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DrawMorseLink

PD Presentation:

X10,1,11,2 X12,4,13,3 X18,5,9,6 X6,9,7,10 X16,12,17,11 X14,8,15,7 X4,14,5,13 X8,16,1,15 X2,17,3,18

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 3q-5/2 - 6q-3/2 + 8q-1/2 - 10q1/2 + 9q3/2 - 9q5/2 + 6q7/2 - 3q9/2 + q11/2

A2 (sl(3)) Invariant:

q-10 - q-8 + 2q-6 + 3 - q2 + 4q4 + 2q8 + q10 - 2q12 + q14 - q16

HOMFLY-PT Polynomial:

3a-3z + 3a-3z3 + a-3z5 - a-1z-1 - 6a-1z - 9a-1z3 - 5a-1z5 - a-1z7 + az-1 + 3az + 3az3 + az5

Kauffman Polynomial:

a-6z2 - a-6z4 - a-5z + 3a-5z3 - 3a-5z5 - 3a-4z2 + 6a-4z4 - 5a-4z6 + 2a-3z - 4a-3z3 + 6a-3z5 - 5a-3z7 - 6a-2z2 + 12a-2z4 - 5a-2z6 - 2a-2z8 - a-1z-1 + 6a-1z - 12a-1z3 + 17a-1z5 - 9a-1z7 + 1 - 5z2 + 11z4 - 3z6 - 2z8 - az-1 + 2az - 3az3 + 7az5 - 4az7 - 3a2z2 + 6a2z4 - 3a2z6 - a3z + 2a3z3 - a3z5

Khovanov Homology:

trqj 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                 2   j = 8               4 1   j = 6             5 2     j = 4           4 4       j = 2         6 5         j = 0       4 6           j = -2     2 4             j = -4   1 4               j = -6   2                 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[9, Alternating, 42]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 42]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[18, 5, 9, 6], X[6, 9, 7, 10], > X[16, 12, 17, 11], X[14, 8, 15, 7], X[4, 14, 5, 13], X[8, 16, 1, 15], > X[2, 17, 3, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -7, 3, -4, 6, -8}, {4, -1, 5, -2, 7, -6, 8, -5, 9, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 3 6 8 3/2 5/2 7/2 -q + ---- - ---- + ------- - 10 Sqrt[q] + 9 q - 9 q + 6 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -8 2 2 4 8 10 12 14 16 3 + q - q + -- - q + 4 q + 2 q + q - 2 q + q - q 6 q
In[8]:=	HOMFLYPT[Link[9, Alternating, 42]][a, z]
Out[8]=	3 3 5 5 7 1 a 3 z 6 z 3 z 9 z 3 z 5 z 5 z -(---) + - + --- - --- + 3 a z + ---- - ---- + 3 a z + -- - ---- + a z - -- a z z 3 a 3 a 3 a a a a a
In[9]:=	Kauffman[Link[9, Alternating, 42]][a, z]
Out[9]=	2 2 2 1 a z 2 z 6 z 3 2 z 3 z 6 z 1 - --- - - - -- + --- + --- + 2 a z - a z - 5 z + -- - ---- - ---- - a z z 5 3 a 6 4 2 a a a a a 3 3 3 4 4 2 2 3 z 4 z 12 z 3 3 3 4 z 6 z > 3 a z + ---- - ---- - ----- - 3 a z + 2 a z + 11 z - -- + ---- + 5 3 a 6 4 a a a a 4 5 5 5 6 12 z 2 4 3 z 6 z 17 z 5 3 5 6 5 z > ----- + 6 a z - ---- + ---- + ----- + 7 a z - a z - 3 z - ---- - 2 5 3 a 4 a a a a 6 7 7 8 5 z 2 6 5 z 9 z 7 8 2 z > ---- - 3 a z - ---- - ---- - 4 a z - 2 z - ---- 2 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 4 2 4 4 2 4 6 + 6 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 5 q t + 4 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 12 5 > 4 q t + 5 q t + 2 q t + 4 q t + q t + 2 q t + q t

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