T(7,3)

[[Image:T(13,2).{{{ext}}}|80px|link=T(13,2)]] T(13,2)

[[Image:T(5,4).{{{ext}}}|80px|link=T(5,4)]] T(5,4)

Visit T(7,3)'s page at Knotilus!

Visit 7₅'s page at the original Knot Atlas!

Knot presentations

Planar diagram presentation	X1,11,2,10 X20,12,21,11 X21,3,22,2 X12,4,13,3 X13,23,14,22 X4,24,5,23 X5,15,6,14 X24,16,25,15 X25,7,26,6 X16,8,17,7 X17,27,18,26 X8,28,9,27 X9,19,10,18 X28,20,1,19
Gauss code	-1, 3, 4, -6, -7, 9, 10, -12, -13, 1, 2, -4, -5, 7, 8, -10, -11, 13, 14, -2, -3, 5, 6, -8, -9, 11, 12, -14
Dowker-Thistlethwaite code	10 -12 14 -16 18 -20 22 -24 26 -28 2 -4 6 -8
Conway Notation	Data:T(7,3)/Conway Notation

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	$t^{6}-t^{5}+t^{3}-t^{2}+1-t^{-2}+t^{-3}-t^{-5}+t^{-6}$
Conway polynomial	$z^{12}+11z^{10}+44z^{8}+78z^{6}+60z^{4}+16z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 1, 8 }
Jones polynomial	$-q^{14}+q^{8}+q^{6}$
HOMFLY-PT polynomial (db, data sources)	$z^{12}a^{-12}+12z^{10}a^{-12}-z^{10}a^{-14}+55z^{8}a^{-12}-11z^{8}a^{-14}+121z^{6}a^{-12}-44z^{6}a^{-14}+z^{6}a^{-16}+132z^{4}a^{-12}-78z^{4}a^{-14}+6z^{4}a^{-16}+66z^{2}a^{-12}-60z^{2}a^{-14}+10z^{2}a^{-16}+12a^{-12}-16a^{-14}+5a^{-16}$
Kauffman polynomial (db, data sources)	$z^{12}a^{-12}+z^{12}a^{-14}+z^{11}a^{-13}+z^{11}a^{-15}-12z^{10}a^{-12}-12z^{10}a^{-14}-11z^{9}a^{-13}-11z^{9}a^{-15}+55z^{8}a^{-12}+55z^{8}a^{-14}+44z^{7}a^{-13}+44z^{7}a^{-15}-121z^{6}a^{-12}-122z^{6}a^{-14}-z^{6}a^{-16}-78z^{5}a^{-13}-78z^{5}a^{-15}+132z^{4}a^{-12}+138z^{4}a^{-14}+6z^{4}a^{-16}+60z^{3}a^{-13}+60z^{3}a^{-15}-66z^{2}a^{-12}-76z^{2}a^{-14}-10z^{2}a^{-16}-16za^{-13}-16za^{-15}+12a^{-12}+16a^{-14}+5a^{-16}$
The A2 invariant	Data:T(7,3)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(7,3)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(7,3) 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(7,3)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{6}-t^{5}+t^{3}-t^{2}+1-t^{-2}+t^{-3}-t^{-5}+t^{-6}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{12}+11z^{10}+44z^{8}+78z^{6}+60z^{4}+16z^{2}+1$

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]=

$-q^{14}+q^{8}+q^{6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{12}a^{-12}+12z^{10}a^{-12}-z^{10}a^{-14}+55z^{8}a^{-12}-11z^{8}a^{-14}+121z^{6}a^{-12}-44z^{6}a^{-14}+z^{6}a^{-16}+132z^{4}a^{-12}-78z^{4}a^{-14}+6z^{4}a^{-16}+66z^{2}a^{-12}-60z^{2}a^{-14}+10z^{2}a^{-16}+12a^{-12}-16a^{-14}+5a^{-16}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{12}a^{-12}+z^{12}a^{-14}+z^{11}a^{-13}+z^{11}a^{-15}-12z^{10}a^{-12}-12z^{10}a^{-14}-11z^{9}a^{-13}-11z^{9}a^{-15}+55z^{8}a^{-12}+55z^{8}a^{-14}+44z^{7}a^{-13}+44z^{7}a^{-15}-121z^{6}a^{-12}-122z^{6}a^{-14}-z^{6}a^{-16}-78z^{5}a^{-13}-78z^{5}a^{-15}+132z^{4}a^{-12}+138z^{4}a^{-14}+6z^{4}a^{-16}+60z^{3}a^{-13}+60z^{3}a^{-15}-66z^{2}a^{-12}-76z^{2}a^{-14}-10z^{2}a^{-16}-16za^{-13}-16za^{-15}+12a^{-12}+16a^{-14}+5a^{-16}$

Vassiliev invariants

V2 and V3:

(16, 56)

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