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DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X16,10,7,9 X2738 X4,16,5,15 X12,5,13,6 X14,11,15,12 X6,13,1,14

Gauss Code:

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

Jones Polynomial:

q-11/2 - 3q-9/2 + 4q-7/2 - 6q-5/2 + 6q-3/2 - 6q-1/2 + 4q1/2 - 3q3/2 + q5/2

A2 (sl(3)) Invariant:

- q-18 + 2q-14 + 2q-10 + q-8 + 2q-4 + 3 + q6 - q8

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-2z2 - a-2z4 - 2a-1z + 5a-1z3 - 3a-1z5 + 3z4 - 3z6 + az-1 - 7az + 13az3 - 6az5 - az7 - a2 - 2a2z2 + 8a2z4 - 6a2z6 + a3z-1 - 7a3z + 13a3z3 - 6a3z5 - a3z7 + 3a4z4 - 3a4z6 - 2a5z + 5a5z3 - 3a5z5 + a6z2 - a6z4

Khovanov Homology:

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

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