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DrawMorseLink

PD Presentation:

X8192 X12,4,13,3 X18,6,19,5 X22,12,7,11 X20,17,21,18 X16,21,17,22 X14,10,15,9 X10,16,11,15 X4,20,5,19 X2738 X6,14,1,13

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 4q-3/2 - 8q-1/2 + 12q1/2 - 17q3/2 + 18q5/2 - 19q7/2 + 16q9/2 - 12q11/2 + 7q13/2 - 3q15/2 + q17/2

A2 (sl(3)) Invariant:

q-8 - q-6 - q-4 + 2q-2 - 3 + 2q2 + 2q4 + 6q8 - q10 + 4q12 - q16 + 3q18 - 3q20 - q26

HOMFLY-PT Polynomial:

a-7z-1 + 2a-7z + a-7z3 - 3a-5z-1 - 7a-5z - 6a-5z3 - 2a-5z5 + 2a-3z-1 + 8a-3z + 8a-3z3 + 4a-3z5 + a-3z7 - 3a-1z - 5a-1z3 - 2a-1z5 + az + az3

Kauffman Polynomial:

a-10z2 - a-10z4 + 2a-9z3 - 3a-9z5 + a-8 - 4a-8z2 + 6a-8z4 - 6a-8z6 - a-7z-1 + 5a-7z - 13a-7z3 + 14a-7z5 - 9a-7z7 + 3a-6 - 7a-6z2 - a-6z4 + 12a-6z6 - 9a-6z8 - 3a-5z-1 + 13a-5z - 27a-5z3 + 23a-5z5 + a-5z7 - 6a-5z9 + 3a-4 + 2a-4z2 - 26a-4z4 + 38a-4z6 - 11a-4z8 - 2a-4z10 - 2a-3z-1 + 10a-3z - 15a-3z3 - 3a-3z5 + 25a-3z7 - 11a-3z9 + 7a-2z2 - 32a-2z4 + 34a-2z6 - 6a-2z8 - 2a-2z10 + 3a-1z - 6a-1z3 - 6a-1z5 + 14a-1z7 - 5a-1z9 + 3z2 - 14z4 + 14z6 - 4z8 + az - 3az3 + 3az5 - az7

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 18                       1 j = 16                     2   j = 14                   5 1   j = 12                 7 2     j = 10               9 5       j = 8             10 7         j = 6           9 10           j = 4         8 9             j = 2       5 10               j = 0     3 7                 j = -2   1 5                   j = -4   3                     j = -6 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, Alternating, 234]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 234]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 4, 13, 3], X[18, 6, 19, 5], X[22, 12, 7, 11], > X[20, 17, 21, 18], X[16, 21, 17, 22], X[14, 10, 15, 9], X[10, 16, 11, 15], > X[4, 20, 5, 19], X[2, 7, 3, 8], X[6, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -9, 3, -11}, > {10, -1, 7, -8, 4, -2, 11, -7, 8, -6, 5, -3, 9, -5, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 4 8 3/2 5/2 7/2 -q + ---- - ------- + 12 Sqrt[q] - 17 q + 18 q - 19 q + 3/2 Sqrt[q] q 9/2 11/2 13/2 15/2 17/2 > 16 q - 12 q + 7 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 -4 2 2 4 8 10 12 16 18 -3 + q - q - q + -- + 2 q + 2 q + 6 q - q + 4 q - q + 3 q - 2 q 20 26 > 3 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 234]][a, z]
Out[8]=	3 3 3 3 1 3 2 2 z 7 z 8 z 3 z z 6 z 8 z 5 z ---- - ---- + ---- + --- - --- + --- - --- + a z + -- - ---- + ---- - ---- + 7 5 3 7 5 3 a 7 5 3 a a z a z a z a a a a a a 5 5 5 7 3 2 z 4 z 2 z z > a z - ---- + ---- - ---- + -- 5 3 a 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 234]][a, z]
Out[9]=	-8 3 3 1 3 2 5 z 13 z 10 z 3 z 2 a + -- + -- - ---- - ---- - ---- + --- + ---- + ---- + --- + a z + 3 z + 6 4 7 5 3 7 5 3 a a a a z a z a z a a a 2 2 2 2 2 3 3 3 3 3 z 4 z 7 z 2 z 7 z 2 z 13 z 27 z 15 z 6 z > --- - ---- - ---- + ---- + ---- + ---- - ----- - ----- - ----- - ---- - 10 8 6 4 2 9 7 5 3 a a a a a a a a a a 4 4 4 4 4 5 5 5 3 4 z 6 z z 26 z 32 z 3 z 14 z 23 z > 3 a z - 14 z - --- + ---- - -- - ----- - ----- - ---- + ----- + ----- - 10 8 6 4 2 9 7 5 a a a a a a a a 5 5 6 6 6 6 7 7 3 z 6 z 5 6 6 z 12 z 38 z 34 z 9 z z > ---- - ---- + 3 a z + 14 z - ---- + ----- + ----- + ----- - ---- + -- + 3 a 8 6 4 2 7 5 a a a a a a a 7 7 8 8 8 9 9 9 25 z 14 z 7 8 9 z 11 z 6 z 6 z 11 z 5 z > ----- + ----- - a z - 4 z - ---- - ----- - ---- - ---- - ----- - ---- - 3 a 6 4 2 5 3 a a a a a a a 10 10 2 z 2 z > ----- - ----- 4 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 3 1 3 5 7 5 q 4 10 q + 8 q + ----- + ----- + ----- + -- + ----- + - + ---- + 9 q t + 6 4 4 3 2 3 2 2 2 t t q t q t q t t q t 6 6 2 8 2 8 3 10 3 10 4 12 4 > 9 q t + 10 q t + 10 q t + 7 q t + 9 q t + 5 q t + 7 q t + 12 5 14 5 14 6 16 6 18 7 > 2 q t + 5 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a234

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