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Knotscape

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DrawMorseLink

Further views:

Floor of the Stock Exchange

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-23/2 + q-21/2 - q-19/2 + q-17/2 - q-15/2 + q-13/2 - q-11/2 - q-7/2

A2 (sl(3)) Invariant:

q-34 + q-32 + q-30 + q-20 + q-18 + 2q-16 + q-14 + q-12

HOMFLY-PT Polynomial:

- a7z-1 - 10a7z - 15a7z3 - 7a7z5 - a7z7 + a9z-1 + 6a9z + 5a9z3 + a9z5

Kauffman Polynomial:

- a7z-1 + 10a7z - 15a7z3 + 7a7z5 - a7z7 + a8 - 6a8z2 + 5a8z4 - a8z6 - a9z-1 + 7a9z - 11a9z3 + 6a9z5 - a9z7 - 3a10z2 + 4a10z4 - a10z6 - a11z + 3a11z3 - a11z5 + 2a12z2 - a12z4 + a13z - a13z3 - a14z2 - a15z

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -6                 1 j = -8                 1 j = -10             1     j = -12                   j = -14         1 1       j = -16                   j = -18     1 1           j = -20                   j = -22 1 1               j = -24 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, 14]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[8, Alternating, 14]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[14, 5, 15, 6], > X[16, 7, 9, 8], X[8, 9, 1, 10], X[4, 13, 5, 14], X[6, 15, 7, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -7, 4, -8, 5, -6}, {6, -1, 2, -3, 7, -4, 8, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) -(21/2) -(19/2) -(17/2) -(15/2) -(13/2) -(11/2) -q + q - q + q - q + q - q - -(7/2) > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 -32 -30 -20 -18 2 -14 -12 q + q + q + q + q + --- + q + q 16 q
In[8]:=	HOMFLYPT[Link[8, Alternating, 14]][a, z]
Out[8]=	7 9 a a 7 9 7 3 9 3 7 5 9 5 7 7 -(--) + -- - 10 a z + 6 a z - 15 a z + 5 a z - 7 a z + a z - a z z z
In[9]:=	Kauffman[Link[8, Alternating, 14]][a, z]
Out[9]=	7 9 8 a a 7 9 11 13 15 8 2 10 2 a - -- - -- + 10 a z + 7 a z - a z + a z - a z - 6 a z - 3 a z + z z 12 2 14 2 7 3 9 3 11 3 13 3 8 4 > 2 a z - a z - 15 a z - 11 a z + 3 a z - a z + 5 a z + 10 4 12 4 7 5 9 5 11 5 8 6 10 6 7 7 > 4 a z - a z + 7 a z + 6 a z - a z - a z - a z - a z - 9 7 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 1 1 1 1 1 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 24 8 22 8 22 7 18 6 18 5 14 4 14 3 q t q t q t q t q t q t q t 1 > ------ 10 2 q t

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