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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X9,14,10,15 X8493 X5,11,6,10 X13,5,14,20 X11,19,12,18 X19,13,20,12 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

- q-3/2 + q-1/2 - 4q1/2 + 4q3/2 - 4q5/2 + 4q7/2 - 3q9/2 + 2q11/2 - q13/2

A2 (sl(3)) Invariant:

q-6 + 2q-4 + 2q-2 + 4 + q2 - q6 - 2q8 - q12 + q14 + q16 + q20

HOMFLY-PT Polynomial:

- a-5z-1 - 2a-5z - a-5z3 + 3a-3z-1 + 6a-3z + 4a-3z3 + a-3z5 - 4a-1z-1 - 5a-1z - 2a-1z3 + 2az-1 + az

Kauffman Polynomial:

- a-7z + 3a-7z3 - a-7z5 + a-6 - 3a-6z2 + 6a-6z4 - 2a-6z6 - a-5z-1 + 2a-5z - 4a-5z3 + 6a-5z5 - 2a-5z7 + 3a-4 - 5a-4z2 + a-4z4 + 2a-4z6 - a-4z8 - 3a-3z-1 + 11a-3z - 19a-3z3 + 12a-3z5 - 3a-3z7 + 3a-2 - 2a-2z2 - 6a-2z4 + 4a-2z6 - a-2z8 - 4a-1z-1 + 11a-1z - 13a-1z3 + 5a-1z5 - a-1z7 + 2 - z4 - 2az-1 + 3az - az3

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 14                 1 j = 12               1   j = 10             2 1   j = 8           2 1     j = 6         2 2       j = 4       2 2         j = 2     2 2           j = 0   1 4             j = -2                   j = -4 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, 6]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 6]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[9, 14, 10, 15], > X[8, 4, 9, 3], X[5, 11, 6, 10], X[13, 5, 14, 20], X[11, 19, 12, 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, -8, 9, -7, 4, 10, -2, 3, 8, > -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(3/2) 1 3/2 5/2 7/2 9/2 11/2 -q + ------- - 4 Sqrt[q] + 4 q - 4 q + 4 q - 3 q + 2 q - Sqrt[q] 13/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 2 2 2 6 8 12 14 16 20 4 + q + -- + -- + q - q - 2 q - q + q + q + q 4 2 q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 6]][a, z]
Out[8]=	3 3 3 5 1 3 4 2 a 2 z 6 z 5 z z 4 z 2 z z -(----) + ---- - --- + --- - --- + --- - --- + a z - -- + ---- - ---- + -- 5 3 a z z 5 3 a 5 3 a 3 a z a z a a a a a
In[9]:=	Kauffman[Link[10, NonAlternating, 6]][a, z]
Out[9]=	-6 3 3 1 3 4 2 a z 2 z 11 z 11 z 2 + a + -- + -- - ---- - ---- - --- - --- - -- + --- + ---- + ---- + 3 a z - 4 2 5 3 a z z 7 5 3 a a a a z a z a a a 2 2 2 3 3 3 3 4 4 3 z 5 z 2 z 3 z 4 z 19 z 13 z 3 4 6 z z > ---- - ---- - ---- + ---- - ---- - ----- - ----- - a z - z + ---- + -- - 6 4 2 7 5 3 a 6 4 a a a a a a a a 4 5 5 5 5 6 6 6 7 7 7 6 z z 6 z 12 z 5 z 2 z 2 z 4 z 2 z 3 z z > ---- - -- + ---- + ----- + ---- - ---- + ---- + ---- - ---- - ---- - -- - 2 7 5 3 a 6 4 2 5 3 a a a a a a a a a a 8 8 z z > -- - -- 4 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 2 4 4 2 6 2 6 3 4 + 2 q + ----- + - + 2 q t + 2 q t + 2 q t + 2 q t + 2 q t + 4 2 t q t 8 3 8 4 10 4 10 5 12 5 14 6 > 2 q t + q t + 2 q t + q t + q t + q t

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