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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 2q-9/2 - 6q-7/2 + 9q-5/2 - 14q-3/2 + 17q-1/2 - 17q1/2 + 16q3/2 - 13q5/2 + 8q7/2 - 4q9/2 + q11/2

A2 (sl(3)) Invariant:

q-18 + 2q-16 + q-14 + 3q-12 + 2q-10 - q-8 + 3q-6 - 3q-4 - q-2 - 3q2 + 4q4 - 2q6 + 3q8 + q10 - 2q12 + 2q14 - q16

HOMFLY-PT Polynomial:

a-3z + 2a-3z3 + a-3z5 - a-1z-1 - 6a-1z - 7a-1z3 - 4a-1z5 - a-1z7 + 3az-1 + 10az + 10az3 + 3az5 - 4a3z-1 - 8a3z - 3a3z3 + 2a5z-1 + a5z

Kauffman Polynomial:

- a-6z4 + 2a-5z3 - 4a-5z5 - a-4z2 + 7a-4z4 - 8a-4z6 - 6a-3z3 + 15a-3z5 - 11a-3z7 + a-2 + a-2z2 - 7a-2z4 + 15a-2z6 - 10a-2z8 - a-1z-1 + 4a-1z - 16a-1z3 + 21a-1z5 - 2a-1z7 - 5a-1z9 + 3 + 4z2 - 28z4 + 38z6 - 13z8 - z10 - 3az-1 + 12az - 22az3 + 9az5 + 12az7 - 7az9 + 3a2 + a2z2 - 19a2z4 + 22a2z6 - 5a2z8 - a2z10 - 4a3z-1 + 15a3z - 23a3z3 + 12a3z5 + 2a3z7 - 2a3z9 + 2a4 - a4z2 - 6a4z4 + 7a4z6 - 2a4z8 - 2a5z-1 + 7a5z - 9a5z3 + 5a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12                       1 j = 10                     3   j = 8                   5 1   j = 6                 8 3     j = 4               8 5       j = 2             9 8         j = 0           9 9           j = -2         5 8             j = -4       4 9               j = -6     2 5                 j = -8   1 5                   j = -10   1                     j = -12 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, 59]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 59]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[14, 8, 15, 7], X[20, 16, 21, 15], > X[18, 9, 19, 10], X[8, 19, 9, 20], X[22, 18, 5, 17], X[16, 22, 17, 21], > X[10, 14, 11, 13], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -6, 5, -9, 11, -2, 9, -3, 4, -8, 7, -5, > 6, -4, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 2 6 9 14 17 3/2 -q + ---- - ---- + ---- - ---- + ------- - 17 Sqrt[q] + 16 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 13 q + 8 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 -14 3 2 -8 3 3 -2 2 4 6 q + --- + q + --- + --- - q + -- - -- - q - 3 q + 4 q - 2 q + 16 12 10 6 4 q q q q q 8 10 12 14 16 > 3 q + q - 2 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 59]][a, z]
Out[8]=	3 5 3 3 1 3 a 4 a 2 a z 6 z 3 5 2 z 7 z -(---) + --- - ---- + ---- + -- - --- + 10 a z - 8 a z + a z + ---- - ---- + a z z z z 3 a 3 a a a 5 5 7 3 3 3 z 4 z 5 z > 10 a z - 3 a z + -- - ---- + 3 a z - -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 59]][a, z]
Out[9]=	3 5 -2 2 4 1 3 a 4 a 2 a 4 z 3 3 + a + 3 a + 2 a - --- - --- - ---- - ---- + --- + 12 a z + 15 a z + a z z z z a 2 2 3 3 3 5 2 z z 2 2 4 2 2 z 6 z 16 z 3 > 7 a z + 4 z - -- + -- + a z - a z + ---- - ---- - ----- - 22 a z - 4 2 5 3 a a a a a 4 4 4 5 3 3 5 3 4 z 7 z 7 z 2 4 4 4 4 z > 23 a z - 9 a z - 28 z - -- + ---- - ---- - 19 a z - 6 a z - ---- + 6 4 2 5 a a a a 5 5 6 6 15 z 21 z 5 3 5 5 5 6 8 z 15 z > ----- + ----- + 9 a z + 12 a z + 5 a z + 38 z - ---- + ----- + 3 a 4 2 a a a 7 7 2 6 4 6 11 z 2 z 7 3 7 5 7 8 > 22 a z + 7 a z - ----- - ---- + 12 a z + 2 a z - a z - 13 z - 3 a a 8 9 10 z 2 8 4 8 5 z 9 3 9 10 2 10 > ----- - 5 a z - 2 a z - ---- - 7 a z - 2 a z - z - a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 5 2 5 4 9 9 + 9 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 5 9 8 2 4 4 2 6 2 6 3 > ----- + - + ---- + 8 q t + 8 q t + 5 q t + 8 q t + 3 q t + 2 2 t 2 q t q t 8 3 8 4 10 4 12 5 > 5 q t + q t + 3 q t + q t

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