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DrawMorseLink

PD Presentation:

X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X11,18,12,19 X20,16,17,15 X16,20,9,19 X17,12,18,13 X2536 X9,1,10,4

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-8 + 2q-6 + 2q-4 + 4q-2 + 9 + 9q2 + 14q4 + 12q6 + 12q8 + 9q10 + 4q12 + 4q14 + q16

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-5z-3 - 3a-5z-1 + 4a-5z - 3a-5z3 - 3a-4z-2 + 6a-4 - 5a-4z2 - a-4z6 + 3a-3z-3 - 6a-3z-1 + 5a-3z + a-3z3 - 2a-3z5 - a-3z7 - 6a-2z-2 + 11a-2 - 6a-2z2 + 6a-2z4 - 5a-2z6 + 3a-1z-3 - 6a-1z-1 + a-1z + 10a-1z3 - 6a-1z5 - a-1z7 - 3z-2 + 6 - z2 + 5z4 - 4z6 + az-3 - 3az-1 + 6az3 - 4az5 - a2z4

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 10               2 j = 8             3 2 j = 6           4     j = 4           3     j = 2       7 4       j = 0     3 6         j = -2   1 1           j = -4   3             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]=	4
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 100]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 100]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 11, 4, 10], X[7, 15, 8, 14], X[13, 5, 14, 8], > X[11, 18, 12, 19], X[20, 16, 17, 15], X[16, 20, 9, 19], X[17, 12, 18, 13], > X[2, 5, 3, 6], X[9, 1, 10, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, -2, 10}, {9, -1, -3, 4}, {-8, 5, 7, -6}, > {-10, 2, -5, 8, -4, 3, 6, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 4 4 3/2 5/2 7/2 9/2 q - ---- + ------- - 7 Sqrt[q] + 4 q - 7 q + 3 q - 2 q 3/2 Sqrt[q] q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 2 2 4 2 4 6 8 10 12 14 9 - q + -- + -- + -- + 9 q + 14 q + 12 q + 12 q + 9 q + 4 q + 4 q + 6 4 2 q q q 16 > q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 100]][a, z]
Out[8]=	3 3 1 3 3 a 1 4 5 2 a z z z 2 z -(-----) + ----- - ---- + -- - ---- + ---- - --- + --- + -- - - - -- + ---- - 5 3 3 3 3 3 5 3 a z z 3 a 3 a a z a z a z z a z a z a a 5 3 z > a z + -- a
In[9]:=	Kauffman[Link[10, NonAlternating, 100]][a, z]
Out[9]=	6 11 1 3 3 a 3 3 6 3 6 6 + -- + -- + ----- + ----- + ---- + -- - -- - ----- - ----- - ---- - ---- - 4 2 5 3 3 3 3 3 2 4 2 2 2 5 3 a a a z a z a z z z a z a z a z a z 2 2 3 3 3 6 3 a 4 z 5 z z 2 5 z 6 z 3 z z 10 z 3 > --- - --- + --- + --- + - - z - ---- - ---- - ---- + -- + ----- + 6 a z + a z z 5 3 a 4 2 5 3 a a a a a a a 4 5 5 6 6 7 7 4 6 z 2 4 2 z 6 z 5 6 z 5 z z z > 5 z + ---- - a z - ---- - ---- - 4 a z - 4 z - -- - ---- - -- - -- 2 3 a 4 2 3 a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 3 1 2 4 2 6 2 6 + 7 q + ----- + ----- + ----- + - + ---- + 4 q t + 3 q t + 4 q t + 6 3 4 2 2 2 t 2 q t q t q t q t 8 3 8 4 10 4 > 3 q t + 2 q t + 2 q t

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