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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-12 + q-10 + 3q-6 - q-2 + 1 - 2q2 + 2q4 + 2q8 + 2q10 - q12 + 2q14 - q18

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-6z2 - a-6z4 - 2a-5z + 4a-5z3 - 3a-5z5 - a-4z2 + 4a-4z4 - 4a-4z6 + a-3z-1 - 5a-3z + 7a-3z3 - a-3z5 - 3a-3z7 - 3a-2z2 + 10a-2z4 - 6a-2z6 - a-2z8 + 2a-1z-1 - 8a-1z + 8a-1z3 + 3a-1z5 - 5a-1z7 + 1 - 3z2 + 9z4 - 4z6 - z8 + 2az-1 - 8az + 8az3 - 2az7 - 2a2z2 + 4a2z4 - 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 j = 12                   1 j = 10                 2   j = 8               3 1   j = 6             5 2     j = 4           3 3       j = 2         5 5         j = 0       4 5           j = -2     1 3             j = -4   1 4               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[9, Alternating, 8]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 8]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[16, 8, 17, 7], X[18, 13, 5, 14], > X[14, 17, 15, 18], X[12, 10, 13, 9], X[8, 16, 9, 15], X[2, 5, 3, 6], > X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9}, {8, -1, 3, -7, 6, -2, 9, -6, 4, -5, 7, -3, 5, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 2 5 7 3/2 5/2 7/2 -q + ---- - ---- + ------- - 8 Sqrt[q] + 8 q - 8 q + 5 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 3 -2 2 4 8 10 12 14 18 1 + q + q + -- - q - 2 q + 2 q + 2 q + 2 q - q + 2 q - q 6 q
In[8]:=	HOMFLYPT[Link[9, Alternating, 8]][a, z]
Out[8]=	3 3 3 1 2 2 a a z 2 z 3 z 3 2 z 2 z -(----) + --- - --- + -- + -- - --- + --- - 3 a z + a z - ---- + ---- - 3 a z z z 5 3 a 3 a a z a a a 5 3 z > 2 a z + -- a
In[9]:=	Kauffman[Link[9, Alternating, 8]][a, z]
Out[9]=	3 2 2 1 2 2 a a 2 z 5 z 8 z 3 2 z z 1 + ---- + --- + --- + -- - --- - --- - --- - 8 a z - 3 a z - 3 z + -- - -- - 3 a z z z 5 3 a 6 4 a z a a a a 2 3 3 3 4 4 3 z 2 2 4 z 7 z 8 z 3 3 3 4 z 4 z > ---- - 2 a z + ---- + ---- + ---- + 8 a z + 3 a z + 9 z - -- + ---- + 2 5 3 a 6 4 a a a a a 4 5 5 5 6 6 10 z 2 4 3 z z 3 z 3 5 6 4 z 6 z 2 6 > ----- + 4 a z - ---- - -- + ---- - a z - 4 z - ---- - ---- - 2 a z - 2 5 3 a 4 2 a a a a a 7 7 8 3 z 5 z 7 8 z > ---- - ---- - 2 a z - z - -- 3 a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 4 1 4 3 2 4 5 + 5 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 5 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 8 4 10 4 12 5 > 3 q t + 5 q t + 2 q t + 3 q t + q t + 2 q t + q t

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