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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X9,21,10,20 X3849 X21,18,22,19 X11,14,12,15 X5,13,6,12 X13,5,14,22 X19,11,20,10 X15,2,16,3

Gauss Code:

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

Jones Polynomial:

q-13/2 - 3q-11/2 + 3q-9/2 - 6q-7/2 + 6q-5/2 - 5q-3/2 + 5q-1/2 - 4q1/2 + 2q3/2 - q5/2

A2 (sl(3)) Invariant:

- q-24 + q-22 + q-18 + 2q-16 + 2q-14 + 4q-12 + q-8 - 2q-6 - 2q-4 + q2 + q6 + q8

HOMFLY-PT Polynomial:

- a-1z-1 - 3a-1z - a-1z3 + 3az-1 + 7az + 8az3 + 2az5 - 4a3z-1 - 8a3z - 8a3z3 - 5a3z5 - a3z7 + 2a5z-1 + 3a5z + 4a5z3 + a5z5 - a7z

Kauffman Polynomial:

- a-1z-1 + 5a-1z - 8a-1z3 + 5a-1z5 - a-1z7 - 1 + 5z2 - 11z4 + 9z6 - 2z8 - 3az-1 + 16az - 28az3 + 16az5 - az9 - 3a2 + 9a2z2 - 23a2z4 + 21a2z6 - 5a2z8 - 4a3z-1 + 23a3z - 37a3z3 + 23a3z5 - 2a3z7 - a3z9 - 2a4 + 4a4z2 - 11a4z4 + 11a4z6 - 3a4z8 - 2a5z-1 + 15a5z - 20a5z3 + 12a5z5 - 3a5z7 - a6 - a6z2 + a6z4 - a6z6 + 3a7z - 3a7z3 - a8z2

Khovanov Homology:

trqj r = -5 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               3 1   j = 0             2 1     j = -2         1 4 3       j = -4         4 3         j = -6       3 3           j = -8   1 2 4             j = -10   2 2               j = -12   2                 j = -14 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, 22]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 22]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[9, 21, 10, 20], > X[3, 8, 4, 9], X[21, 18, 22, 19], X[11, 14, 12, 15], X[5, 13, 6, 12], > X[13, 5, 14, 22], X[19, 11, 20, 10], X[15, 2, 16, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -5, -3}, {-8, -1, 2, 5, -4, 10, -7, 8, -9, 7, -11, -2, 3, 6, > -10, 4, -6, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 3 3 6 6 5 5 3/2 q - ----- + ---- - ---- + ---- - ---- + ------- - 4 Sqrt[q] + 2 q - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 -18 2 2 4 -8 2 2 2 6 8 -q + q + q + --- + --- + --- + q - -- - -- + q + q + q 16 14 12 6 4 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 22]][a, z]
Out[8]=	3 5 3 1 3 a 4 a 2 a 3 z 3 5 7 z -(---) + --- - ---- + ---- - --- + 7 a z - 8 a z + 3 a z - a z - -- + a z z z z a a 3 3 3 5 3 5 3 5 5 5 3 7 > 8 a z - 8 a z + 4 a z + 2 a z - 5 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 22]][a, z]
Out[9]=	3 5 2 4 6 1 3 a 4 a 2 a 5 z 3 -1 - 3 a - 2 a - a - --- - --- - ---- - ---- + --- + 16 a z + 23 a z + a z z z z a 3 5 7 2 2 2 4 2 6 2 8 2 8 z > 15 a z + 3 a z + 5 z + 9 a z + 4 a z - a z - a z - ---- - a 3 3 3 5 3 7 3 4 2 4 4 4 > 28 a z - 37 a z - 20 a z - 3 a z - 11 z - 23 a z - 11 a z + 5 6 4 5 z 5 3 5 5 5 6 2 6 4 6 > a z + ---- + 16 a z + 23 a z + 12 a z + 9 z + 21 a z + 11 a z - a 7 6 6 z 3 7 5 7 8 2 8 4 8 9 3 9 > a z - -- - 2 a z - 3 a z - 2 z - 5 a z - 3 a z - a z - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 1 2 2 1 2 2 4 3 -- + -- + ------ + ------ + ------ + ----- + ------ + ----- + ----- + ----- + 4 2 14 5 12 4 10 4 8 4 10 3 8 3 8 2 6 2 q q q t q t q t q t q t q t q t q t 3 4 1 3 t 2 2 2 2 3 4 3 6 4 > ---- + ---- + ---- + 2 t + --- + t + 3 q t + q t + q t + q t 6 4 2 2 q t q t q t q

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