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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 3q-3/2 - 7q-1/2 + 10q1/2 - 14q3/2 + 16q5/2 - 17q7/2 + 14q9/2 - 11q11/2 + 7q13/2 - 3q15/2 + q17/2

A2 (sl(3)) Invariant:

q-8 + 3q-2 - 1 + 2q2 + q4 - 2q6 + 4q8 - q10 + 4q12 + q14 - q16 + 2q18 - 3q20 - q26

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-10z2 - a-10z4 + 2a-9z3 - 3a-9z5 + a-8 - 5a-8z2 + 7a-8z4 - 6a-8z6 - a-7z-1 + 5a-7z - 11a-7z3 + 12a-7z5 - 8a-7z7 + 2a-6 - 5a-6z2 + 2a-6z4 + 7a-6z6 - 7a-6z8 - 3a-5z-1 + 14a-5z - 27a-5z3 + 26a-5z5 - 4a-5z7 - 4a-5z9 + 9a-4z2 - 30a-4z4 + 37a-4z6 - 12a-4z8 - a-4z10 - 3a-3z-1 + 16a-3z - 31a-3z3 + 18a-3z5 + 10a-3z7 - 7a-3z9 - 2a-2 + 12a-2z2 - 36a-2z4 + 35a-2z6 - 8a-2z8 - a-2z10 - 2a-1z-1 + 11a-1z - 23a-1z3 + 11a-1z5 + 5a-1z7 - 3a-1z9 + 4z2 - 12z4 + 11z6 - 3z8 - az-1 + 4az - 6az3 + 4az5 - 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                 6 2     j = 10               8 5       j = 8             9 6         j = 6           7 8           j = 4         7 9             j = 2       5 9               j = 0     2 5                 j = -2   1 5                   j = -4   2                     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, 66]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 66]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 8, 15, 7], X[22, 12, 5, 11], > X[20, 17, 21, 18], X[10, 16, 11, 15], X[8, 20, 9, 19], X[18, 10, 19, 9], > X[16, 21, 17, 22], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -7, 8, -6, 4, -2, 11, -3, 6, -9, 5, -8, > 7, -5, 9, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 3 7 3/2 5/2 7/2 -q + ---- - ------- + 10 Sqrt[q] - 14 q + 16 q - 17 q + 3/2 Sqrt[q] q 9/2 11/2 13/2 15/2 17/2 > 14 q - 11 q + 7 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 3 2 4 6 8 10 12 14 16 18 -1 + q + -- + 2 q + q - 2 q + 4 q - q + 4 q + q - q + 2 q - 2 q 20 26 > 3 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 66]][a, z]
Out[8]=	3 3 1 3 3 2 a 2 z 6 z 7 z 5 z z 6 z ---- - ---- + ---- - --- + - + --- - --- + --- - --- + 2 a z + -- - ---- + 7 5 3 a z z 7 5 3 a 7 5 a z a z a z a a a a a 3 3 5 5 5 7 7 z 6 z 3 2 z 4 z 2 z z > ---- - ---- + a z - ---- + ---- - ---- + -- 3 a 5 3 a 3 a a a a
In[9]:=	Kauffman[Link[11, Alternating, 66]][a, z]
Out[9]=	-8 2 2 1 3 3 2 a 5 z 14 z 16 z 11 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 2 3 3 3 2 z 5 z 5 z 9 z 12 z 2 z 11 z 27 z > 4 a z + 4 z + --- - ---- - ---- + ---- + ----- + ---- - ----- - ----- - 10 8 6 4 2 9 7 5 a a a a a a a a 3 3 4 4 4 4 4 5 31 z 23 z 3 4 z 7 z 2 z 30 z 36 z 3 z > ----- - ----- - 6 a z - 12 z - --- + ---- + ---- - ----- - ----- - ---- + 3 a 10 8 6 4 2 9 a a a a a a a 5 5 5 5 6 6 6 12 z 26 z 18 z 11 z 5 6 6 z 7 z 37 z > ----- + ----- + ----- + ----- + 4 a z + 11 z - ---- + ---- + ----- + 7 5 3 a 8 6 4 a a a a a a 6 7 7 7 7 8 8 8 35 z 8 z 4 z 10 z 5 z 7 8 7 z 12 z 8 z > ----- - ---- - ---- + ----- + ---- - a z - 3 z - ---- - ----- - ---- - 2 7 5 3 a 6 4 2 a a a a a a a 9 9 9 10 10 4 z 7 z 3 z z z > ---- - ---- - ---- - --- - --- 5 3 a 4 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 2 1 2 5 5 5 q 4 6 9 q + 7 q + ----- + ----- + ----- + -- + ----- + - + ---- + 9 q t + 7 q t + 6 4 4 3 2 3 2 2 2 t t q t q t q t t q t 6 2 8 2 8 3 10 3 10 4 12 4 12 5 > 8 q t + 9 q t + 6 q t + 8 q t + 5 q t + 6 q t + 2 q t + 14 5 14 6 16 6 18 7 > 5 q t + q t + 2 q t + q t

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