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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X12,8,13,7 X14,10,5,9 X8,14,9,13 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

- q-1/2 + q1/2 - 3q3/2 + 2q5/2 - 3q7/2 + 3q9/2 - 2q11/2 + q13/2

A2 (sl(3)) Invariant:

q-2 + 1 + 2q2 + 3q4 + 2q6 + 3q8 - q14 - q16 - q20

HOMFLY-PT Polynomial:

a-5z-1 + 2a-5z + a-5z3 - 3a-3z-1 - 5a-3z - 4a-3z3 - a-3z5 + 2a-1z-1 + 3a-1z + a-1z3

Kauffman Polynomial:

- a-8z2 - 2a-7z3 - a-6 + 3a-6z2 - 3a-6z4 + a-5z-1 - 4a-5z + 6a-5z3 - 3a-5z5 - 3a-4 + 6a-4z2 - a-4z4 - a-4z6 + 3a-3z-1 - 9a-3z + 12a-3z3 - 4a-3z5 - 3a-2 + 2a-2z2 + 2a-2z4 - a-2z6 + 2a-1z-1 - 5a-1z + 4a-1z3 - a-1z5

Khovanov Homology:

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

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