T(8,3)

T(15,2)

T(17,2)

Visit 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	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
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

Knot presentations

Planar diagram presentation	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
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

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.

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)

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 $s=$ 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

0

27

1

0

25

1

0

23

1

0

21

1

0

19

1

17

1

15

1

13

1

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 3 4 6 7 -1 + t - t + t - t + - + t - t + t - t + t t
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 17 2 21 3 19 4 21 4 23 5 25 5 q + q + q t + q t + q t + q t + q t + q t + 23 6 27 7 25 8 27 8 29 9 31 9 29 10 q t + q t + q t + q t + q t + q t + q t + 33 11 q t

T(8,3)

Contents

Knot presentations

Knot presentations

Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

Navigation menu

Page actions

Page actions

Personal tools

Navigation

Search

Tools