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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X17,1,18,4 X9,14,10,15 X3849 X5,11,6,10 X11,5,12,20 X13,19,14,18 X19,13,20,12 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-12 + q-10 + q-8 + 2q-6 + q-4 + q-2 + 1 - q2 - q4 - 2q6 + q10 + q12 + 2q14 + q16

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-5z-1 + 3a-5z - a-5z3 - a-4 + 2a-4z2 - a-4z4 - 2a-3z-1 + 11a-3z - 14a-3z3 + 6a-3z5 - a-3z7 - 3a-2 + 9a-2z2 - 13a-2z4 + 6a-2z6 - a-2z8 - a-1z-1 + 8a-1z - 17a-1z3 + 10a-1z5 - 2a-1z7 - 2 + 7z2 - 9z4 + 5z6 - z8 + az-1 - 3az + 3az5 - az7 - a2 + 3a2z4 - a2z6 + a3z-1 - 3a3z + 4a3z3 - 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                   j = 6             2 1   j = 4           1       j = 2         2 2       j = 0       2 3         j = -2       1 1         j = -4   1 2             j = -6                   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, NonAlternating, 8]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 8]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[17, 1, 18, 4], X[9, 14, 10, 15], > X[3, 8, 4, 9], X[5, 11, 6, 10], X[11, 5, 12, 20], X[13, 19, 14, 18], > X[19, 13, 20, 12], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -5, 3}, {-6, -1, 2, 5, -4, 6, -7, 9, -8, 4, 10, -2, -3, 8, > -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) -(5/2) 2 2 3/2 5/2 7/2 9/2 -q + q - ---- + ------- - 3 Sqrt[q] + 3 q - 2 q + q - q 3/2 Sqrt[q] q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 -8 2 -4 -2 2 4 6 10 12 14 1 + q + q + q + -- + q + q - q - q - 2 q + q + q + 2 q + 6 q 16 > q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 8]][a, z]
Out[8]=	3 3 1 2 1 a a 2 z z 3 z 3 -(----) + ---- - --- - - + -- + --- - - - 2 a z + a z - -- - a z 5 3 a z z z 3 a a a z a z a
In[9]:=	Kauffman[Link[10, NonAlternating, 8]][a, z]
Out[9]=	3 -4 3 2 1 2 1 a a 3 z 11 z 8 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 3 3 3 4 3 2 2 z 9 z z 14 z 17 z 3 3 4 z > 3 a z + 7 z + ---- + ---- - -- - ----- - ----- + 4 a z - 9 z - -- - 4 2 5 3 a 4 a a a a a 4 5 5 6 13 z 2 4 6 z 10 z 5 3 5 6 6 z 2 6 > ----- + 3 a z + ---- + ----- + 3 a z - a z + 5 z + ---- - a z - 2 3 a 2 a a a 7 7 8 z 2 z 7 8 z > -- - ---- - a z - z - -- 3 a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	-2 2 1 1 2 2 1 2 4 6 2 3 + q + 2 q + ----- + ----- + ----- + - + ---- + 2 q t + q t + 2 q t + 8 4 4 3 4 2 t 2 q t q t q t q t 6 3 10 4 > q t + q t

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