T(4,3)

T(7,2)

T(9,2)

Visit T(4,3)'s page at Knotilus! Visit T(4,3)'s page at the original Knot Atlas! See also 8_19.

T(4,3) Further Notes and Views

Knot presentations

Planar diagram presentation	X5,11,6,10 X16,12,1,11 X1726 X12,8,13,7 X13,3,14,2 X8493 X9,15,10,14 X4,16,5,15
Gauss code	-3, 5, 6, -8, -1, 3, 4, -6, -7, 1, 2, -4, -5, 7, 8, -2
Dowker-Thistlethwaite code	6 -8 10 -12 14 -16 2 -4
Conway Notation	Data:T(4,3)/Conway Notation

Knot presentations

Planar diagram presentation	X5,11,6,10 X16,12,1,11 X1726 X12,8,13,7 X13,3,14,2 X8493 X9,15,10,14 X4,16,5,15
Gauss code	$\{-3,5,6,-8,-1,3,4,-6,-7,1,2,-4,-5,7,8,-2\}$
Dowker-Thistlethwaite code	6 -8 10 -12 14 -16 2 -4

Polynomial invariants

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

Further Quantum Invariants

T(4,3) 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(4,3)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

$z^{6}+5z^{4}+5z^{2}+1$

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, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+q^{5}+q^{3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{6}a^{-6}+z^{6}a^{-8}+z^{5}a^{-7}+z^{5}a^{-9}-6z^{4}a^{-6}-6z^{4}a^{-8}-5z^{3}a^{-7}-5z^{3}a^{-9}+10z^{2}a^{-6}+10z^{2}a^{-8}+5za^{-7}+5za^{-9}-5a^{-6}-5a^{-8}-a^{-10}$

Vassiliev invariants

V2 and V3:

(5, 10)

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

\   r    \    j   \	0	1	2	3	4	5	χ
17						1	-1
15						1	-1
13				1	1		0
11					1		1
9			1				1
7	1						1
5	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[4, 3]]
Out[2]=	8
In[3]:=	PD[TorusKnot[4, 3]]
Out[3]=	PD[X[5, 11, 6, 10], X[16, 12, 1, 11], X[1, 7, 2, 6], X[12, 8, 13, 7], X[13, 3, 14, 2], X[8, 4, 9, 3], X[9, 15, 10, 14], X[4, 16, 5, 15]]
In[4]:=	GaussCode[TorusKnot[4, 3]]
Out[4]=	GaussCode[-3, 5, 6, -8, -1, 3, 4, -6, -7, 1, 2, -4, -5, 7, 8, -2]
In[5]:=	BR[TorusKnot[4, 3]]
Out[5]=	BR[3, {1, 2, 1, 2, 1, 2, 1, 2}]
In[6]:=	alex = Alexander[TorusKnot[4, 3]][t]
Out[6]=	-3 -2 2 3 1 + t - t - t + t
In[7]:=	Conway[TorusKnot[4, 3]][z]
Out[7]=	2 4 6 1 + 5 z + 5 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 19]}
In[9]:=	{KnotDet[TorusKnot[4, 3]], KnotSignature[TorusKnot[4, 3]]}
Out[9]=	{3, 6}
In[10]:=	J=Jones[TorusKnot[4, 3]][q]
Out[10]=	3 5 8 q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[8, 19]}
In[12]:=	A2Invariant[TorusKnot[4, 3]][q]
Out[12]=	10 12 14 16 18 22 24 26 28 32 q + q + 2 q + 2 q + 2 q - q - 2 q - 2 q - q + q
In[13]:=	Kauffman[TorusKnot[4, 3]][a, z]
Out[13]=	2 2 3 3 4 -10 5 5 5 z 5 z 10 z 10 z 5 z 5 z 6 z -a - -- - -- + --- + --- + ----- + ----- - ---- - ---- - ---- - 8 6 9 7 8 6 9 7 8 a a a a a a a a a 4 5 5 6 6 6 z z z z z ---- + -- + -- + -- + -- 6 9 7 8 6 a a a a a
In[14]:=	{Vassiliev[2][TorusKnot[4, 3]], Vassiliev[3][TorusKnot[4, 3]]}
Out[14]=	{0, 10}
In[15]:=	Kh[TorusKnot[4, 3]][q, t]
Out[15]=	5 7 9 2 13 3 11 4 13 4 15 5 17 5 q + q + q t + q t + q t + q t + q t + q t