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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X17,1,18,4 X9,14,10,15 X3849 X5,11,6,10 X11,20,12,21 X19,22,20,5 X13,19,14,18 X21,12,22,13 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-24 - q-22 - q-18 + q-16 + q-14 + 2q-10 + q-8 + 2q-6 + q-4 + 2 + q4 + q6

HOMFLY-PT Polynomial:

- a-1z-1 - a-1z + 2az-1 + 4az + 2az3 - 3a3z-1 - 7a3z - 4a3z3 - a3z5 + 3a5z-1 + 5a5z + 2a5z3 - a7z-1 - a7z

Kauffman Polynomial:

- a-1z-1 + 2a-1z - a-1z3 + 3z2 - 2z4 - 2az-1 + 9az - 10az3 + 4az5 - az7 - 2a2 + 11a2z2 - 19a2z4 + 10a2z6 - 2a2z8 - 3a3z-1 + 16a3z - 21a3z3 + 6a3z5 + 2a3z7 - a3z9 + 8a4z2 - 24a4z4 + 18a4z6 - 4a4z8 - 3a5z-1 + 12a5z - 19a5z3 + 10a5z5 + a5z7 - a5z9 + 2a6 - 3a6z2 - 3a6z4 + 7a6z6 - 2a6z8 - a7z-1 + 3a7z - 7a7z3 + 8a7z5 - 2a7z7 + a8 - 3a8z2 + 4a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 4                   1 j = 2                 1   j = 0               3 1   j = -2             3 3     j = -4           3 2 1     j = -6         3 4         j = -8       2 3           j = -10     1 3 1           j = -12   1 2               j = -14   1                 j = -16 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, NonAlternating, 8]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 8]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[17, 1, 18, 4], X[9, 14, 10, 15], > X[3, 8, 4, 9], X[5, 11, 6, 10], X[11, 20, 12, 21], X[19, 22, 20, 5], > X[13, 19, 14, 18], X[21, 12, 22, 13], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, -5, 3}, {-6, -1, 2, 5, -4, 6, -7, 10, -9, 4, 11, -2, -3, 9, > -8, 7, -10, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 2 3 4 5 6 4 4 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 2 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 -18 -16 -14 2 -8 2 -4 4 6 2 - q - q - q + q + q + --- + q + -- + q + q + q 10 6 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 8]][a, z]
Out[8]=	3 5 7 1 2 a 3 a 3 a a z 3 5 7 3 -(---) + --- - ---- + ---- - -- - - + 4 a z - 7 a z + 5 a z - a z + 2 a z - a z z z z z a 3 3 5 3 3 5 > 4 a z + 2 a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 8]][a, z]
Out[9]=	3 5 7 2 6 8 1 2 a 3 a 3 a a 2 z 3 -2 a + 2 a + a - --- - --- - ---- - ---- - -- + --- + 9 a z + 16 a z + a z z z z z a 3 5 7 2 2 2 4 2 6 2 8 2 z > 12 a z + 3 a z + 3 z + 11 a z + 8 a z - 3 a z - 3 a z - -- - a 3 3 3 5 3 7 3 4 2 4 4 4 > 10 a z - 21 a z - 19 a z - 7 a z - 2 z - 19 a z - 24 a z - 6 4 8 4 5 3 5 5 5 7 5 2 6 > 3 a z + 4 a z + 4 a z + 6 a z + 10 a z + 8 a z + 10 a z + 4 6 6 6 8 6 7 3 7 5 7 7 7 2 8 > 18 a z + 7 a z - a z - a z + 2 a z + a z - 2 a z - 2 a z - 4 8 6 8 3 9 5 9 > 4 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 3 1 1 1 2 1 3 2 3 + q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 q q t q t q t q t q t q t q t 1 3 3 4 3 2 3 2 4 2 > ------ + ----- + ----- + ----- + ----- + ---- + ---- + t + q t + q t 10 3 8 3 6 3 6 2 4 2 4 2 q t q t q t q t q t q t q t

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