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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X10,3,11,4 X14,6,15,5 X16,11,17,12 X20,17,7,18 X18,14,19,13 X12,20,13,19 X4,16,5,15 X2738 X6,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 3q-11/2 - 6q-9/2 + 9q-7/2 - 12q-5/2 + 12q-3/2 - 12q-1/2 + 9q1/2 - 6q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant:

q-20 - q-18 + q-16 + 3q-14 - 2q-12 + 2q-10 - q-6 + 2q-4 - q-2 + 4 + 2q6 - 2q8 + q12

HOMFLY-PT Polynomial:

- a-3z - a-1z-1 + a-1z + 2a-1z3 + 2az-1 + 2az - az5 - 2a3z-1 - 4a3z - 2a3z3 - a3z5 + a5z-1 + a5z + a5z3

Kauffman Polynomial:

- a-3z + 2a-3z3 - a-3z5 - 3a-2z2 + 6a-2z4 - 3a-2z6 - a-1z-1 + 2a-1z - 2a-1z3 + 6a-1z5 - 4a-1z7 - 2z2 + 6z4 - 3z8 - 2az-1 + 12az - 22az3 + 21az5 - 8az7 - az9 - a2 + 3a2z2 - 9a2z4 + 12a2z6 - 7a2z8 - 2a3z-1 + 14a3z - 31a3z3 + 27a3z5 - 9a3z7 - a3z9 - 3a4z4 + 6a4z6 - 4a4z8 - a5z-1 + 5a5z - 11a5z3 + 12a5z5 - 5a5z7 - 2a6z2 + 6a6z4 - 3a6z6 + 2a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                     1 j = 6                   2   j = 4                 4 1   j = 2               5 2     j = 0             7 4       j = -2           6 6         j = -4         6 6           j = -6       4 7             j = -8     2 5               j = -10   1 4                 j = -12   2                   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[10, Alternating, 57]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 57]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[14, 6, 15, 5], X[16, 11, 17, 12], > X[20, 17, 7, 18], X[18, 14, 19, 13], X[12, 20, 13, 19], X[4, 16, 5, 15], > X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -8, 3, -10}, > {9, -1, 10, -2, 4, -7, 6, -3, 8, -4, 5, -6, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 3 6 9 12 12 12 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 9 Sqrt[q] - 6 q + 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 7/2 > 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 -16 3 2 2 -6 2 -2 6 8 12 4 + q - q + q + --- - --- + --- - q + -- - q + 2 q - 2 q + q 14 12 10 4 q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 57]][a, z]
Out[8]=	3 5 3 1 2 a 2 a a z z 3 5 2 z 3 3 -(---) + --- - ---- + -- - -- + - + 2 a z - 4 a z + a z + ---- - 2 a z + a z z z z 3 a a a 5 3 5 3 5 > a z - a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 57]][a, z]
Out[9]=	3 5 2 1 2 a 2 a a z 2 z 3 5 2 -a - --- - --- - ---- - -- - -- + --- + 12 a z + 14 a z + 5 a z - 2 z - a z z z z 3 a a 2 3 3 3 z 2 2 6 2 2 z 2 z 3 3 3 5 3 > ---- + 3 a z - 2 a z + ---- - ---- - 22 a z - 31 a z - 11 a z + 2 3 a a a 4 5 5 7 3 4 6 z 2 4 4 4 6 4 z 6 z 5 > 2 a z + 6 z + ---- - 9 a z - 3 a z + 6 a z - -- + ---- + 21 a z + 2 3 a a a 6 7 3 5 5 5 7 5 3 z 2 6 4 6 6 6 4 z > 27 a z + 12 a z - a z - ---- + 12 a z + 6 a z - 3 a z - ---- - 2 a a 7 3 7 5 7 8 2 8 4 8 9 3 9 > 8 a z - 9 a z - 5 a z - 3 z - 7 a z - 4 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 1 2 1 4 2 5 4 7 7 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 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 6 6 6 2 2 2 4 2 4 3 6 3 > ----- + ---- + ---- + 4 t + 5 q t + 2 q t + 4 q t + q t + 2 q t + 4 2 4 2 q t q t q t 8 4 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a57

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