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DrawMorseLink

PD Presentation:

X8192 X16,6,17,5 X18,10,19,9 X14,19,15,20 X10,16,11,15 X20,11,7,12 X4758 X2,14,3,13 X12,4,13,3 X6,18,1,17

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-10 - 2q-8 + 3q-6 - q-2 + 5 - 3q2 + 3q4 - 2q6 + 3q10 - 2q12 + 4q14 - q18 + q20

HOMFLY-PT Polynomial:

- a-5z-1 - a-5z - a-5z3 + 2a-3z-1 + 5a-3z + 5a-3z3 + 2a-3z5 - 2a-1z-1 - 5a-1z - 7a-1z3 - 4a-1z5 - a-1z7 + az-1 + 2az + 2az3 + az5

Kauffman Polynomial:

a-7z3 - a-7z5 - a-6z2 + 5a-6z4 - 4a-6z6 - a-5z-1 + 4a-5z - 9a-5z3 + 14a-5z5 - 8a-5z7 + a-4z2 - 4a-4z4 + 11a-4z6 - 8a-4z8 - 2a-3z-1 + 12a-3z - 31a-3z3 + 36a-3z5 - 12a-3z7 - 3a-3z9 - a-2 + 4a-2z2 - 18a-2z4 + 30a-2z6 - 16a-2z8 - 2a-1z-1 + 12a-1z - 31a-1z3 + 36a-1z5 - 12a-1z7 - 3a-1z9 + z2 - 4z4 + 11z6 - 8z8 - az-1 + 4az - 9az3 + 14az5 - 8az7 - a2z2 + 5a2z4 - 4a2z6 + a3z3 - a3z5

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 j = 14                     1 j = 12                   3   j = 10                 6 1   j = 8               7 3     j = 6             10 6       j = 4           9 8         j = 2         8 9           j = 0       6 10             j = -2     3 7               j = -4   1 6                 j = -6   3                   j = -8 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[10, Alternating, 86]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 86]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 6, 17, 5], X[18, 10, 19, 9], X[14, 19, 15, 20], > X[10, 16, 11, 15], X[20, 11, 7, 12], X[4, 7, 5, 8], X[2, 14, 3, 13], > X[12, 4, 13, 3], X[6, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 9, -7, 2, -10}, > {7, -1, 3, -5, 6, -9, 8, -4, 5, -2, 10, -3, 4, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 4 9 13 3/2 5/2 7/2 -q + ---- - ---- + ------- - 17 Sqrt[q] + 18 q - 17 q + 13 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 > 9 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 2 3 -2 2 4 6 10 12 14 18 5 + q - -- + -- - q - 3 q + 3 q - 2 q + 3 q - 2 q + 4 q - q + 8 6 q q 20 > q
In[8]:=	HOMFLYPT[Link[10, Alternating, 86]][a, z]
Out[8]=	3 3 3 1 2 2 a z 5 z 5 z z 5 z 7 z 3 -(----) + ---- - --- + - - -- + --- - --- + 2 a z - -- + ---- - ---- + 2 a z + 5 3 a z z 5 3 a 5 3 a a z a z a a a a 5 5 7 2 z 4 z 5 z > ---- - ---- + a z - -- 3 a a a
In[9]:=	Kauffman[Link[10, Alternating, 86]][a, z]
Out[9]=	2 2 -2 1 2 2 a 4 z 12 z 12 z 2 z z -a - ---- - ---- - --- - - + --- + ---- + ---- + 4 a z + z - -- + -- + 5 3 a z z 5 3 a 6 4 a z a z a a a a 2 3 3 3 3 4 4 z 2 2 z 9 z 31 z 31 z 3 3 3 4 5 z > ---- - a z + -- - ---- - ----- - ----- - 9 a z + a z - 4 z + ---- - 2 7 5 3 a 6 a a a a a 4 4 5 5 5 5 4 z 18 z 2 4 z 14 z 36 z 36 z 5 3 5 > ---- - ----- + 5 a z - -- + ----- + ----- + ----- + 14 a z - a z + 4 2 7 5 3 a a a a a a 6 6 6 7 7 7 6 4 z 11 z 30 z 2 6 8 z 12 z 12 z 7 > 11 z - ---- + ----- + ----- - 4 a z - ---- - ----- - ----- - 8 a z - 6 4 2 5 3 a a a a a a 8 8 9 9 8 8 z 16 z 3 z 3 z > 8 z - ---- - ----- - ---- - ---- 4 2 3 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 6 3 6 7 2 10 + 8 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 9 q t + 8 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 4 2 6 2 6 3 8 3 8 4 10 4 > 9 q t + 8 q t + 10 q t + 6 q t + 7 q t + 3 q t + 6 q t + 10 5 12 5 14 6 > q t + 3 q t + q t

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