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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X12,10,13,9 X8493 X10,5,11,6 X16,11,5,12 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-18 + 2q-14 - q-12 + q-10 - q-6 + 2q-4 + 4 + q2 + q4 + 2q6 - q8

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-2z4 - a-1z-1 + 5a-1z3 - 4a-1z5 + 1 - 2z2 + 7z4 - 5z6 - az-1 - 2az + 11az3 - 5az5 - 2az7 - 5a2z2 + 14a2z4 - 9a2z6 - 4a3z + 10a3z3 - 4a3z5 - 2a3z7 - 2a4z2 + 5a4z4 - 4a4z6 - 2a5z + 4a5z3 - 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               3   j = 2             2 1   j = 0           5 3     j = -2         4 4       j = -4       3 3         j = -6     2 4           j = -8   1 3             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, 1]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[8, Alternating, 1]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[12, 10, 13, 9], > X[8, 4, 9, 3], X[10, 5, 11, 6], X[16, 11, 5, 12], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 5, -3}, {6, -1, 2, -5, 4, -6, 7, -4, 8, -2, 3, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 3 5 7 7 7 3/2 5/2 q - ---- + ---- - ---- + ---- - ------- + 5 Sqrt[q] - 4 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 -12 -10 -6 2 2 4 6 8 4 - q + --- - q + q - q + -- + q + q + 2 q - q 14 4 q q
In[8]:=	HOMFLYPT[Link[8, Alternating, 1]][a, z]
Out[8]=	3 1 a 3 5 z 3 3 3 5 -(---) + - - a z + 2 a z - a z + -- - 2 a z + 2 a z - a z a z z a
In[9]:=	Kauffman[Link[8, Alternating, 1]][a, z]
Out[9]=	1 a 3 5 2 2 2 4 2 6 2 1 - --- - - - 2 a z - 4 a z - 2 a z - 2 z - 5 a z - 2 a z + a z + a z z 3 4 5 z 3 3 3 5 3 4 z 2 4 4 4 > ---- + 11 a z + 10 a z + 4 a z + 7 z - -- + 14 a z + 5 a z - a 2 a 5 6 4 4 z 5 3 5 5 5 6 2 6 4 6 > a z - ---- - 5 a z - 4 a z - 3 a z - 5 z - 9 a z - 4 a z - a 7 3 7 > 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 2 1 3 2 4 3 3 5 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 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 4 2 2 2 4 2 6 3 > ---- + 3 t + 2 q t + q t + 3 q t + q t 2 q t

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