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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X12,5,7,6 X10,3,11,4 X4,11,5,12 X2738 X6,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-15/2 + q-13/2 - 2q-11/2 + 2q-9/2 - 2q-7/2 + q-5/2 - q-3/2

A2 (sl(3)) Invariant:

q-24 + q-22 + q-20 + 2q-18 + q-16 + q-14 + q-8 + q-4

HOMFLY-PT Polynomial:

- 2a3z - a3z3 - a5z-1 - 2a5z - a5z3 + a7z-1 + a7z

Kauffman Polynomial:

2a3z - a3z3 + a4z2 - a4z4 + a5z-1 - 3a5z + 2a5z3 - a5z5 - a6 + 2a6z2 - 2a6z4 + a7z-1 - 3a7z + 2a7z3 - a7z5 + a8z2 - a8z4 + 2a9z - a9z3

Khovanov Homology:

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

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