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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X9,12,10,13 X3849 X5,11,6,10 X11,5,12,16 X13,2,14,3

Gauss Code:

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

Jones Polynomial:

- 2q-11/2 + 2q-9/2 - 2q-7/2 + 2q-5/2 - 2q-3/2 + q-1/2 - q1/2

A2 (sl(3)) Invariant:

q-22 + q-20 + 2q-18 + q-16 - q-10 + q-8 + q-4 + q-2 + 1 + q2

HOMFLY-PT Polynomial:

- az-1 - 3az - az3 + 2a3z-1 + 4a3z + 4a3z3 + a3z5 - 2a5z-1 - 3a5z - a5z3 + a7z-1

Kauffman Polynomial:

az-1 - 4az + 4az3 - az5 - a2z2 + 3a2z4 - a2z6 + 2a3z-1 - 8a3z + 10a3z3 - 3a3z5 + a4 - a4z2 + 2a4z4 - a4z6 + 2a5z-1 - 7a5z + 6a5z3 - 2a5z5 - a6z4 + a7z-1 - 3a7z

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2             1 j = 0               j = -2         2 1   j = -4       1 1     j = -6     1 1       j = -8   1 1         j = -10 1 1           j = -12 2

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, NonAlternating, 1]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[8, NonAlternating, 1]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[9, 12, 10, 13], > X[3, 8, 4, 9], X[5, 11, 6, 10], X[11, 5, 12, 16], X[13, 2, 14, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 8, -5, -3}, {-6, -1, 2, 5, -4, 6, -7, 4, -8, -2, 3, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-2 2 2 2 2 1 ----- + ---- - ---- + ---- - ---- + ------- - Sqrt[q] 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 2 -16 -10 -8 -4 -2 2 1 + q + q + --- + q - q + q + q + q + q 18 q
In[8]:=	HOMFLYPT[Link[8, NonAlternating, 1]][a, z]
Out[8]=	3 5 7 a 2 a 2 a a 3 5 3 3 3 5 3 -(-) + ---- - ---- + -- - 3 a z + 4 a z - 3 a z - a z + 4 a z - a z + z z z z 3 5 > a z
In[9]:=	Kauffman[Link[8, NonAlternating, 1]][a, z]
Out[9]=	3 5 7 4 a 2 a 2 a a 3 5 7 2 2 4 2 a + - + ---- + ---- + -- - 4 a z - 8 a z - 7 a z - 3 a z - a z - a z + z z z z 3 3 3 5 3 2 4 4 4 6 4 5 3 5 > 4 a z + 10 a z + 6 a z + 3 a z + 2 a z - a z - a z - 3 a z - 5 5 2 6 4 6 > 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 2 2 1 1 1 1 1 1 1 q + -- + ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 2 12 4 10 4 10 3 8 3 8 2 6 2 6 4 q q t q t q t q t q t q t q t q t t 2 2 > -- + q t 2 q

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