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

A2 (sl(3)) Invariant:

q-8 + q-6 + 2q-4 + 2q-2 + 1 + q2 - q10 + q16 + q18

HOMFLY-PT Polynomial:

- a-5z-1 - a-5z + 3a-3z-1 + 6a-3z + 2a-3z3 - 4a-1z-1 - 8a-1z - 5a-1z3 - a-1z5 + 2az-1 + 3az + az3

Kauffman Polynomial:

- a-7z + a-6 - 2a-6z2 - a-5z-1 + 2a-5z3 - a-5z5 + 3a-4 - 9a-4z2 + 8a-4z4 - 2a-4z6 - 3a-3z-1 + 11a-3z - 13a-3z3 + 9a-3z5 - 2a-3z7 + 3a-2 - 7a-2z2 + 2a-2z4 + 3a-2z6 - a-2z8 - 4a-1z-1 + 17a-1z - 26a-1z3 + 16a-1z5 - 3a-1z7 + 2 - 6z4 + 5z6 - z8 - 2az-1 + 7az - 11az3 + 6az5 - az7

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 12                 1 j = 10               1   j = 8             2 2   j = 6           1 1     j = 4         2 2 1     j = 2       2 3         j = 0       1 1         j = -2   1 2             j = -4                   j = -6 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, 5]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 5]]
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[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]=	-(5/2) -(3/2) 2 3/2 5/2 7/2 9/2 -q + q - ------- + 2 Sqrt[q] - 3 q + 2 q - 2 q + 2 q - Sqrt[q] 11/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 2 2 2 10 16 18 1 + q + q + -- + -- + q - q + q + q 4 2 q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 5]][a, z]
Out[8]=	3 3 5 1 3 4 2 a z 6 z 8 z 2 z 5 z 3 z -(----) + ---- - --- + --- - -- + --- - --- + 3 a z + ---- - ---- + a z - -- 5 3 a z z 5 3 a 3 a a a z a z a a a
In[9]:=	Kauffman[Link[10, NonAlternating, 5]][a, z]
Out[9]=	2 -6 3 3 1 3 4 2 a z 11 z 17 z 2 z 2 + a + -- + -- - ---- - ---- - --- - --- - -- + ---- + ---- + 7 a z - ---- - 4 2 5 3 a z z 7 3 a 6 a a a z a z a a a 2 2 3 3 3 4 4 5 9 z 7 z 2 z 13 z 26 z 3 4 8 z 2 z z > ---- - ---- + ---- - ----- - ----- - 11 a z - 6 z + ---- + ---- - -- + 4 2 5 3 a 4 2 5 a a a a a a a 5 5 6 6 7 7 8 9 z 16 z 5 6 2 z 3 z 2 z 3 z 7 8 z > ---- + ----- + 6 a z + 5 z - ---- + ---- - ---- - ---- - a z - z - -- 3 a 4 2 3 a 2 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 2 1 2 q 4 6 4 2 1 + 3 q + 2 q + ----- + ----- + ----- + - + ---- + 2 q t + q t + q t + 6 4 2 3 2 2 t t q t q t q t 6 2 8 2 8 3 10 3 12 4 > q t + 2 q t + 2 q t + q t + q t

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