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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X14,10,15,9 X8493 X10,5,11,6 X20,13,5,14 X18,11,19,12 X12,19,13,20 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

q-17/2 - 4q-15/2 + 8q-13/2 - 11q-11/2 + 14q-9/2 - 15q-7/2 + 13q-5/2 - 11q-3/2 + 6q-1/2 - 4q1/2 + q3/2

A2 (sl(3)) Invariant:

- q-26 + q-24 + q-22 - 2q-20 + 2q-18 - 3q-16 - q-14 - 2q-10 + 5q-8 + 5q-4 + 3q-2 + 2q2 - q4

HOMFLY-PT Polynomial:

- 2az-1 - az + 2az3 + az5 + 3a3z-1 - 5a3z3 - 4a3z5 - a3z7 - a5z-1 + 2a5z + 5a5z3 + 2a5z5 - a7z - a7z3

Kauffman Polynomial:

2z4 - z6 + 2az-1 - az - 9az3 + 12az5 - 4az7 - 3a2 + 3a2z2 - 9a2z4 + 13a2z6 - 5a2z8 + 3a3z-1 + a3z - 22a3z3 + 27a3z5 - 6a3z7 - 2a3z9 - 3a4 + 4a4z2 - 12a4z4 + 23a4z6 - 11a4z8 + a5z-1 + 4a5z - 20a5z3 + 30a5z5 - 11a5z7 - 2a5z9 - a6 - 2a6z2 + 9a6z4 + a6z6 - 6a6z8 + 2a7z - 5a7z3 + 11a7z5 - 9a7z7 - 3a8z2 + 9a8z4 - 8a8z6 + 2a9z3 - 4a9z5 - a10z4

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                     1 j = 2                   3   j = 0                 3 1   j = -2               8 3     j = -4             7 5       j = -6           8 6         j = -8         6 7           j = -10       5 8             j = -12     3 6               j = -14   1 5                 j = -16   3                   j = -18 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, 2]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 2]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[14, 10, 15, 9], > X[8, 4, 9, 3], X[10, 5, 11, 6], X[20, 13, 5, 14], X[18, 11, 19, 12], > X[12, 19, 13, 20], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -3}, {6, -1, 2, -5, 4, -6, 8, -9, 7, -4, 10, -2, 3, -8, > 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 4 8 11 14 15 13 11 6 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q 3/2 > 4 Sqrt[q] + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -24 -22 2 2 3 -14 2 5 5 3 2 4 -q + q + q - --- + --- - --- - q - --- + -- + -- + -- + 2 q - q 20 18 16 10 8 4 2 q q q q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 2]][a, z]
Out[8]=	3 5 -2 a 3 a a 5 7 3 3 3 5 3 7 3 ---- + ---- - -- - a z + 2 a z - a z + 2 a z - 5 a z + 5 a z - a z + z z z 5 3 5 5 5 3 7 > a z - 4 a z + 2 a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 2]][a, z]
Out[9]=	3 5 2 4 6 2 a 3 a a 3 5 7 2 2 -3 a - 3 a - a + --- + ---- + -- - a z + a z + 4 a z + 2 a z + 3 a z + z z z 4 2 6 2 8 2 3 3 3 5 3 7 3 > 4 a z - 2 a z - 3 a z - 9 a z - 22 a z - 20 a z - 5 a z + 9 3 4 2 4 4 4 6 4 8 4 10 4 > 2 a z + 2 z - 9 a z - 12 a z + 9 a z + 9 a z - a z + 5 3 5 5 5 7 5 9 5 6 2 6 > 12 a z + 27 a z + 30 a z + 11 a z - 4 a z - z + 13 a z + 4 6 6 6 8 6 7 3 7 5 7 7 7 > 23 a z + a z - 8 a z - 4 a z - 6 a z - 11 a z - 9 a z - 2 8 4 8 6 8 3 9 5 9 > 5 a z - 11 a z - 6 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 8 1 3 1 5 3 6 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 10 4 q q q t q t q t q t q t q t q t 8 6 7 8 6 7 3 t 2 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 3 t + --- + t + 3 q t + 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q 4 3 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a2

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