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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X5,10,6,11 X8493 X13,22,14,5 X11,20,12,21 X21,12,22,13 X19,14,20,15 X9,18,10,19 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

q-21/2 - 2q-19/2 + 3q-17/2 - 3q-15/2 + 4q-13/2 - 4q-11/2 + 3q-9/2 - 3q-7/2 - q-3/2

A2 (sl(3)) Invariant:

- q-32 - q-28 - q-26 - 2q-22 - q-20 - q-18 + q-16 + 3q-14 + 3q-12 + 4q-10 + 3q-8 + q-6 + q-4

HOMFLY-PT Polynomial:

- 2a3z-1 - 3a3z - a3z3 + 2a5z-1 - a5z3 + a7z-1 + 5a7z + 4a7z3 + a7z5 - a9z-1 - 2a9z - a9z3

Kauffman Polynomial:

- 2a3z-1 + 3a3z - a3z3 + 3a4 - 2a4z2 - 2a5z-1 + 2a5z3 - a5z5 + 8a6z2 - 13a6z4 + 6a6z6 - a6z8 + a7z-1 - 4a7z + 5a7z3 - 8a7z5 + 5a7z7 - a7z9 - 3a8 + 13a8z2 - 23a8z4 + 15a8z6 - 3a8z8 + a9z-1 - 4a9z3 + a9z5 + 3a9z7 - a9z9 - 6a10z4 + 8a10z6 - 2a10z8 + a11z - 6a11z3 + 8a11z5 - 2a11z7 + a12 - 3a12z2 + 4a12z4 - a12z6

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -2                   1 j = -4                 1 2 j = -6               2   1 j = -8             2 2     j = -10           3 2       j = -12         2 3 1       j = -14       2 3           j = -16     1 2 1           j = -18   1 2               j = -20   1                 j = -22 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, 12]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 12]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[5, 10, 6, 11], > X[8, 4, 9, 3], X[13, 22, 14, 5], X[11, 20, 12, 21], X[21, 12, 22, 13], > X[19, 14, 20, 15], X[9, 18, 10, 19], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {-4, -1, 2, -5, -10, 4, -7, 8, -6, 9, 11, -2, 3, 10, > -9, 7, -8, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 2 3 3 4 4 3 3 -(3/2) q - ----- + ----- - ----- + ----- - ----- + ---- - ---- - q 19/2 17/2 15/2 13/2 11/2 9/2 7/2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -28 -26 2 -20 -18 -16 3 3 4 3 -6 -q - q - q - --- - q - q + q + --- + --- + --- + -- + q + 22 14 12 10 8 q q q q q -4 > q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 12]][a, z]
Out[8]=	3 5 7 9 -2 a 2 a a a 3 7 9 3 3 5 3 7 3 ----- + ---- + -- - -- - 3 a z + 5 a z - 2 a z - a z - a z + 4 a z - z z z z 9 3 7 5 > a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 12]][a, z]
Out[9]=	3 5 7 9 4 8 12 2 a 2 a a a 3 7 11 4 2 3 a - 3 a + a - ---- - ---- + -- + -- + 3 a z - 4 a z + a z - 2 a z + z z z z 6 2 8 2 12 2 3 3 5 3 7 3 9 3 > 8 a z + 13 a z - 3 a z - a z + 2 a z + 5 a z - 4 a z - 11 3 6 4 8 4 10 4 12 4 5 5 7 5 > 6 a z - 13 a z - 23 a z - 6 a z + 4 a z - a z - 8 a z + 9 5 11 5 6 6 8 6 10 6 12 6 7 7 > a z + 8 a z + 6 a z + 15 a z + 8 a z - a z + 5 a z + 9 7 11 7 6 8 8 8 10 8 7 9 9 9 > 3 a z - 2 a z - a z - 3 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 2 -2 1 1 1 2 1 2 2 q + -- + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q t q t q t q t q t q t q t 1 3 2 3 3 1 2 2 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 16 5 14 5 12 5 12 4 10 4 12 3 10 3 8 3 q t q t q t q t q t q t q t q t 2 2 1 > ----- + ----- + ---- 8 2 6 2 4 q t q t q t

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