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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 2q-11/2 - 5q-9/2 + 7q-7/2 - 9q-5/2 + 9q-3/2 - 8q-1/2 + 6q1/2 - 4q3/2 + q5/2

A2 (sl(3)) Invariant:

q-22 + 2q-20 + q-16 + 2q-14 - 2q-12 + q-10 + 2q-4 - q-2 + 3 - q2 + 2q6 - q8

HOMFLY-PT Polynomial:

a-1z3 - az-1 - 3az - 2az3 - az5 + 2a3z-1 + 4a3z + 3a3z3 - 2a5z-1 - 3a5z + a7z-1

Kauffman Polynomial:

- a-2z4 + 4a-1z3 - 4a-1z5 - 2z2 + 8z4 - 6z6 + az-1 - 4az + 7az3 - 4az7 - 7a2z2 + 17a2z4 - 9a2z6 - a2z8 + 2a3z-1 - 8a3z + 7a3z3 + 5a3z5 - 6a3z7 + a4 - 7a4z2 + 12a4z4 - 5a4z6 - a4z8 + 2a5z-1 - 7a5z + 7a5z3 - 2a5z7 - 2a6z2 + 4a6z4 - 2a6z6 + a7z-1 - 3a7z + 3a7z3 - a7z5

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9a11

L9a10 L9a12