T(5,4)

[[Image:T(7,3).{{{ext}}}|80px|link=T(7,3)]] T(7,3)

[[Image:T(15,2).{{{ext}}}|80px|link=T(15,2)]] T(15,2)

Visit T(5,4)'s page at Knotilus!

Visit T(5,4)'s page at the original Knot Atlas!

Knot presentations

Planar diagram presentation	PD[X[17, 25, 18, 24], X[10, 26, 11, 25], X[3, 27, 4, 26], X[11, 19, 12, 18], X[4, 20, 5, 19], X[27, 21, 28, 20], X[5, 13, 6, 12], X[28, 14, 29, 13], X[21, 15, 22, 14], X[29, 7, 30, 6], X[22, 8, 23, 7], X[15, 9, 16, 8], X[23, 1, 24, 30], X[16, 2, 17, 1], X[9, 3, 10, 2]]
Gauss code	-1, 7, -2, 1, -3, 5, -4, 6, -7, 2, -6, 3, -5, 4
Dowker-Thistlethwaite code	4 10 12 14 2 8 6

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	2
Bridge index (super bridge index)	2 (4)
Nakanishi index	1

Polynomial invariants

Alexander polynomial	$t^{6}-t^{5}+t^{2}-1+t^{-2}-t^{-5}+t^{-6}$
Conway polynomial	$z^{12}+11z^{10}+44z^{8}+77z^{6}+56z^{4}+15z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 5, 8 }
Jones polynomial	$-q^{13}-q^{11}+q^{10}+q^{8}+q^{6}$
HOMFLY-PT polynomial (db, data sources)	$z^{12}a^{-12}+12z^{10}a^{-12}-z^{10}a^{-14}+55z^{8}a^{-12}-11z^{8}a^{-14}+121z^{6}a^{-12}-45z^{6}a^{-14}+z^{6}a^{-16}+133z^{4}a^{-12}-84z^{4}a^{-14}+7z^{4}a^{-16}+70z^{2}a^{-12}-70z^{2}a^{-14}+15z^{2}a^{-16}+14a^{-12}-21a^{-14}+9a^{-16}-a^{-18}$
Kauffman polynomial (db, data sources)	$z^{12}a^{-12}+z^{12}a^{-14}+z^{11}a^{-13}+z^{11}a^{-15}-12z^{10}a^{-12}-12z^{10}a^{-14}-11z^{9}a^{-13}-11z^{9}a^{-15}+55z^{8}a^{-12}+56z^{8}a^{-14}+z^{8}a^{-16}+45z^{7}a^{-13}+46z^{7}a^{-15}+z^{7}a^{-17}-121z^{6}a^{-12}-129z^{6}a^{-14}-8z^{6}a^{-16}-84z^{5}a^{-13}-91z^{5}a^{-15}-7z^{5}a^{-17}+133z^{4}a^{-12}+154z^{4}a^{-14}+21z^{4}a^{-16}+70z^{3}a^{-13}+84z^{3}a^{-15}+14z^{3}a^{-17}-70z^{2}a^{-12}-91z^{2}a^{-14}-22z^{2}a^{-16}-z^{2}a^{-18}-21za^{-13}-28za^{-15}-8za^{-17}-za^{-19}+14a^{-12}+21a^{-14}+9a^{-16}+a^{-18}$
The A2 invariant	Data:T(5,4)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(5,4)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(5,4) 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(5,4)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{6}-t^{5}+t^{2}-1+t^{-2}-t^{-5}+t^{-6}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{12}+11z^{10}+44z^{8}+77z^{6}+56z^{4}+15z^{2}+1$

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]=

{ 5, 8 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{13}-q^{11}+q^{10}+q^{8}+q^{6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{12}a^{-12}+12z^{10}a^{-12}-z^{10}a^{-14}+55z^{8}a^{-12}-11z^{8}a^{-14}+121z^{6}a^{-12}-45z^{6}a^{-14}+z^{6}a^{-16}+133z^{4}a^{-12}-84z^{4}a^{-14}+7z^{4}a^{-16}+70z^{2}a^{-12}-70z^{2}a^{-14}+15z^{2}a^{-16}+14a^{-12}-21a^{-14}+9a^{-16}-a^{-18}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{12}a^{-12}+z^{12}a^{-14}+z^{11}a^{-13}+z^{11}a^{-15}-12z^{10}a^{-12}-12z^{10}a^{-14}-11z^{9}a^{-13}-11z^{9}a^{-15}+55z^{8}a^{-12}+56z^{8}a^{-14}+z^{8}a^{-16}+45z^{7}a^{-13}+46z^{7}a^{-15}+z^{7}a^{-17}-121z^{6}a^{-12}-129z^{6}a^{-14}-8z^{6}a^{-16}-84z^{5}a^{-13}-91z^{5}a^{-15}-7z^{5}a^{-17}+133z^{4}a^{-12}+154z^{4}a^{-14}+21z^{4}a^{-16}+70z^{3}a^{-13}+84z^{3}a^{-15}+14z^{3}a^{-17}-70z^{2}a^{-12}-91z^{2}a^{-14}-22z^{2}a^{-16}-z^{2}a^{-18}-21za^{-13}-28za^{-15}-8za^{-17}-za^{-19}+14a^{-12}+21a^{-14}+9a^{-16}+a^{-18}$

Vassiliev invariants

V2 and V3:

(15, 50)

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(5,4)/V 2,1 Data:T(5,4)/V 3,1 Data:T(5,4)/V 4,1 Data:T(5,4)/V 4,2 Data:T(5,4)/V 4,3 Data:T(5,4)/V 5,1 Data:T(5,4)/V 5,2 Data:T(5,4)/V 5,3 Data:T(5,4)/V 5,4 Data:T(5,4)/V 6,1 Data:T(5,4)/V 6,2 Data:T(5,4)/V 6,3 Data:T(5,4)/V 6,4 Data:T(5,4)/V 6,5 Data:T(5,4)/V 6,6 Data:T(5,4)/V 6,7 Data:T(5,4)/V 6,8 Data:T(5,4)/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.