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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 2q-3/2 - 7q-1/2 + 10q1/2 - 15q3/2 + 18q5/2 - 19q7/2 + 17q9/2 - 13q11/2 + 9q13/2 - 4q15/2 + q17/2

A2 (sl(3)) Invariant:

q-8 + q-6 + q-4 + 5q-2 + 1 + 4q2 + 3q4 - 3q6 + 3q8 - 4q10 + q12 - q14 - 3q16 + 2q18 - 3q20 + q22 + q24 - q26

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-10z4 + a-9z3 - 4a-9z5 + a-8 - 4a-8z2 + 9a-8z4 - 9a-8z6 + 2a-7z - 8a-7z3 + 16a-7z5 - 12a-7z7 - 3a-6z2 + 5a-6z4 + 9a-6z6 - 10a-6z8 + a-5z-1 + 2a-5z - 20a-5z3 + 32a-5z5 - 7a-5z7 - 5a-5z9 - 3a-4 + 15a-4z2 - 38a-4z4 + 46a-4z6 - 16a-4z8 - a-4z10 + a-3z-1 - 3a-3z - 5a-3z3 + 3a-3z5 + 14a-3z7 - 8a-3z9 + 10a-2z2 - 36a-2z4 + 34a-2z6 - 8a-2z8 - a-2z10 - 2a-1z-1 + 4a-1z - 3a-1z3 - 4a-1z5 + 8a-1z7 - 3a-1z9 + 3 - 4z2 - 3z4 + 6z6 - 2z8 - 2az-1 + 7az - 9az3 + 5az5 - 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                     3   j = 14                   6 1   j = 12                 7 3     j = 10               10 6       j = 8             9 7         j = 6           9 10           j = 4         6 9             j = 2       5 10               j = 0     2 5                 j = -2     5                   j = -4 1 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, 71]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 71]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[16, 8, 17, 7], X[18, 10, 19, 9], > X[8, 18, 9, 17], X[22, 20, 5, 19], X[20, 14, 21, 13], X[14, 22, 15, 21], > X[10, 16, 11, 15], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -5, 4, -9, 11, -2, 7, -8, 9, -3, 5, -4, > 6, -7, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 2 7 3/2 5/2 7/2 -q + ---- - ------- + 10 Sqrt[q] - 15 q + 18 q - 19 q + 3/2 Sqrt[q] q 9/2 11/2 13/2 15/2 17/2 > 17 q - 13 q + 9 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 -4 5 2 4 6 8 10 12 14 1 + q + q + q + -- + 4 q + 3 q - 3 q + 3 q - 4 q + q - q - 2 q 16 18 20 22 24 26 > 3 q + 2 q - 3 q + q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 71]][a, z]
Out[8]=	3 3 3 3 1 1 2 2 a z z 5 z z 4 z 2 z 6 z ---- - ---- - --- + --- + -- - -- - --- + 3 a z + -- - ---- + ---- - ---- + 5 3 a z z 7 3 a 7 5 3 a a z a z a a a a a 5 5 5 7 3 2 z 3 z 2 z z > a z - ---- + ---- - ---- + -- 5 3 a 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 71]][a, z]
Out[9]=	-8 3 1 1 2 2 a 2 z 2 z 3 z 4 z 2 3 + a - -- + ---- + ---- - --- - --- + --- + --- - --- + --- + 7 a z - 4 z - 4 5 3 a z z 7 5 3 a a a z a z a a a 2 2 2 2 3 3 3 3 3 4 z 3 z 15 z 10 z z 8 z 20 z 5 z 3 z 3 > ---- - ---- + ----- + ----- + -- - ---- - ----- - ---- - ---- - 9 a z - 8 6 4 2 9 7 5 3 a a a a a a a a a 4 4 4 4 4 5 5 5 5 4 z 9 z 5 z 38 z 36 z 4 z 16 z 32 z 3 z > 3 z - --- + ---- + ---- - ----- - ----- - ---- + ----- + ----- + ---- - 10 8 6 4 2 9 7 5 3 a a a a a a a a a 5 6 6 6 6 7 7 7 4 z 5 6 9 z 9 z 46 z 34 z 12 z 7 z 14 z > ---- + 5 a z + 6 z - ---- + ---- + ----- + ----- - ----- - ---- + ----- + a 8 6 4 2 7 5 3 a a a a a a a 7 8 8 8 9 9 9 10 10 8 z 7 8 10 z 16 z 8 z 5 z 8 z 3 z z z > ---- - a z - 2 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]=	2 2 4 1 1 2 2 5 5 5 q 4 10 q + 6 q + ----- + ----- + ----- + -- + ----- + - + ---- + 9 q t + 6 4 4 4 4 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 + 9 q t + 7 q t + 10 q t + 6 q t + 7 q t + 12 5 14 5 14 6 16 6 18 7 > 3 q t + 6 q t + q t + 3 q t + q t

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