T(5,3)

T(9,2)

T(11,2)

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	T(5,3) Quick Notes

T(5,3) Further Notes and Views

Knot presentations

Planar diagram presentation	X7,1,8,20 X14,2,15,1 X15,9,16,8 X2,10,3,9 X3,17,4,16 X10,18,11,17 X11,5,12,4 X18,6,19,5 X19,13,20,12 X6,14,7,13
Gauss code	2, -4, -5, 7, 8, -10, -1, 3, 4, -6, -7, 9, 10, -2, -3, 5, 6, -8, -9, 1
Dowker-Thistlethwaite code	14 -16 18 -20 2 -4 6 -8 10 -12
Conway Notation	Data:T(5,3)/Conway Notation

Polynomial invariants

Alexander polynomial	$t^{4}-t^{3}+t-1+t^{-1}-t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+7z^{6}+14z^{4}+8z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 1, 8 }
Jones polynomial	$-q^{10}+q^{6}+q^{4}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-8}+8z^{6}a^{-8}-z^{6}a^{-10}+21z^{4}a^{-8}-7z^{4}a^{-10}+21z^{2}a^{-8}-14z^{2}a^{-10}+z^{2}a^{-12}+7a^{-8}-8a^{-10}+2a^{-12}$
Kauffman polynomial (db, data sources)	$z^{8}a^{-8}+z^{8}a^{-10}+z^{7}a^{-9}+z^{7}a^{-11}-8z^{6}a^{-8}-8z^{6}a^{-10}-7z^{5}a^{-9}-7z^{5}a^{-11}+21z^{4}a^{-8}+21z^{4}a^{-10}+14z^{3}a^{-9}+14z^{3}a^{-11}-21z^{2}a^{-8}-22z^{2}a^{-10}-z^{2}a^{-12}-8za^{-9}-8za^{-11}+7a^{-8}+8a^{-10}+2a^{-12}$
The A2 invariant	Data:T(5,3)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(5,3)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(5,3) 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(5,3)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+7z^{6}+14z^{4}+8z^{2}+1$

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]=

$-q^{10}+q^{6}+q^{4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}a^{-8}+8z^{6}a^{-8}-z^{6}a^{-10}+21z^{4}a^{-8}-7z^{4}a^{-10}+21z^{2}a^{-8}-14z^{2}a^{-10}+z^{2}a^{-12}+7a^{-8}-8a^{-10}+2a^{-12}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{8}a^{-8}+z^{8}a^{-10}+z^{7}a^{-9}+z^{7}a^{-11}-8z^{6}a^{-8}-8z^{6}a^{-10}-7z^{5}a^{-9}-7z^{5}a^{-11}+21z^{4}a^{-8}+21z^{4}a^{-10}+14z^{3}a^{-9}+14z^{3}a^{-11}-21z^{2}a^{-8}-22z^{2}a^{-10}-z^{2}a^{-12}-8za^{-9}-8za^{-11}+7a^{-8}+8a^{-10}+2a^{-12}$

Vassiliev invariants

V2 and V3:

(8, 20)

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(5,3)/V 2,1 Data:T(5,3)/V 3,1 Data:T(5,3)/V 4,1 Data:T(5,3)/V 4,2 Data:T(5,3)/V 4,3 Data:T(5,3)/V 5,1 Data:T(5,3)/V 5,2 Data:T(5,3)/V 5,3 Data:T(5,3)/V 5,4 Data:T(5,3)/V 6,1 Data:T(5,3)/V 6,2 Data:T(5,3)/V 6,3 Data:T(5,3)/V 6,4 Data:T(5,3)/V 6,5 Data:T(5,3)/V 6,6 Data:T(5,3)/V 6,7 Data:T(5,3)/V 6,8 Data:T(5,3)/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(5,3). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 χ 21               1 -1 19           1     -1 17           1 1   0 15       1 1       0 13         1       1 11     1           1 9 1               1 7 1               1

Integral Khovanov Homology (db, data source)

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

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[5, 3]]
Out[2]=	10
In[3]:=	PD[TorusKnot[5, 3]]
Out[3]=	PD[X[7, 1, 8, 20], X[14, 2, 15, 1], X[15, 9, 16, 8], X[2, 10, 3, 9], X[3, 17, 4, 16], X[10, 18, 11, 17], X[11, 5, 12, 4], X[18, 6, 19, 5], X[19, 13, 20, 12], X[6, 14, 7, 13]]
In[4]:=	GaussCode[TorusKnot[5, 3]]
Out[4]=	GaussCode[2, -4, -5, 7, 8, -10, -1, 3, 4, -6, -7, 9, 10, -2, -3, 5, 6, -8, -9, 1]
In[5]:=	BR[TorusKnot[5, 3]]
Out[5]=	BR[3, {1, 2, 1, 2, 1, 2, 1, 2, 1, 2}]
In[6]:=	alex = Alexander[TorusKnot[5, 3]][t]
Out[6]=	-4 -3 1 -1 + Alternating - Alternating + ----------- + Alternating - Alternating 3 4 Alternating + Alternating
In[7]:=	Conway[TorusKnot[5, 3]][z]
Out[7]=	2 4 6 8 1 + 8 z + 14 z + 7 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 124]}
In[9]:=	{KnotDet[TorusKnot[5, 3]], KnotSignature[TorusKnot[5, 3]]}
Out[9]=	{1, 8}
In[10]:=	J=Jones[TorusKnot[5, 3]][q]
Out[10]=	4 6 10 q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 124]}
In[12]:=	A2Invariant[TorusKnot[5, 3]][q]
Out[12]=	14 16 18 20 22 24 28 30 32 34 q + q + 2 q + 2 q + 2 q + q - 2 q - 2 q - 2 q - q + 40 q
In[13]:=	Kauffman[TorusKnot[5, 3]][a, z]
Out[13]=	2 2 2 3 3 2 8 7 8 z 8 z z 22 z 21 z 14 z 14 z --- + --- + -- - --- - --- - --- - ----- - ----- + ----- + ----- + 12 10 8 11 9 12 10 8 11 9 a a a a a a a a a a 4 4 5 5 6 6 7 7 8 8 21 z 21 z 7 z 7 z 8 z 8 z z z z z ----- + ----- - ---- - ---- - ---- - ---- + --- + -- + --- + -- 10 8 11 9 10 8 11 9 10 8 a a a a a a a a a a
In[14]:=	{Vassiliev[2][TorusKnot[5, 3]], Vassiliev[3][TorusKnot[5, 3]]}
Out[14]=	{0, 20}
In[15]:=	Kh[TorusKnot[5, 3]][q, t]
Out[15]=	7 9 2 11 4 13 3 15 q + q + Alternating q + Alternating q + Alternating q + 4 15 5 17 6 17 Alternating q + Alternating q + Alternating q + 5 19 7 21 Alternating q + Alternating q