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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X22,16,9,15 X18,12,19,11 X14,20,15,19 X20,14,21,13 X12,22,13,21 X8,18,5,17 X16,8,17,7 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-3 - 2q-2 + 7q-1 - 10 + 16q - 16q2 + 18q3 - 16q4 + 11q5 - 7q6 + 3q7 - q8

A2 (sl(3)) Invariant:

q-10 + q-8 + q-6 + 6q-4 + 3q-2 + 5 + 9q2 + 3q4 + 8q6 + q8 - q12 - 6q14 + q16 - 3q18 - 2q20 + q22 - q24

HOMFLY-PT Polynomial:

- a-6z-2 - 2a-6 - 2a-6z2 - a-6z4 + 3a-4z-2 + 6a-4 + 6a-4z2 + 3a-4z4 + a-4z6 - 2a-2z-2 - 3a-2 - 2a-2z2 + a-2z4 + a-2z6 - z-2 - 3 - 4z2 - 2z4 + a2z-2 + 2a2 + a2z2

Kauffman Polynomial:

- 2a-9z3 + a-9z5 - 5a-8z4 + 3a-8z6 + 2a-7z-1 - 7a-7z + 14a-7z3 - 14a-7z5 + 6a-7z7 - a-6z-2 + a-6 - 5a-6z2 + 17a-6z4 - 16a-6z6 + 7a-6z8 + 8a-5z-1 - 27a-5z + 40a-5z3 - 21a-5z5 + a-5z7 + 4a-5z9 - 3a-4z-2 + 5a-4 - 9a-4z2 + 23a-4z4 - 25a-4z6 + 10a-4z8 + a-4z10 + 10a-3z-1 - 34a-3z + 43a-3z3 - 23a-3z5 - 2a-3z7 + 6a-3z9 - 2a-2z-2 + 4a-2 - 4a-2z2 + 2a-2z4 - 11a-2z6 + 6a-2z8 + a-2z10 + 2a-1z-1 - 10a-1z + 19a-1z3 - 21a-1z5 + 5a-1z7 + 2a-1z9 + z-2 - 3 + 6z2 - 3z4 - 4z6 + 3z8 - 2az-1 + 4az - 4az5 + 2az7 + a2z-2 - 4a2 + 6a2z2 - 4a2z4 + a2z6

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 = 17                       1 j = 15                     2   j = 13                   5 1   j = 11                 6 2     j = 9               10 5       j = 7             8 6         j = 5           8 10           j = 3         8 8             j = 1       5 11               j = -1     2 5                 j = -3     5                   j = -5 1 2                     j = -7 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 394]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 394]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[22, 16, 9, 15], X[18, 12, 19, 11], > X[14, 20, 15, 19], X[20, 14, 21, 13], X[12, 22, 13, 21], X[8, 18, 5, 17], > X[16, 8, 17, 7], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 9, -8}, > {11, -2, 4, -7, 6, -5, 3, -9, 8, -4, 5, -6, 7, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-3 2 7 2 3 4 5 6 7 8 -10 + q - -- + - + 16 q - 16 q + 18 q - 16 q + 11 q - 7 q + 3 q - q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -8 -6 6 3 2 4 6 8 12 14 16 5 + q + q + q + -- + -- + 9 q + 3 q + 8 q + q - q - 6 q + q - 4 2 q q 18 20 22 24 > 3 q - 2 q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 394]][a, z]
Out[8]=	2 2 2 6 3 2 -2 1 3 2 a 2 2 z -3 - -- + -- - -- + 2 a - z - ----- + ----- - ----- + -- - 4 z - ---- + 6 4 2 6 2 4 2 2 2 2 6 a a a a z a z a z z a 2 2 4 4 4 6 6 6 z 2 z 2 2 4 z 3 z z z z > ---- - ---- + a z - 2 z - -- + ---- + -- + -- + -- 4 2 6 4 2 4 2 a a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 394]][a, z]
Out[9]=	2 -6 5 4 2 -2 1 3 2 a 2 8 -3 + a + -- + -- - 4 a + z - ----- - ----- - ----- + -- + ---- + ---- + 4 2 6 2 4 2 2 2 2 7 5 a a a z a z a z z a z a z 2 2 10 2 2 a 7 z 27 z 34 z 10 z 2 5 z 9 z > ---- + --- - --- - --- - ---- - ---- - ---- + 4 a z + 6 z - ---- - ---- - 3 a z z 7 5 3 a 6 4 a z a a a a a 2 3 3 3 3 3 4 4 z 2 2 2 z 14 z 40 z 43 z 19 z 4 5 z > ---- + 6 a z - ---- + ----- + ----- + ----- + ----- - 3 z - ---- + 2 9 7 5 3 a 8 a a a a a a 4 4 4 5 5 5 5 5 17 z 23 z 2 z 2 4 z 14 z 21 z 23 z 21 z > ----- + ----- + ---- - 4 a z + -- - ----- - ----- - ----- - ----- - 6 4 2 9 7 5 3 a a a a a a a a 6 6 6 6 7 7 7 5 6 3 z 16 z 25 z 11 z 2 6 6 z z 2 z > 4 a z - 4 z + ---- - ----- - ----- - ----- + a z + ---- + -- - ---- + 8 6 4 2 7 5 3 a a a a a a a 7 8 8 8 9 9 9 10 10 5 z 7 8 7 z 10 z 6 z 4 z 6 z 2 z z z > ---- + 2 a z + 3 z + ---- + ----- + ---- + ---- + ---- + ---- + --- + --- a 6 4 2 5 3 a 4 2 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 2 5 2 5 5 q 3 11 q + 8 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 8 q t + 7 4 5 4 5 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 8 q t + 10 q t + 8 q t + 6 q t + 10 q t + 5 q t + 6 q t + 11 5 13 5 13 6 15 6 17 7 > 2 q t + 5 q t + q t + 2 q t + q t

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