+14z^{12}a^{-14}-z^{12}a^{-16}+78z^{10}a^{-14}-13z^{10}a^{-16}+221z^{8}a^{-14}-65z^{8}a^{-16}+z^{8}a^{-18}+338z^{6}a^{-14}-157z^{6}a^{-16}+8z^{6}a^{-18}+273z^{4}a^{-14}-189z^{4}a^{-16}+21z^{4}a^{-18}+105z^{2}a^{-14}-105z^{2}a^{-16}+21z^{2}a^{-18}+15a^{-14}-21a^{-16}+7a^{-18}$
Kauffman polynomial (db, data sources)	$z^{14}a^{-14}+z^{14}a^{-16}+z^{13}a^{-15}+z^{13}a^{-17}-14z^{12}a^{-14}-14z^{12}a^{-16}-13z^{11}a^{-15}-13z^{11}a^{-17}+78z^{10}a^{-14}+78z^{10}a^{-16}+65z^{9}a^{-15}+65z^{9}a^{-17}-221z^{8}a^{-14}-222z^{8}a^{-16}-z^{8}a^{-18}-157z^{7}a^{-15}-157z^{7}a^{-17}+338z^{6}a^{-14}+346z^{6}a^{-16}+8z^{6}a^{-18}+189z^{5}a^{-15}+189z^{5}a^{-17}-273z^{4}a^{-14}-294z^{4}a^{-16}-21z^{4}a^{-18}-105z^{3}a^{-15}-105z^{3}a^{-17}+105z^{2}a^{-14}+126z^{2}a^{-16}+21z^{2}a^{-18}+21za^{-15}+21za^{-17}-15a^{-14}-21a^{-16}-7a^{-18}$
The A2 invariant	Data:T(8,3)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(8,3)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(8,3) 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(8,3)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

$z^{14}+13z^{12}+65z^{10}+157z^{8}+189z^{6}+105z^{4}+21z^{2}+1$

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]=

{ 3, 10 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{16}+q^{9}+q^{7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{14}a^{-14}+14z^{12}a^{-14}-z^{12}a^{-16}+78z^{10}a^{-14}-13z^{10}a^{-16}+221z^{8}a^{-14}-65z^{8}a^{-16}+z^{8}a^{-18}+338z^{6}a^{-14}-157z^{6}a^{-16}+8z^{6}a^{-18}+273z^{4}a^{-14}-189z^{4}a^{-16}+21z^{4}a^{-18}+105z^{2}a^{-14}-105z^{2}a^{-16}+21z^{2}a^{-18}+15a^{-14}-21a^{-16}+7a^{-18}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{14}a^{-14}+z^{14}a^{-16}+z^{13}a^{-15}+z^{13}a^{-17}-14z^{12}a^{-14}-14z^{12}a^{-16}-13z^{11}a^{-15}-13z^{11}a^{-17}+78z^{10}a^{-14}+78z^{10}a^{-16}+65z^{9}a^{-15}+65z^{9}a^{-17}-221z^{8}a^{-14}-222z^{8}a^{-16}-z^{8}a^{-18}-157z^{7}a^{-15}-157z^{7}a^{-17}+338z^{6}a^{-14}+346z^{6}a^{-16}+8z^{6}a^{-18}+189z^{5}a^{-15}+189z^{5}a^{-17}-273z^{4}a^{-14}-294z^{4}a^{-16}-21z^{4}a^{-18}-105z^{3}a^{-15}-105z^{3}a^{-17}+105z^{2}a^{-14}+126z^{2}a^{-16}+21z^{2}a^{-18}+21za^{-15}+21za^{-17}-15a^{-14}-21a^{-16}-7a^{-18}$

Vassiliev invariants

V2 and V3:

(21, 84)

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(8,3)/V 2,1 Data:T(8,3)/V 3,1 Data:T(8,3)/V 4,1 Data:T(8,3)/V 4,2 Data:T(8,3)/V 4,3 Data:T(8,3)/V 5,1 Data:T(8,3)/V 5,2 Data:T(8,3)/V 5,3 Data:T(8,3)/V 5,4 Data:T(8,3)/V 6,1 Data:T(8,3)/V 6,2 Data:T(8,3)/V 6,3 Data:T(8,3)/V 6,4 Data:T(8,3)/V 6,5 Data:T(8,3)/V 6,6 Data:T(8,3)/V 6,7 Data:T(8,3)/V 6,8 Data:T(8,3)/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=$10 is the signature of T(8,3). 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 10 11 χ 33                       1 -1 31                   1     -1 29                   1 1   0 27               1 1       0 25           1     1       0 23           1 1           0 21       1 1               0 19         1               1 17     1                   1 15 1                       1 13 1                       1

Integral Khovanov Homology (db, data source)

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

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[8, 3]]
Out[2]=	16
In[3]:=	PD[TorusKnot[8, 3]]
Out[3]=	PD[X[11, 1, 12, 32], X[22, 2, 23, 1], X[23, 13, 24, 12], X[2, 14, 3, 13], X[3, 25, 4, 24], X[14, 26, 15, 25], X[15, 5, 16, 4], X[26, 6, 27, 5], X[27, 17, 28, 16], X[6, 18, 7, 17], X[7, 29, 8, 28], X[18, 30, 19, 29], X[19, 9, 20, 8], X[30, 10, 31, 9], X[31, 21, 32, 20], X[10, 22, 11, 21]]
In[4]:=	GaussCode[TorusKnot[8, 3]]
Out[4]=	GaussCode[2, -4, -5, 7, 8, -10, -11, 13, 14, -16, -1, 3, 4, -6, -7, 9, 10, -12, -13, 15, 16, -2, -3, 5, 6, -8, -9, 11, 12, -14, -15, 1]
In[5]:=	BR[TorusKnot[8, 3]]
Out[5]=	BR[3, {1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2}]
In[6]:=	alex = Alexander[TorusKnot[8, 3]][t]
Out[6]=	-7 -6 -4 -3 -1 + Alternating - Alternating + Alternating - Alternating + 1 3 4 ----------- + Alternating - Alternating + Alternating - Alternating 6 7 Alternating + Alternating
In[7]:=	Conway[TorusKnot[8, 3]][z]
Out[7]=	2 4 6 8 10 12 14 1 + 21 z + 105 z + 189 z + 157 z + 65 z + 13 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[TorusKnot[8, 3]], KnotSignature[TorusKnot[8, 3]]}
Out[9]=	{3, 10}
In[10]:=	J=Jones[TorusKnot[8, 3]][q]
Out[10]=	7 9 16 q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[TorusKnot[8, 3]][q]
Out[12]=	NotAvailable
In[13]:=	Kauffman[TorusKnot[8, 3]][a, z]
Out[13]=	2 2 2 3 -7 21 15 21 z 21 z 21 z 126 z 105 z 105 z --- - --- - --- + ---- + ---- + ----- + ------ + ------ - ------ - 18 16 14 17 15 18 16 14 17 a a a a a a a a a 3 4 4 4 5 5 6 6 105 z 21 z 294 z 273 z 189 z 189 z 8 z 346 z ------ - ----- - ------ - ------ + ------ + ------ + ---- + ------ + 15 18 16 14 17 15 18 16 a a a a a a a a 6 7 7 8 8 8 9 9 338 z 157 z 157 z z 222 z 221 z 65 z 65 z ------ - ------ - ------ - --- - ------ - ------ + ----- + ----- + 14 17 15 18 16 14 17 15 a a a a a a a a 10 10 11 11 12 12 13 13 78 z 78 z 13 z 13 z 14 z 14 z z z ------ + ------ - ------ - ------ - ------ - ------ + --- + --- + 16 14 17 15 16 14 17 15 a a a a a a a a 14 14 z z --- + --- 16 14 a a
In[14]:=	{Vassiliev[2][TorusKnot[8, 3]], Vassiliev[3][TorusKnot[8, 3]]}
Out[14]=	{0, 84}
In[15]:=	Kh[TorusKnot[8, 3]][q, t]
Out[15]=	13 15 2 17 4 19 3 21 q + q + Alternating q + Alternating q + Alternating q + 4 21 5 23 6 23 Alternating q + Alternating q + Alternating q + 5 25 8 25 7 27 Alternating q + Alternating q + Alternating q + 8 27 9 29 10 29 Alternating q + Alternating q + Alternating q + 9 31 11 33 Alternating q + Alternating q