T(4,3)

T(7,2)

T(9,2)

	See other torus knots Visit T(4,3) at Knotilus!
	Edit T(4,3) Quick Notes See also 8_19.

Edit 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

Braid presentation

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]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_19,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {8_19,}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["T(4,3)"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

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

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

Out[4]=

{ $t^{3}-t^{2}+1-t^{-2}+t^{-3}$, $-q^{8}+q^{5}+q^{3}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{8_19,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{8_19,}

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

Integral Khovanov Homology (db, data source)

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

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Torus Knot Page master template (intermediate). See/edit the Torus Knot_Splice_Base (expert). Back to the top.

T(7,2) T(9,2)