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L11n117 L11n119

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DrawMorseLink

PD Presentation:

X6172 X3,15,4,14 X9,16,10,17 X11,21,12,20 X21,9,22,8 X7,19,8,18 X19,13,20,12 X15,10,16,11 X17,5,18,22 X2536 X13,1,14,4

Gauss Code:

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

Jones Polynomial:

- 2q-3/2 + 5q-1/2 - 9q1/2 + 11q3/2 - 13q5/2 + 11q7/2 - 10q9/2 + 7q11/2 - 3q13/2 + q15/2

A2 (sl(3)) Invariant:

2q-6 + q-4 - 2q-2 + 3 - q2 + q4 + 3q6 + 4q10 - q12 + 2q14 + q16 - 3q18 + q20 - q22 - q24

HOMFLY-PT Polynomial:

a-7z-1 + a-7z - 3a-5z-1 - 5a-5z - 3a-5z3 + 3a-3z-1 + 8a-3z + 6a-3z3 + 2a-3z5 - 2a-1z-1 - 6a-1z - 4a-1z3 + az-1 + 2az

Kauffman Polynomial:

a-8 - 3a-8z2 + 3a-8z4 - a-8z6 - a-7z-1 + 3a-7z - 6a-7z3 + 8a-7z5 - 3a-7z7 + 2a-6 - 3a-6z2 + a-6z4 + 8a-6z6 - 4a-6z8 - 3a-5z-1 + 12a-5z - 25a-5z3 + 25a-5z5 - 4a-5z7 - 2a-5z9 + 7a-4z2 - 22a-4z4 + 27a-4z6 - 10a-4z8 - 3a-3z-1 + 18a-3z - 39a-3z3 + 30a-3z5 - 6a-3z7 - 2a-3z9 - 2a-2 + 11a-2z2 - 24a-2z4 + 17a-2z6 - 6a-2z8 - 2a-1z-1 + 13a-1z - 23a-1z3 + 13a-1z5 - 5a-1z7 + 4z2 - 4z4 - z6 - az-1 + 4az - 3az3

Khovanov Homology:

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

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, 118]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 118]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 15, 4, 14], X[9, 16, 10, 17], X[11, 21, 12, 20], > X[21, 9, 22, 8], X[7, 19, 8, 18], X[19, 13, 20, 12], X[15, 10, 16, 11], > X[17, 5, 18, 22], X[2, 5, 3, 6], X[13, 1, 14, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -6, 5, -3, 8, -4, 7, -11, 2, -8, 3, -9, 6, > -7, 4, -5, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-2 5 3/2 5/2 7/2 9/2 11/2 ---- + ------- - 9 Sqrt[q] + 11 q - 13 q + 11 q - 10 q + 7 q - 3/2 Sqrt[q] q 13/2 15/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 -4 2 2 4 6 10 12 14 16 18 20 3 + -- + q - -- - q + q + 3 q + 4 q - q + 2 q + q - 3 q + q - 6 2 q q 22 24 > q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 118]][a, z]
Out[8]=	3 3 1 3 3 2 a z 5 z 8 z 6 z 3 z 6 z ---- - ---- + ---- - --- + - + -- - --- + --- - --- + 2 a z - ---- + ---- - 7 5 3 a z z 7 5 3 a 5 3 a z a z a z a a a a a 3 5 4 z 2 z > ---- + ---- a 3 a
In[9]:=	Kauffman[Link[11, NonAlternating, 118]][a, z]
Out[9]=	-8 2 2 1 3 3 2 a 3 z 12 z 18 z 13 z a + -- - -- - ---- - ---- - ---- - --- - - + --- + ---- + ---- + ---- + 6 2 7 5 3 a z z 7 5 3 a a a a z a z a z a a a 2 2 2 2 3 3 3 3 2 3 z 3 z 7 z 11 z 6 z 25 z 39 z 23 z > 4 a z + 4 z - ---- - ---- + ---- + ----- - ---- - ----- - ----- - ----- - 8 6 4 2 7 5 3 a a a a a a a a 4 4 4 4 5 5 5 5 3 4 3 z z 22 z 24 z 8 z 25 z 30 z 13 z > 3 a z - 4 z + ---- + -- - ----- - ----- + ---- + ----- + ----- + ----- - 8 6 4 2 7 5 3 a a a a a a a a 6 6 6 6 7 7 7 7 8 8 6 z 8 z 27 z 17 z 3 z 4 z 6 z 5 z 4 z 10 z > z - -- + ---- + ----- + ----- - ---- - ---- - ---- - ---- - ---- - ----- - 8 6 4 2 7 5 3 a 6 4 a a a a a a a a a 8 9 9 6 z 2 z 2 z > ---- - ---- - ---- 2 5 3 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 2 3 2 4 4 2 6 2 6 3 6 + 5 q + ----- + - + ---- + 7 q t + 4 q t + 6 q t + 7 q t + 5 q t + 4 2 t 2 q t q t 8 3 8 4 10 4 10 5 12 5 12 6 14 6 > 6 q t + 5 q t + 5 q t + 2 q t + 5 q t + q t + 2 q t + 16 7 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n118

L11n117 L11n119