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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X18,7,19,8 X19,1,20,4 X5,12,6,13 X3849 X9,16,10,17 X13,22,14,5 X15,10,16,11 X21,14,22,15 X11,20,12,21 X2,18,3,17

Gauss Code:

{{1, -11, -5, 3}, {-4, -1, 2, 5, -6, 8, -10, 4, -7, 9, -8, 6, 11, -2, -3, 10, -9, 7}}

Jones Polynomial:

q-19/2 - q-17/2 + 2q-15/2 - 2q-13/2 + 2q-11/2 - 3q-9/2 + q-7/2 - 2q-5/2 + q-3/2 - q-1/2

A2 (sl(3)) Invariant:

- q-30 - q-28 - q-26 - 2q-24 - q-22 - q-20 + 2q-16 + 3q-14 + 4q-12 + 3q-10 + 2q-8 + q-6 + 1

HOMFLY-PT Polynomial:

- az - 2a3z-1 - 2a3z + 2a5z-1 + 2a5z + a5z3 + a7z-1 + 2a7z + a7z3 - a9z-1 - a9z

Kauffman Polynomial:

- az - a2z2 - 2a3z-1 + 5a3z - 2a3z3 + 2a4 - 4a4z2 + 4a4z4 - a4z6 - 2a5z-1 + 9a5z - 14a5z3 + 10a5z5 - 2a5z7 - 4a6 + 13a6z2 - 18a6z4 + 11a6z6 - 2a6z8 + a7z-1 + 3a7z - 8a7z3 + 4a7z7 - a7z9 - 9a8 + 30a8z2 - 38a8z4 + 19a8z6 - 3a8z8 + a9z-1 + 4a9z3 - 10a9z5 + 6a9z7 - a9z9 - 4a10 + 14a10z2 - 16a10z4 + 7a10z6 - a10z8

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = 0                   1 j = -2                 2 2 j = -4               1 1 1 j = -6             2 3     j = -8           2 1 1     j = -10         1 3 1       j = -12       2 2           j = -14       1 1           j = -16   1 2               j = -18                     j = -20 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[11, NonAlternating, 30]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 30]]
Out[4]=	PD[X[6, 1, 7, 2], X[18, 7, 19, 8], X[19, 1, 20, 4], X[5, 12, 6, 13], > X[3, 8, 4, 9], X[9, 16, 10, 17], X[13, 22, 14, 5], X[15, 10, 16, 11], > X[21, 14, 22, 15], X[11, 20, 12, 21], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, -5, 3}, {-4, -1, 2, 5, -6, 8, -10, 4, -7, 9, -8, 6, 11, -2, > -3, 10, -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) -(17/2) 2 2 2 3 -(7/2) 2 -(3/2) q - q + ----- - ----- + ----- - ---- + q - ---- + q - 15/2 13/2 11/2 9/2 5/2 q q q q q 1 > ------- Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 -26 2 -22 -20 2 3 4 3 2 -6 1 - q - q - q - --- - q - q + --- + --- + --- + --- + -- + q 24 16 14 12 10 8 q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 30]][a, z]
Out[8]=	3 5 7 9 -2 a 2 a a a 3 5 7 9 5 3 7 3 ----- + ---- + -- - -- - a z - 2 a z + 2 a z + 2 a z - a z + a z + a z z z z z
In[9]:=	Kauffman[Link[11, NonAlternating, 30]][a, z]
Out[9]=	3 5 7 9 4 6 8 10 2 a 2 a a a 3 5 2 a - 4 a - 9 a - 4 a - ---- - ---- + -- + -- - a z + 5 a z + 9 a z + z z z z 7 2 2 4 2 6 2 8 2 10 2 3 3 > 3 a z - a z - 4 a z + 13 a z + 30 a z + 14 a z - 2 a z - 5 3 7 3 9 3 4 4 6 4 8 4 10 4 > 14 a z - 8 a z + 4 a z + 4 a z - 18 a z - 38 a z - 16 a z + 5 5 9 5 4 6 6 6 8 6 10 6 5 7 > 10 a z - 10 a z - a z + 11 a z + 19 a z + 7 a z - 2 a z + 7 7 9 7 6 8 8 8 10 8 7 9 9 9 > 4 a z + 6 a z - 2 a z - 3 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 2 1 1 2 1 2 1 2 1 + q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 20 9 16 8 16 7 14 6 12 6 14 5 12 5 q q t q t q t q t q t q t q t 1 3 2 1 1 2 1 3 1 > ------ + ------ + ----- + ------ + ----- + ----- + ----- + ----- + ----- + 10 5 10 4 8 4 10 3 8 3 6 3 8 2 6 2 4 2 q t q t q t q t q t q t q t q t q t 1 2 > ---- + ---- 4 2 q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n30

L11n29 L11n31