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L11n225 L11n227

Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X18,11,19,12 X6,9,7,10 X7,21,8,20 X19,1,20,8 X22,15,9,16 X3,12,4,13 X16,6,17,5 X13,4,14,5 X14,21,15,22 X2,18,3,17

Gauss Code:

{{1, -11, -7, 9, 8, -3, -4, 5}, {3, -1, 2, 7, -9, -10, 6, -8, 11, -2, -5, 4, 10, -6}}

Jones Polynomial:

q-15/2 - q-13/2 - q-3/2 - q1/2

A2 (sl(3)) Invariant:

- q-24 - q-14 + 2q-8 + 2q-6 + 2q-4 + 2q-2 + 2 + q2

HOMFLY-PT Polynomial:

- 2az-1 - 4az - az3 + 3a3z-1 + 6a3z + 5a3z3 + a3z5 - a5z-1 - a5z - a7z

Kauffman Polynomial:

2az-1 - 7az + 5az3 - az5 - 3a2 + 4a2z2 - a2z4 + 3a3z-1 - 11a3z + 15a3z3 - 7a3z5 + a3z7 - 3a4 + 10a4z2 - 6a4z4 + a4z6 + a5z-1 - 2a5z + a5z3 - a6 + a6z2 + 2a7z - 9a7z3 + 6a7z5 - a7z7 - 5a8z2 + 5a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                   1 j = 0               1     j = -2             1 3 1   j = -4           1 1 1     j = -6         1 2 1       j = -8       1 2 1         j = -10       1 1           j = -12   1 1 1             j = -14                     j = -16 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 226]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 226]]
Out[4]=	PD[X[10, 1, 11, 2], X[18, 11, 19, 12], X[6, 9, 7, 10], X[7, 21, 8, 20], > X[19, 1, 20, 8], X[22, 15, 9, 16], X[3, 12, 4, 13], X[16, 6, 17, 5], > X[13, 4, 14, 5], X[14, 21, 15, 22], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, -7, 9, 8, -3, -4, 5}, > {3, -1, 2, 7, -9, -10, 6, -8, 11, -2, -5, 4, 10, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) -(13/2) -(3/2) q - q - q - Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -14 2 2 2 2 2 2 - q - q + -- + -- + -- + -- + q 8 6 4 2 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 226]][a, z]
Out[8]=	3 5 -2 a 3 a a 3 5 7 3 3 3 3 5 ---- + ---- - -- - 4 a z + 6 a z - a z - a z - a z + 5 a z + a z z z z
In[9]:=	Kauffman[Link[11, NonAlternating, 226]][a, z]
Out[9]=	3 5 2 4 6 2 a 3 a a 3 5 7 -3 a - 3 a - a + --- + ---- + -- - 7 a z - 11 a z - 2 a z + 2 a z + z z z 2 2 4 2 6 2 8 2 3 3 3 5 3 > 4 a z + 10 a z + a z - 5 a z + 5 a z + 15 a z + a z - 7 3 2 4 4 4 8 4 5 3 5 7 5 4 6 > 9 a z - a z - 6 a z + 5 a z - a z - 7 a z + 6 a z + a z - 8 6 3 7 7 7 > a z + a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 3 1 1 1 1 1 1 1 1 + q + -- + ------ + ------ + ------ + ------ + ------ + ----- + ------ + 2 16 7 12 6 12 5 12 4 10 4 8 4 10 3 q q t q t q t q t q t q t q t 2 1 1 2 1 1 1 1 t 2 2 > ----- + ----- + ----- + ----- + ----- + ---- + ---- + ---- + -- + q t 8 3 6 3 8 2 6 2 4 2 6 4 2 2 q t q t q t q t q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n226

L11n225 L11n227