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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X18,11,19,12 X12,6,13,5 X4,18,5,17 X14,7,15,8 X16,14,17,13 X20,15,7,16 X6,19,1,20

Gauss Code:

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

Jones Polynomial:

q-17/2 - 3q-15/2 + 6q-13/2 - 10q-11/2 + 12q-9/2 - 13q-7/2 + 12q-5/2 - 10q-3/2 + 6q-1/2 - 4q1/2 + q3/2

A2 (sl(3)) Invariant:

- q-26 + q-22 - q-20 + 3q-18 - q-16 + q-12 - 2q-10 + 4q-8 - q-6 + 3q-4 + 2q-2 + 2q2 - q4

HOMFLY-PT Polynomial:

- az-1 - az + 2az3 + az5 + a3z-1 - 2a3z - 5a3z3 - 4a3z5 - a3z7 + 4a5z + 6a5z3 + 2a5z5 - 2a7z - a7z3

Kauffman Polynomial:

2z4 - z6 + az-1 - az - 8az3 + 12az5 - 4az7 - a2 + 3a2z2 - 10a2z4 + 14a2z6 - 5a2z8 + a3z-1 + 2a3z - 19a3z3 + 23a3z5 - 4a3z7 - 2a3z9 + 6a4z2 - 19a4z4 + 25a4z6 - 10a4z8 + 5a5z - 18a5z3 + 21a5z5 - 6a5z7 - 2a5z9 - a6z4 + 5a6z6 - 5a6z8 + a7z - 4a7z3 + 7a7z5 - 6a7z7 - 2a8z2 + 5a8z4 - 5a8z6 - a9z + 3a9z3 - 3a9z5 + a10z2 - a10z4

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                     1 j = 2                   3   j = 0                 3 1   j = -2               7 3     j = -4             6 4       j = -6           7 6         j = -8         6 7           j = -10       4 6             j = -12     2 6               j = -14   1 4                 j = -16   2                   j = -18 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, 54]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 54]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[18, 11, 19, 12], > X[12, 6, 13, 5], X[4, 18, 5, 17], X[14, 7, 15, 8], X[16, 14, 17, 13], > X[20, 15, 7, 16], X[6, 19, 1, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -6, 5, -10}, > {7, -1, 2, -3, 4, -5, 8, -7, 9, -8, 6, -4, 10, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 3 6 10 12 13 12 10 6 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q 3/2 > 4 Sqrt[q] + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -22 -20 3 -16 -12 2 4 -6 3 2 2 4 -q + q - q + --- - q + q - --- + -- - q + -- + -- + 2 q - q 18 10 8 4 2 q q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 54]][a, z]
Out[8]=	3 a a 3 5 7 3 3 3 5 3 -(-) + -- - a z - 2 a z + 4 a z - 2 a z + 2 a z - 5 a z + 6 a z - z z 7 3 5 3 5 5 5 3 7 > a z + a z - 4 a z + 2 a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 54]][a, z]
Out[9]=	3 2 a a 3 5 7 9 2 2 4 2 -a + - + -- - a z + 2 a z + 5 a z + a z - a z + 3 a z + 6 a z - z z 8 2 10 2 3 3 3 5 3 7 3 9 3 > 2 a z + a z - 8 a z - 19 a z - 18 a z - 4 a z + 3 a z + 4 2 4 4 4 6 4 8 4 10 4 5 > 2 z - 10 a z - 19 a z - a z + 5 a z - a z + 12 a z + 3 5 5 5 7 5 9 5 6 2 6 4 6 > 23 a z + 21 a z + 7 a z - 3 a z - z + 14 a z + 25 a z + 6 6 8 6 7 3 7 5 7 7 7 2 8 > 5 a z - 5 a z - 4 a z - 4 a z - 6 a z - 6 a z - 5 a z - 4 8 6 8 3 9 5 9 > 10 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 7 1 2 1 4 2 6 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 10 4 q q q t q t q t q t q t q t q t 6 6 7 7 6 6 3 t 2 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 3 t + --- + t + 3 q t + 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q 4 3 > q t

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