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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X11,19,12,18 X7,17,8,16 X17,9,18,8 X15,5,16,22 X21,13,22,12 X13,21,14,20 X19,15,20,14 X2536 X4,9,1,10

Gauss Code:

{{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 7, -8, 9, -6, 4, -5, 3, -9, 8, -7, 6}}

Jones Polynomial:

- q-5/2 + q-3/2 - 2q-1/2 - q5/2 + q7/2 - 2q9/2 + 2q11/2 - q13/2 + q15/2

A2 (sl(3)) Invariant:

q-8 + q-6 + q-4 + q-2 + 3 + 3q2 + 2q4 + 3q6 + q8 - q12 - 2q14 - q16 - 2q18 - q24

HOMFLY-PT Polynomial:

a-7z + 2a-5z-1 + 3a-5z - 4a-3z-1 - 8a-3z - 6a-3z3 - a-3z5 + a-1z-1 - a-1z - 4a-1z3 - a-1z5 + az-1 + 3az + az3

Kauffman Polynomial:

2a-8 - 7a-8z2 + 5a-8z4 - a-8z6 - 3a-7z3 + 4a-7z5 - a-7z7 + a-6 - 6a-6z2 + 8a-6z4 - 2a-6z6 + 2a-5z-1 - 7a-5z + 9a-5z3 - 2a-5z5 - 6a-4 + 21a-4z2 - 20a-4z4 + 8a-4z6 - a-4z8 + 4a-3z-1 - 17a-3z + 27a-3z3 - 21a-3z5 + 8a-3z7 - a-3z9 - 5a-2 + 18a-2z2 - 27a-2z4 + 14a-2z6 - 2a-2z8 + a-1z-1 - 4a-1z + 5a-1z3 - 9a-1z5 + 6a-1z7 - a-1z9 + 1 - 2z2 - 4z4 + 5z6 - z8 - az-1 + 6az - 10az3 + 6az5 - az7

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 16                       1 j = 14                         j = 12                   2 1   j = 10               1 1       j = 8               1 2       j = 6           2 3 1         j = 4         1 1 1           j = 2       1 3 2             j = 0     1 1 2               j = -2     1                   j = -4 1 1                     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[11, NonAlternating, 75]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 75]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[11, 19, 12, 18], X[7, 17, 8, 16], > X[17, 9, 18, 8], X[15, 5, 16, 22], X[21, 13, 22, 12], X[13, 21, 14, 20], > X[19, 15, 20, 14], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 7, -8, 9, -6, 4, -5, 3, > -9, 8, -7, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) -(3/2) 2 5/2 7/2 9/2 11/2 13/2 15/2 -q + q - ------- - q + q - 2 q + 2 q - q + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 -4 -2 2 4 6 8 12 14 16 3 + q + q + q + q + 3 q + 2 q + 3 q + q - q - 2 q - q - 18 24 > 2 q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 75]][a, z]
Out[8]=	3 3 5 2 4 1 a z 3 z 8 z z 6 z 4 z 3 z ---- - ---- + --- + - + -- + --- - --- - - + 3 a z - ---- - ---- + a z - -- - 5 3 a z z 7 5 3 a 3 a 3 a z a z a a a a a 5 z > -- a
In[9]:=	Kauffman[Link[11, NonAlternating, 75]][a, z]
Out[9]=	2 -6 6 5 2 4 1 a 7 z 17 z 4 z 1 + -- + a - -- - -- + ---- + ---- + --- - - - --- - ---- - --- + 6 a z - 8 4 2 5 3 a z z 5 3 a a a a a z a z a a 2 2 2 2 3 3 3 3 2 7 z 6 z 21 z 18 z 3 z 9 z 27 z 5 z 3 > 2 z - ---- - ---- + ----- + ----- - ---- + ---- + ----- + ---- - 10 a z - 8 6 4 2 7 5 3 a a a a a a a a 4 4 4 4 5 5 5 5 4 5 z 8 z 20 z 27 z 4 z 2 z 21 z 9 z 5 > 4 z + ---- + ---- - ----- - ----- + ---- - ---- - ----- - ---- + 6 a z + 8 6 4 2 7 5 3 a a a a a a a a 6 6 6 6 7 7 7 8 6 z 2 z 8 z 14 z z 8 z 6 z 7 8 z > 5 z - -- - ---- + ---- + ----- - -- + ---- + ---- - a z - z - -- - 8 6 4 2 7 3 a 4 a a a a a a a 8 9 9 2 z z z > ---- - -- - -- 2 3 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 1 -2 1 1 q 2 4 2 + 3 q + q + ----- + ----- + ----- + t + ----- + - + -- + 2 q t + q t + 6 4 4 4 4 3 2 2 t t q t q t q t q t 6 4 2 6 2 6 3 8 3 10 3 8 4 10 4 > 2 q t + q t + 3 q t + q t + q t + q t + 2 q t + q t + 12 5 12 6 16 7 > 2 q t + q t + q t

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