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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 2q-5/2 - 6q-3/2 + 9q-1/2 - 11q1/2 + 12q3/2 - 12q5/2 + 9q7/2 - 6q9/2 + 3q11/2 - q13/2

A2 (sl(3)) Invariant:

q-12 + q-10 + 4q-6 + q-4 - q-2 + 2 - 3q2 + q4 - q6 + 2q10 - 2q12 + 3q14 + q16 - q18 + q20

HOMFLY-PT Polynomial:

- a-5z-1 - a-5z - a-5z3 + 2a-3z-1 + 4a-3z + 2a-3z3 + a-3z5 - a-1z-1 - 2a-1z + a-1z5 - az-1 - 2az - 2az3 + a3z-1 + a3z

Kauffman Polynomial:

2a-7z3 - a-7z5 - 2a-6z2 + 6a-6z4 - 3a-6z6 - a-5z-1 + 5a-5z - 11a-5z3 + 12a-5z5 - 5a-5z7 - a-4 + a-4z2 - 3a-4z4 + 6a-4z6 - 4a-4z8 - 2a-3z-1 + 15a-3z - 33a-3z3 + 28a-3z5 - 9a-3z7 - a-3z9 - 3a-2 + 11a-2z2 - 16a-2z4 + 14a-2z6 - 7a-2z8 - a-1z-1 + 10a-1z - 21a-1z3 + 19a-1z5 - 7a-1z7 - a-1z9 - 2 + 8z2 - 4z4 + 3z6 - 3z8 + az-1 - 3az + 2az3 + 3az5 - 3az7 - a2 + 3a2z4 - 2a2z6 + a3z-1 - 3a3z + 3a3z3 - a3z5

Khovanov Homology:

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

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