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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X20,8,21,7 X22,17,5,18 X18,21,19,22 X16,13,17,14 X14,9,15,10 X10,15,11,16 X8,20,9,19 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

q-21/2 - 3q-19/2 + 7q-17/2 - 13q-15/2 + 17q-13/2 - 20q-11/2 + 20q-9/2 - 18q-7/2 + 13q-5/2 - 8q-3/2 + 3q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-34 - 2q-32 + q-30 + q-28 - q-26 + 6q-24 - q-20 + 2q-18 - 4q-16 + 3q-14 - 2q-12 + 2q-10 + 4q-8 - 3q-6 + 4q-4 - 1 + q2

HOMFLY-PT Polynomial:

- az - az3 - 2a3z-1 - 4a3z - a3z3 + a3z5 + 5a5z-1 + 10a5z + 8a5z3 + 3a5z5 - 6a7z-1 - 11a7z - 6a7z3 + 4a9z-1 + 4a9z - a11z-1

Kauffman Polynomial:

- az + 2az3 - az5 - a2z2 + 4a2z4 - 3a2z6 - 2a3z-1 + 7a3z - 10a3z3 + 10a3z5 - 6a3z7 + a4 - 5a4z4 + 9a4z6 - 7a4z8 - 5a5z-1 + 23a5z - 41a5z3 + 33a5z5 - 8a5z7 - 4a5z9 + a6 - 15a6z4 + 28a6z6 - 14a6z8 - a6z10 - 6a7z-1 + 29a7z - 52a7z3 + 41a7z5 - 4a7z7 - 7a7z9 + 3a8 - 7a8z2 - 4a8z4 + 22a8z6 - 11a8z8 - a8z10 - 4a9z-1 + 18a9z - 31a9z3 + 27a9z5 - 5a9z7 - 3a9z9 + 3a10 - 9a10z2 + 5a10z4 + 5a10z6 - 4a10z8 - a11z-1 + 4a11z - 8a11z3 + 8a11z5 - 3a11z7 + a12 - 3a12z2 + 3a12z4 - a12z6

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                       1 j = 0                     2   j = -2                   6 1   j = -4                 8 3     j = -6               10 5       j = -8             10 8         j = -10           10 10           j = -12         8 11             j = -14       5 9               j = -16     2 8                 j = -18   1 5                   j = -20   2                     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, Alternating, 92]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 92]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[20, 8, 21, 7], X[22, 17, 5, 18], > X[18, 21, 19, 22], X[16, 13, 17, 14], X[14, 9, 15, 10], X[10, 15, 11, 16], > X[8, 20, 9, 19], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -9, 7, -8, 11, -2, 6, -7, 8, -6, 4, -5, > 9, -3, 5, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 3 7 13 17 20 20 18 13 8 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q 3 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 2 -30 -28 -26 6 -20 2 4 3 2 -1 - q - --- + q + q - q + --- - q + --- - --- + --- - --- + 32 24 18 16 14 12 q q q q q q 2 4 3 4 2 > --- + -- - -- + -- + q 10 8 6 4 q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 92]][a, z]
Out[8]=	3 5 7 9 11 -2 a 5 a 6 a 4 a a 3 5 7 9 ----- + ---- - ---- + ---- - --- - a z - 4 a z + 10 a z - 11 a z + 4 a z - z z z z z 3 3 3 5 3 7 3 3 5 5 5 > a z - a z + 8 a z - 6 a z + a z + 3 a z
In[9]:=	Kauffman[Link[11, Alternating, 92]][a, z]
Out[9]=	3 5 7 9 11 4 6 8 10 12 2 a 5 a 6 a 4 a a 3 a + a + 3 a + 3 a + a - ---- - ---- - ---- - ---- - --- - a z + 7 a z + z z z z z 5 7 9 11 2 2 8 2 10 2 > 23 a z + 29 a z + 18 a z + 4 a z - a z - 7 a z - 9 a z - 12 2 3 3 3 5 3 7 3 9 3 11 3 > 3 a z + 2 a z - 10 a z - 41 a z - 52 a z - 31 a z - 8 a z + 2 4 4 4 6 4 8 4 10 4 12 4 5 > 4 a z - 5 a z - 15 a z - 4 a z + 5 a z + 3 a z - a z + 3 5 5 5 7 5 9 5 11 5 2 6 4 6 > 10 a z + 33 a z + 41 a z + 27 a z + 8 a z - 3 a z + 9 a z + 6 6 8 6 10 6 12 6 3 7 5 7 7 7 > 28 a z + 22 a z + 5 a z - a z - 6 a z - 8 a z - 4 a z - 9 7 11 7 4 8 6 8 8 8 10 8 5 9 > 5 a z - 3 a z - 7 a z - 14 a z - 11 a z - 4 a z - 4 a z - 7 9 9 9 6 10 8 10 > 7 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 6 1 2 1 5 2 8 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q q t q t q t q t q t q t q t 9 8 11 10 10 10 8 10 5 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q t q t q t q t q t q t q t q t q t 8 t 2 2 > ---- + 2 t + -- + q t 4 2 q t q

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