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L11n118 L11n120

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DrawMorseLink

PD Presentation:

X6172 X3,15,4,14 X16,10,17,9 X11,21,12,20 X21,9,22,8 X7,19,8,18 X19,13,20,12 X10,16,11,15 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:

- 3q3/2 + 4q5/2 - 7q7/2 + 7q9/2 - 7q11/2 + 6q13/2 - 4q15/2 + 2q17/2

A2 (sl(3)) Invariant:

3q4 + q6 + 4q8 + 3q10 + q12 + 3q14 - q16 + q18 - 2q20 - 2q22 - 2q26

HOMFLY-PT Polynomial:

2a-7z-1 + 4a-7z + 2a-7z3 - 5a-5z-1 - 11a-5z - 8a-5z3 - 2a-5z5 + 3a-3z-1 + 7a-3z + 3a-3z3

Kauffman Polynomial:

- a-10 + 5a-10z2 - 3a-10z4 + 4a-9z3 - a-9z5 - a-9z7 + 3a-8z4 - a-8z6 - a-8z8 - 2a-7z-1 + 7a-7z - 12a-7z3 + 11a-7z5 - 5a-7z7 + 5a-6 - 14a-6z2 + 12a-6z4 - 4a-6z6 - a-6z8 - 5a-5z-1 + 17a-5z - 22a-5z3 + 12a-5z5 - 4a-5z7 + 5a-4 - 9a-4z2 + 6a-4z4 - 3a-4z6 - 3a-3z-1 + 10a-3z - 6a-3z3

Khovanov Homology:

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 18               2 j = 16             2   j = 14           4 2   j = 12         3 2     j = 10       4 4       j = 8     3 3         j = 6   1 4           j = 4 2 3             j = 2 3

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, 119]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 119]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 15, 4, 14], X[16, 10, 17, 9], X[11, 21, 12, 20], > X[21, 9, 22, 8], X[7, 19, 8, 18], X[19, 13, 20, 12], X[10, 16, 11, 15], > 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]=	3/2 5/2 7/2 9/2 11/2 13/2 15/2 17/2 -3 q + 4 q - 7 q + 7 q - 7 q + 6 q - 4 q + 2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	4 6 8 10 12 14 16 18 20 22 26 3 q + q + 4 q + 3 q + q + 3 q - q + q - 2 q - 2 q - 2 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 119]][a, z]
Out[8]=	3 3 3 5 2 5 3 4 z 11 z 7 z 2 z 8 z 3 z 2 z ---- - ---- + ---- + --- - ---- + --- + ---- - ---- + ---- - ---- 7 5 3 7 5 3 7 5 3 5 a z a z a z a a a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 119]][a, z]
Out[9]=	2 2 -10 5 5 2 5 3 7 z 17 z 10 z 5 z 14 z -a + -- + -- - ---- - ---- - ---- + --- + ---- + ---- + ---- - ----- - 6 4 7 5 3 7 5 3 10 6 a a a z a z a z a a a a a 2 3 3 3 3 4 4 4 4 5 9 z 4 z 12 z 22 z 6 z 3 z 3 z 12 z 6 z z > ---- + ---- - ----- - ----- - ---- - ---- + ---- + ----- + ---- - -- + 4 9 7 5 3 10 8 6 4 9 a a a a a a a a a a 5 5 6 6 6 7 7 7 8 8 11 z 12 z z 4 z 3 z z 5 z 4 z z z > ----- + ----- - -- - ---- - ---- - -- - ---- - ---- - -- - -- 7 5 8 6 4 9 7 5 8 6 a a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 4 6 6 2 8 2 8 3 10 3 3 q + 2 q + 3 q t + q t + 4 q t + 3 q t + 3 q t + 4 q t + 10 4 12 4 12 5 14 5 14 6 16 6 18 7 > 4 q t + 3 q t + 2 q t + 4 q t + 2 q t + 2 q t + 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n119

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