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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X16,10,7,9 X2738 X14,12,15,11 X12,5,13,6 X4,13,5,14 X6,16,1,15

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 2q-5/2 - 4q-3/2 + 4q-1/2 - 6q1/2 + 5q3/2 - 4q5/2 + 3q7/2 - q9/2

A2 (sl(3)) Invariant:

q-12 + q-10 + 2q-6 + q-4 + 2q-2 + 3 + q4 - 2q6 - q12 + q14

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-5z3 + 2a-4z2 - 3a-4z4 - 2a-3z + 4a-3z3 - 4a-3z5 + 2a-2z2 + a-2z4 - 3a-2z6 - 6a-1z + 12a-1z3 - 5a-1z5 - a-1z7 - 2z2 + 9z4 - 5z6 + az-1 - 7az + 10az3 - 2az5 - az7 - a2 - 2a2z2 + 5a2z4 - 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 j = 10                 1 j = 8               2   j = 6             2 1   j = 4           3 2     j = 2         3 2       j = 0       2 4         j = -2     2 2           j = -4   1 3             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[8, Alternating, 8]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[8, Alternating, 8]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[16, 10, 7, 9], X[2, 7, 3, 8], > X[14, 12, 15, 11], X[12, 5, 13, 6], X[4, 13, 5, 14], X[6, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -7, 6, -8}, {4, -1, 3, -2, 5, -6, 7, -5, 8, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 2 4 4 3/2 5/2 7/2 9/2 -q + ---- - ---- + ------- - 6 Sqrt[q] + 5 q - 4 q + 3 q - q 5/2 3/2 Sqrt[q] q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 2 -4 2 4 6 12 14 3 + q + q + -- + q + -- + q - 2 q - q + q 6 2 q q
In[8]:=	HOMFLYPT[Link[8, Alternating, 8]][a, z]
Out[8]=	3 3 3 5 a a z 3 z 3 z 3 z 3 z -(-) + -- - -- + --- - 4 a z + a z - -- + ---- - 2 a z + -- z z 3 a 3 a a a a
In[9]:=	Kauffman[Link[8, Alternating, 8]][a, z]
Out[9]=	3 2 2 3 2 a a 2 z 6 z 3 2 2 z 2 z 2 2 z -a + - + -- - --- - --- - 7 a z - 3 a z - 2 z + ---- + ---- - 2 a z - -- + z z 3 a 4 2 5 a a a a 3 3 4 4 5 4 z 12 z 3 3 3 4 3 z z 2 4 4 z > ---- + ----- + 10 a z + 3 a z + 9 z - ---- + -- + 5 a z - ---- - 3 a 4 2 3 a a a a 5 6 7 5 z 5 3 5 6 3 z 2 6 z 7 > ---- - 2 a z - a z - 5 z - ---- - 2 a z - -- - a z a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 3 2 2 2 2 4 4 + 3 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 2 q t + 3 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 10 4 > 2 q t + 2 q t + q t + 2 q t + q t

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