T(9,2)

T(4,3)

T(5,3)

Visit T(9,2)'s page at Knotilus! Visit T(9,2)'s page at the original Knot Atlas! See also 9_1.

T(9,2) Further Notes and Views

Knot presentations

Planar diagram presentation	X7,17,8,16 X17,9,18,8 X9,1,10,18 X1,11,2,10 X11,3,12,2 X3,13,4,12 X13,5,14,4 X5,15,6,14 X15,7,16,6
Gauss code	-4, 5, -6, 7, -8, 9, -1, 2, -3, 4, -5, 6, -7, 8, -9, 1, -2, 3
Dowker-Thistlethwaite code	10 12 14 16 18 2 4 6 8
Conway Notation	Data:T(9,2)/Conway Notation

Knot presentations

Planar diagram presentation	X7,17,8,16 X17,9,18,8 X9,1,10,18 X1,11,2,10 X11,3,12,2 X3,13,4,12 X13,5,14,4 X5,15,6,14 X15,7,16,6
Gauss code	$\{-4,5,-6,7,-8,9,-1,2,-3,4,-5,6,-7,8,-9,1,-2,3\}$
Dowker-Thistlethwaite code	10 12 14 16 18 2 4 6 8

Polynomial invariants

Alexander polynomial	$t^{4}-t^{3}+t^{2}-t+1-t^{-1}+t^{-2}-t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+7z^{6}+15z^{4}+10z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 9, 8 }
Jones polynomial	$-q^{13}+q^{12}-q^{11}+q^{10}-q^{9}+q^{8}-q^{7}+q^{6}+q^{4}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-8}+8z^{6}a^{-8}-z^{6}a^{-10}+21z^{4}a^{-8}-6z^{4}a^{-10}+20z^{2}a^{-8}-10z^{2}a^{-10}+5a^{-8}-4a^{-10}$
Kauffman polynomial (db, data sources)	$z^{8}a^{-8}+z^{8}a^{-10}+z^{7}a^{-9}+z^{7}a^{-11}-8z^{6}a^{-8}-7z^{6}a^{-10}+z^{6}a^{-12}-6z^{5}a^{-9}-5z^{5}a^{-11}+z^{5}a^{-13}+21z^{4}a^{-8}+16z^{4}a^{-10}-4z^{4}a^{-12}+z^{4}a^{-14}+10z^{3}a^{-9}+6z^{3}a^{-11}-3z^{3}a^{-13}+z^{3}a^{-15}-20z^{2}a^{-8}-14z^{2}a^{-10}+3z^{2}a^{-12}-2z^{2}a^{-14}+z^{2}a^{-16}-4za^{-9}-za^{-11}+za^{-13}-za^{-15}+za^{-17}+5a^{-8}+4a^{-10}$
The A2 invariant	Data:T(9,2)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(9,2)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(9,2) 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(9,2)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-t^{3}+t^{2}-t+1-t^{-1}+t^{-2}-t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+7z^{6}+15z^{4}+10z^{2}+1$

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]=

{ 9, 8 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{13}+q^{12}-q^{11}+q^{10}-q^{9}+q^{8}-q^{7}+q^{6}+q^{4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}a^{-8}+8z^{6}a^{-8}-z^{6}a^{-10}+21z^{4}a^{-8}-6z^{4}a^{-10}+20z^{2}a^{-8}-10z^{2}a^{-10}+5a^{-8}-4a^{-10}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{8}a^{-8}+z^{8}a^{-10}+z^{7}a^{-9}+z^{7}a^{-11}-8z^{6}a^{-8}-7z^{6}a^{-10}+z^{6}a^{-12}-6z^{5}a^{-9}-5z^{5}a^{-11}+z^{5}a^{-13}+21z^{4}a^{-8}+16z^{4}a^{-10}-4z^{4}a^{-12}+z^{4}a^{-14}+10z^{3}a^{-9}+6z^{3}a^{-11}-3z^{3}a^{-13}+z^{3}a^{-15}-20z^{2}a^{-8}-14z^{2}a^{-10}+3z^{2}a^{-12}-2z^{2}a^{-14}+z^{2}a^{-16}-4za^{-9}-za^{-11}+za^{-13}-za^{-15}+za^{-17}+5a^{-8}+4a^{-10}$

Vassiliev invariants

V2 and V3:

(10, 30)

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(9,2)/V 2,1 Data:T(9,2)/V 3,1 Data:T(9,2)/V 4,1 Data:T(9,2)/V 4,2 Data:T(9,2)/V 4,3 Data:T(9,2)/V 5,1 Data:T(9,2)/V 5,2 Data:T(9,2)/V 5,3 Data:T(9,2)/V 5,4 Data:T(9,2)/V 6,1 Data:T(9,2)/V 6,2 Data:T(9,2)/V 6,3 Data:T(9,2)/V 6,4 Data:T(9,2)/V 6,5 Data:T(9,2)/V 6,6 Data:T(9,2)/V 6,7 Data:T(9,2)/V 6,8 Data:T(9,2)/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=$8 is the signature of T(9,2). 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	χ
27										1	-1
25											0
23								1	1		0
21											0
19						1	1				0
17											0
15				1	1						0
13											0
11			1								1
9	1										1
7	1										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[9, 2]]
Out[2]=	9
In[3]:=	PD[TorusKnot[9, 2]]
Out[3]=	PD[X[7, 17, 8, 16], X[17, 9, 18, 8], X[9, 1, 10, 18], X[1, 11, 2, 10], X[11, 3, 12, 2], X[3, 13, 4, 12], X[13, 5, 14, 4], X[5, 15, 6, 14], X[15, 7, 16, 6]]
In[4]:=	GaussCode[TorusKnot[9, 2]]
Out[4]=	GaussCode[-4, 5, -6, 7, -8, 9, -1, 2, -3, 4, -5, 6, -7, 8, -9, 1, -2, 3]
In[5]:=	BR[TorusKnot[9, 2]]
Out[5]=	BR[2, {1, 1, 1, 1, 1, 1, 1, 1, 1}]
In[6]:=	alex = Alexander[TorusKnot[9, 2]][t]
Out[6]=	-4 -3 -2 1 2 3 4 1 + t - t + t - - - t + t - t + t t
In[7]:=	Conway[TorusKnot[9, 2]][z]
Out[7]=	2 4 6 8 1 + 10 z + 15 z + 7 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 1]}
In[9]:=	{KnotDet[TorusKnot[9, 2]], KnotSignature[TorusKnot[9, 2]]}
Out[9]=	{9, 8}
In[10]:=	J=Jones[TorusKnot[9, 2]][q]
Out[10]=	4 6 7 8 9 10 11 12 13 q + q - q + q - q + q - q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 1]}
In[12]:=	A2Invariant[TorusKnot[9, 2]][q]
Out[12]=	14 16 18 20 22 34 36 38 q + q + 2 q + q + q - q - q - q
In[13]:=	Kauffman[TorusKnot[9, 2]][a, z]
Out[13]=	2 2 2 2 4 5 z z z z 4 z z 2 z 3 z 14 z --- + -- + --- - --- + --- - --- - --- + --- - ---- + ---- - ----- - 10 8 17 15 13 11 9 16 14 12 10 a a a a a a a a a a a 2 3 3 3 3 4 4 4 4 20 z z 3 z 6 z 10 z z 4 z 16 z 21 z ----- + --- - ---- + ---- + ----- + --- - ---- + ----- + ----- + 8 15 13 11 9 14 12 10 8 a a a a a a a a a 5 5 5 6 6 6 7 7 8 8 z 5 z 6 z z 7 z 8 z z z z z --- - ---- - ---- + --- - ---- - ---- + --- + -- + --- + -- 13 11 9 12 10 8 11 9 10 8 a a a a a a a a a a
In[14]:=	{Vassiliev[2][TorusKnot[9, 2]], Vassiliev[3][TorusKnot[9, 2]]}
Out[14]=	{0, 30}
In[15]:=	Kh[TorusKnot[9, 2]][q, t]
Out[15]=	7 9 11 2 15 3 15 4 19 5 19 6 23 7 q + q + q t + q t + q t + q t + q t + q t + 23 8 27 9 q t + q t