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DrawMorseLink

PD Presentation:

X8192 X12,3,13,4 X20,10,7,9 X10,14,11,13 X16,6,17,5 X18,16,19,15 X14,20,15,19 X2738 X4,11,5,12 X6,18,1,17

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 2q-5/2 - 6q-3/2 + 9q-1/2 - 12q1/2 + 13q3/2 - 13q5/2 + 10q7/2 - 7q9/2 + 4q11/2 - q13/2

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-7z3 - a-7z5 - 2a-6z2 + 7a-6z4 - 4a-6z6 + a-5z - 5a-5z3 + 11a-5z5 - 6a-5z7 - 6a-4z2 + 11a-4z4 - 4a-4z8 + 5a-3z - 21a-3z3 + 27a-3z5 - 11a-3z7 - a-3z9 - 2a-2z2 + 8a-2z6 - 7a-2z8 + 5a-1z - 17a-1z3 + 19a-1z5 - 8a-1z7 - a-1z9 + 2z2 - z4 + 2z6 - 3z8 + az-1 - 2az + az3 + 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                   3   j = 10                 4 1   j = 8               6 3     j = 6             7 4       j = 4           6 6         j = 2         6 7           j = 0       4 7             j = -2     2 5               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, 82]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 82]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[20, 10, 7, 9], X[10, 14, 11, 13], > X[16, 6, 17, 5], X[18, 16, 19, 15], X[14, 20, 15, 19], X[2, 7, 3, 8], > X[4, 11, 5, 12], X[6, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9, 5, -10}, > {8, -1, 3, -4, 9, -2, 4, -7, 6, -5, 10, -6, 7, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 2 6 9 3/2 5/2 7/2 -q + ---- - ---- + ------- - 12 Sqrt[q] + 13 q - 13 q + 10 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 > 7 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 4 -4 -2 2 4 10 12 14 3 + q + q + -- + q - q - 2 q + 2 q + 2 q - 3 q + 2 q - 6 q 18 20 > 2 q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 82]][a, z]
Out[8]=	3 3 3 5 5 a a z z 3 z z 3 z z -(-) + -- + -- - - - 2 a z + a z - -- + -- - 2 a z + -- + -- z z 3 a 5 3 3 a a a a a
In[9]:=	Kauffman[Link[10, Alternating, 82]][a, z]
Out[9]=	3 2 2 2 2 a a z 5 z 5 z 3 2 2 z 6 z 2 z -a + - + -- + -- + --- + --- - 2 a z - 3 a z + 2 z - ---- - ---- - ---- + z z 5 3 a 6 4 2 a a a a a 3 3 3 3 4 4 z 5 z 21 z 17 z 3 3 3 4 7 z 11 z 2 4 > -- - ---- - ----- - ----- + a z + 3 a z - z + ---- + ----- + 3 a z - 7 5 3 a 6 4 a a a a a 5 5 5 5 6 6 z 11 z 27 z 19 z 5 3 5 6 4 z 8 z > -- + ----- + ----- + ----- + 3 a z - a z + 2 z - ---- + ---- - 7 5 3 a 6 2 a a a a a 7 7 7 8 8 9 9 2 6 6 z 11 z 8 z 7 8 4 z 7 z z z > 2 a z - ---- - ----- - ---- - 3 a z - 3 z - ---- - ---- - -- - -- 5 3 a 4 2 3 a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 5 2 4 5 2 4 7 + 6 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 7 q t + 6 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 > 6 q t + 7 q t + 4 q t + 6 q t + 3 q t + 4 q t + q t + 12 5 14 6 > 3 q t + q t

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