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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X14,8,15,7 X16,10,17,9 X18,12,5,11 X8,16,9,15 X10,18,11,17 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

- q1/2 + q3/2 - 3q5/2 + 2q7/2 - 4q9/2 + 4q11/2 - 3q13/2 + 3q15/2 - 2q17/2 + q19/2

A2 (sl(3)) Invariant:

q2 + q4 + 2q6 + 3q8 + 3q10 + 4q12 + q14 + q16 - 2q18 - 2q20 - q22 - q24 - q28

HOMFLY-PT Polynomial:

2a-7z-1 + 4a-7z + 4a-7z3 + a-7z5 - 5a-5z-1 - 11a-5z - 12a-5z3 - 6a-5z5 - a-5z7 + 3a-3z-1 + 7a-3z + 5a-3z3 + a-3z5

Kauffman Polynomial:

- a-12z2 - 2a-11z3 - a-10 + 3a-10z2 - 3a-10z4 + 4a-9z3 - 3a-9z5 + 6a-8z4 - 3a-8z6 - 2a-7z-1 + 7a-7z - 12a-7z3 + 11a-7z5 - 3a-7z7 + 5a-6 - 12a-6z2 + 8a-6z4 + a-6z6 - a-6z8 - 5a-5z-1 + 17a-5z - 30a-5z3 + 20a-5z5 - 4a-5z7 + 5a-4 - 8a-4z2 - a-4z4 + 4a-4z6 - a-4z8 - 3a-3z-1 + 10a-3z - 12a-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               2 1   j = 14             1 1     j = 12           3 2       j = 10         1 1         j = 8       1 3           j = 6     2 1             j = 4   1 3               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, 14]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 14]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 8, 15, 7], X[16, 10, 17, 9], > X[18, 12, 5, 11], X[8, 16, 9, 15], X[10, 18, 11, 17], X[2, 5, 3, 6], > X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9}, {8, -1, 3, -6, 4, -7, 5, -2, 9, -3, 6, -4, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	3/2 5/2 7/2 9/2 11/2 13/2 15/2 -Sqrt[q] + q - 3 q + 2 q - 4 q + 4 q - 3 q + 3 q - 17/2 19/2 > 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 4 6 8 10 12 14 16 18 20 22 24 q + q + 2 q + 3 q + 3 q + 4 q + q + q - 2 q - 2 q - q - q - 28 > q
In[8]:=	HOMFLYPT[Link[9, Alternating, 14]][a, z]
Out[8]=	3 3 3 5 5 5 2 5 3 4 z 11 z 7 z 4 z 12 z 5 z z 6 z z ---- - ---- + ---- + --- - ---- + --- + ---- - ----- + ---- + -- - ---- + -- - 7 5 3 7 5 3 7 5 3 7 5 3 a z a z a z a a a a a a a a a 7 z > -- 5 a
In[9]:=	Kauffman[Link[9, Alternating, 14]][a, z]
Out[9]=	2 2 2 -10 5 5 2 5 3 7 z 17 z 10 z z 3 z 12 z -a + -- + -- - ---- - ---- - ---- + --- + ---- + ---- - --- + ---- - ----- - 6 4 7 5 3 7 5 3 12 10 6 a a a z a z a z a a a a a a 2 3 3 3 3 3 4 4 4 4 8 z 2 z 4 z 12 z 30 z 12 z 3 z 6 z 8 z z > ---- - ---- + ---- - ----- - ----- - ----- - ---- + ---- + ---- - -- - 4 11 9 7 5 3 10 8 6 4 a a a a a a a a a a 5 5 5 5 6 6 6 7 7 7 8 8 3 z 11 z 20 z 6 z 3 z z 4 z 3 z 4 z z z z > ---- + ----- + ----- + ---- - ---- + -- + ---- - ---- - ---- - -- - -- - -- 9 7 5 3 8 6 4 7 5 3 6 4 a a a 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 3 q + 2 q + t + -- + q t + q t + 3 q t + q t + q t + 3 q t + t 12 4 14 4 14 5 16 5 16 6 18 6 20 7 > 2 q t + q t + q t + 2 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9a14

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