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Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X12,4,13,3 X18,12,9,11 X14,6,15,5 X16,8,17,7 X2,9,3,10 X4,14,5,13 X6,16,7,15 X8,18,1,17

Gauss Code:

{{1, -6, 2, -7, 4, -8, 5, -9}, {6, -1, 3, -2, 7, -4, 8, -5, 9, -3}}

Jones Polynomial:

- q1/2 + q3/2 - 2q5/2 + 2q7/2 - 3q9/2 + 3q11/2 - 3q13/2 + 2q15/2 - 2q17/2 + q19/2

A2 (sl(3)) Invariant:

q2 + q4 + q6 + q8 + q12 + 2q16 + q18 + q20 + q22 - q28

HOMFLY-PT Polynomial:

3a-7z + 4a-7z3 + a-7z5 - a-5z-1 - 7a-5z - 11a-5z3 - 6a-5z5 - a-5z7 + a-3z-1 + 6a-3z + 5a-3z3 + a-3z5

Kauffman Polynomial:

- a-12z2 + a-11z - 2a-11z3 + a-10z2 - 2a-10z4 + a-9z + 2a-9z3 - 2a-9z5 + 4a-8z4 - 2a-8z6 + 2a-7z - 6a-7z3 + 7a-7z5 - 2a-7z7 - 2a-6z2 + 3a-6z6 - a-6z8 - a-5z-1 + 9a-5z - 21a-5z3 + 15a-5z5 - 3a-5z7 + a-4 - 6a-4z4 + 5a-4z6 - a-4z8 - a-3z-1 + 7a-3z - 11a-3z3 + 6a-3z5 - a-3z7

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 20                   1 j = 18                 1   j = 16               1 1   j = 14             2 1     j = 12           2 2       j = 10         1 1         j = 8       1 2           j = 6     1 1             j = 4   1 2               j = 2                     j = 0 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[9, Alternating, 36]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 36]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[18, 12, 9, 11], X[14, 6, 15, 5], > X[16, 8, 17, 7], X[2, 9, 3, 10], X[4, 14, 5, 13], X[6, 16, 7, 15], > X[8, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -6, 2, -7, 4, -8, 5, -9}, {6, -1, 3, -2, 7, -4, 8, -5, 9, -3}]
In[6]:=	Jones[L][q]
Out[6]=	3/2 5/2 7/2 9/2 11/2 13/2 15/2 -Sqrt[q] + q - 2 q + 2 q - 3 q + 3 q - 3 q + 2 q - 17/2 19/2 > 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 4 6 8 12 16 18 20 22 28 q + q + q + q + q + 2 q + q + q + q - q
In[8]:=	HOMFLYPT[Link[9, Alternating, 36]][a, z]
Out[8]=	3 3 3 5 5 5 7 1 1 3 z 7 z 6 z 4 z 11 z 5 z z 6 z z z -(----) + ---- + --- - --- + --- + ---- - ----- + ---- + -- - ---- + -- - -- 5 3 7 5 3 7 5 3 7 5 3 5 a z a z a a a a a a a a a a
In[9]:=	Kauffman[Link[9, Alternating, 36]][a, z]
Out[9]=	2 2 2 3 -4 1 1 z z 2 z 9 z 7 z z z 2 z 2 z a - ---- - ---- + --- + -- + --- + --- + --- - --- + --- - ---- - ---- + 5 3 11 9 7 5 3 12 10 6 11 a z a z a a a a a a a a a 3 3 3 3 4 4 4 5 5 5 2 z 6 z 21 z 11 z 2 z 4 z 6 z 2 z 7 z 15 z > ---- - ---- - ----- - ----- - ---- + ---- - ---- - ---- + ---- + ----- + 9 7 5 3 10 8 4 9 7 5 a a a a a a a a a a 5 6 6 6 7 7 7 8 8 6 z 2 z 3 z 5 z 2 z 3 z z z z > ---- - ---- + ---- + ---- - ---- - ---- - -- - -- - -- 3 8 6 4 7 5 3 6 4 a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	4 4 6 -2 q 6 8 8 2 10 2 10 3 12 3 2 q + q + t + -- + q t + q t + 2 q t + q t + q t + 2 q t + t 12 4 14 4 14 5 16 5 16 6 18 6 20 7 > 2 q t + 2 q t + q t + q t + q t + q t + q t

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