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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X20,13,21,14 X7,17,8,16 X19,9,20,8 X9,19,10,18 X17,11,18,10 X22,15,5,16 X14,21,15,22 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

- 2q-11/2 + 3q-9/2 - 4q-7/2 + 4q-5/2 - 5q-3/2 + 4q-1/2 - 3q1/2 + 2q3/2 - q5/2

A2 (sl(3)) Invariant:

q-22 + q-20 + 2q-18 + q-16 - q-14 - q-10 + 2q-8 + 2q-6 + q-4 + q-2 - 1 + q8

HOMFLY-PT Polynomial:

- 2a-1z - a-1z3 - az-1 + az + 3az3 + az5 + 2a3z-1 + 2a3z + 3a3z3 + a3z5 - 2a5z-1 - 3a5z - a5z3 + a7z-1

Kauffman Polynomial:

2a-1z - 7a-1z3 + 5a-1z5 - a-1z7 + 5z2 - 14z4 + 10z6 - 2z8 + az-1 - 2az - az3 + 3az7 - az9 + 4a2z2 - 12a2z4 + 12a2z6 - 3a2z8 + 2a3z-1 - 10a3z + 17a3z3 - 9a3z5 + 4a3z7 - a3z9 + a4 - 2a4z2 + a4z4 + 2a4z6 - a4z8 + 2a5z-1 - 9a5z + 11a5z3 - 4a5z5 - a6z2 - a6z4 + a7z-1 - 3a7z

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 6                 1 j = 4               1   j = 2             2 1   j = 0           2 1     j = -2         3 2       j = -4       2 3         j = -6     2 2           j = -8   1 2             j = -10 1 2               j = -12 2

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, NonAlternating, 100]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 100]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[20, 13, 21, 14], X[7, 17, 8, 16], > X[19, 9, 20, 8], X[9, 19, 10, 18], X[17, 11, 18, 10], X[22, 15, 5, 16], > X[14, 21, 15, 22], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -4, 5, -6, 7, 11, -2, 3, -9, 8, 4, -7, 6, > -5, -3, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-2 3 4 4 5 4 3/2 5/2 ----- + ---- - ---- + ---- - ---- + ------- - 3 Sqrt[q] + 2 q - q 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 2 -16 -14 -10 2 2 -4 -2 8 -1 + q + q + --- + q - q - q + -- + -- + q + q + q 18 8 6 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 100]][a, z]
Out[8]=	3 5 7 3 a 2 a 2 a a 2 z 3 5 z 3 3 3 -(-) + ---- - ---- + -- - --- + a z + 2 a z - 3 a z - -- + 3 a z + 3 a z - z z z z a a 5 3 5 3 5 > a z + a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 100]][a, z]
Out[9]=	3 5 7 4 a 2 a 2 a a 2 z 3 5 7 2 a + - + ---- + ---- + -- + --- - 2 a z - 10 a z - 9 a z - 3 a z + 5 z + z z z z a 3 2 2 4 2 6 2 7 z 3 3 3 5 3 4 > 4 a z - 2 a z - a z - ---- - a z + 17 a z + 11 a z - 14 z - a 5 2 4 4 4 6 4 5 z 3 5 5 5 6 2 6 > 12 a z + a z - a z + ---- - 9 a z - 4 a z + 10 z + 12 a z + a 7 4 6 z 7 3 7 8 2 8 4 8 9 3 9 > 2 a z - -- + 3 a z + 4 a z - 2 z - 3 a z - a z - a z - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 2 1 2 1 2 2 2 2 -- + -- + ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 4 2 12 4 10 4 10 3 8 3 8 2 6 2 6 4 q q q t q t q t q t q t q t q t q t 2 t 2 2 2 2 3 4 3 6 4 > 2 t + --- + t + 2 q t + q t + q t + q t 2 q

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