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DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X20,10,7,9 X2738 X15,5,16,4 X5,13,6,12 X11,16,12,17 X6,18,1,17 X14,19,15,20 X18,13,19,14

Gauss Code:

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

Jones Polynomial:

q-7/2 - 2q-5/2 + 3q-3/2 - 4q-1/2 + 3q1/2 - 4q3/2 + 2q5/2 - 2q7/2 + q9/2

A2 (sl(3)) Invariant:

- q-12 - q-10 - q-6 + q-4 + q-2 + 2 + 3q2 + 2q4 + 3q6 + q8 - q14

HOMFLY-PT Polynomial:

2a-3z + a-3z3 - 2a-1z-1 - 5a-1z - 4a-1z3 - a-1z5 + 3az-1 + 5az + 2az3 - a3z-1 - a3z

Kauffman Polynomial:

- 3a-4z2 + 4a-4z4 - a-4z6 + 5a-3z - 11a-3z3 + 9a-3z5 - 2a-3z7 - 5a-2z2 + 4a-2z4 + 2a-2z6 - a-2z8 - 2a-1z-1 + 14a-1z - 25a-1z3 + 18a-1z5 - 4a-1z7 + 3 - 7z2 + 3z4 + 2z6 - z8 - 3az-1 + 11az - 16az3 + 9az5 - 2az7 + 3a2 - 6a2z2 + 3a2z4 - a2z6 - a3z-1 + 2a3z - 2a3z3 + a4 - a4z2

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                 1 j = 8               1   j = 6             1 1   j = 4           3 1     j = 2         1 2       j = 0       3 2         j = -2     1 2           j = -4   1 2             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[10, NonAlternating, 40]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 40]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[20, 10, 7, 9], X[2, 7, 3, 8], > X[15, 5, 16, 4], X[5, 13, 6, 12], X[11, 16, 12, 17], X[6, 18, 1, 17], > X[14, 19, 15, 20], X[18, 13, 19, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, 5, -6, -8}, > {4, -1, 3, -2, -7, 6, 10, -9, -5, 7, 8, -10, 9, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 2 3 4 3/2 5/2 7/2 9/2 q - ---- + ---- - ------- + 3 Sqrt[q] - 4 q + 2 q - 2 q + q 5/2 3/2 Sqrt[q] q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 -6 -4 -2 2 4 6 8 14 2 - q - q - q + q + q + 3 q + 2 q + 3 q + q - q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 40]][a, z]
Out[8]=	3 3 3 5 -2 3 a a 2 z 5 z 3 z 4 z 3 z --- + --- - -- + --- - --- + 5 a z - a z + -- - ---- + 2 a z - -- a z z z 3 a 3 a a a a
In[9]:=	Kauffman[Link[10, NonAlternating, 40]][a, z]
Out[9]=	3 2 2 4 2 3 a a 5 z 14 z 3 2 3 z 3 + 3 a + a - --- - --- - -- + --- + ---- + 11 a z + 2 a z - 7 z - ---- - a z z z 3 a 4 a a 2 3 3 4 5 z 2 2 4 2 11 z 25 z 3 3 3 4 4 z > ---- - 6 a z - a z - ----- - ----- - 16 a z - 2 a z + 3 z + ---- + 2 3 a 4 a a a 4 5 5 6 6 7 4 z 2 4 9 z 18 z 5 6 z 2 z 2 6 2 z > ---- + 3 a z + ---- + ----- + 9 a z + 2 z - -- + ---- - a z - ---- - 2 3 a 4 2 3 a a a a a 7 8 4 z 7 8 z > ---- - 2 a z - z - -- a 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 2 1 2 2 2 4 2 3 + -- + ----- + ----- + ----- + ---- + ---- + 2 t + q t + 2 q t + 3 q t + 2 8 3 6 2 4 2 4 2 q q t q t q t q t q t 4 3 6 3 6 4 8 4 10 5 > q t + q t + q t + q t + q t

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