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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X8,9,1,10 X18,13,9,14 X14,8,15,7 X16,6,17,5 X6,16,7,15 X4,18,5,17

Gauss Code:

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

Jones Polynomial:

q-13/2 - 2q-11/2 + 3q-9/2 - 4q-7/2 + 3q-5/2 - 4q-3/2 + 3q-1/2 - 2q1/2 + q3/2 - q5/2

A2 (sl(3)) Invariant:

- q-20 + q-12 + 2q-8 + q-6 + q-4 + q-2 + q2 + q4 + q6 + q8

HOMFLY-PT Polynomial:

- a-1z-1 - 3a-1z - a-1z3 + az-1 + 4az + 4az3 + az5 + a3z + 3a3z3 + a3z5 - 2a5z - a5z3

Kauffman Polynomial:

- a-1z-1 + 6a-1z - 11a-1z3 + 6a-1z5 - a-1z7 + 1 + 2z2 - 7z4 + 5z6 - z8 - az-1 + 8az - 19az3 + 14az5 - 3az7 - 2a2z2 + 2a2z4 + 2a2z6 - a2z8 - 2a3z + 2a3z3 + 4a3z5 - 2a3z7 - a4z2 + 6a4z4 - 3a4z6 - 4a5z + 8a5z3 - 4a5z5 + 2a6z2 - 3a6z4 - 2a7z3 - a8z2

Khovanov Homology:

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

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