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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X4,9,5,10 X14,6,15,5 X18,14,9,13 X16,8,17,7 X6,16,7,15 X8,18,1,17

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 2q-5/2 - 3q-3/2 + 4q-1/2 - 6q1/2 + 5q3/2 - 5q5/2 + 3q7/2 - 2q9/2 + q11/2

A2 (sl(3)) Invariant:

q-10 + q-6 + 2 + 3q4 + q6 + 2q8 + q10 - q12 - q16

HOMFLY-PT Polynomial:

4a-3z + 4a-3z3 + a-3z5 - a-1z-1 - 8a-1z - 12a-1z3 - 6a-1z5 - a-1z7 + az-1 + 4az + 4az3 + az5

Kauffman Polynomial:

2a-6z2 - a-6z4 - a-5z + 4a-5z3 - 2a-5z5 - a-4z2 + 3a-4z4 - 2a-4z6 + 4a-3z - 8a-3z3 + 5a-3z5 - 2a-3z7 - 3a-2z2 + a-2z6 - a-2z8 - a-1z-1 + 10a-1z - 21a-1z3 + 14a-1z5 - 4a-1z7 + 1 - 3z2 + 2z4 + z6 - z8 - az-1 + 4az - 6az3 + 6az5 - 2az7 - 3a2z2 + 6a2z4 - 2a2z6 - a3z + 3a3z3 - a3z5

Khovanov Homology:

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

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