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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X16,13,17,14 X14,7,15,8 X8,15,9,16 X20,18,5,17 X18,12,19,11 X12,20,13,19 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-15/2 - 3q-13/2 + 6q-11/2 - 10q-9/2 + 11q-7/2 - 13q-5/2 + 12q-3/2 - 10q-1/2 + 6q1/2 - 3q3/2 + q5/2

A2 (sl(3)) Invariant:

- q-24 - q-22 + q-20 - 2q-18 + 2q-16 + 4q-14 + q-12 + 5q-10 + q-6 - 2q-2 + 3 - 2q2 + q6 - q8

HOMFLY-PT Polynomial:

a-1z + a-1z3 + az-1 - az3 - az5 - 4a3z-1 - 8a3z - 6a3z3 - 2a3z5 + 4a5z-1 + 6a5z + 3a5z3 - a7z-1 - a7z

Kauffman Polynomial:

a-2z2 - a-2z4 - a-1z + 3a-1z3 - 3a-1z5 + 1 - 3z2 + 5z4 - 5z6 - az-1 + 2az - 5az3 + 7az5 - 6az7 + 4a2 - 15a2z2 + 15a2z4 - 2a2z6 - 4a2z8 - 4a3z-1 + 16a3z - 31a3z3 + 33a3z5 - 12a3z7 - a3z9 + 7a4 - 21a4z2 + 16a4z4 + 7a4z6 - 7a4z8 - 4a5z-1 + 17a5z - 32a5z3 + 32a5z5 - 9a5z7 - a5z9 + 4a6 - 13a6z2 + 10a6z4 + 3a6z6 - 3a6z8 - a7z-1 + 4a7z - 9a7z3 + 9a7z5 - 3a7z7 + a8 - 3a8z2 + 3a8z4 - a8z6

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 = 6                     1 j = 4                   2   j = 2                 4 1   j = 0               6 2     j = -2             7 5       j = -4           6 5         j = -6         5 7           j = -8       5 6             j = -10     2 6               j = -12   1 4                 j = -14   2                   j = -16 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, 13]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 13]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 13, 17, 14], X[14, 7, 15, 8], > X[8, 15, 9, 16], X[20, 18, 5, 17], X[18, 12, 19, 11], X[12, 20, 13, 19], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 4, -5, 10, -2, 7, -8, 3, -4, 5, -3, 6, -7, > 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 6 10 11 13 12 10 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 6 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 -20 2 2 4 -12 5 -6 2 2 6 3 - q - q + q - --- + --- + --- + q + --- + q - -- - 2 q + q - 18 16 14 10 2 q q q q q 8 > q
In[8]:=	HOMFLYPT[Link[10, Alternating, 13]][a, z]
Out[8]=	3 5 7 3 a 4 a 4 a a z 3 5 7 z 3 3 3 - - ---- + ---- - -- + - - 8 a z + 6 a z - a z + -- - a z - 6 a z + z z z z a a 5 3 5 3 5 > 3 a z - a z - 2 a z
In[9]:=	Kauffman[Link[10, Alternating, 13]][a, z]
Out[9]=	3 5 7 2 4 6 8 a 4 a 4 a a z 3 1 + 4 a + 7 a + 4 a + a - - - ---- - ---- - -- - - + 2 a z + 16 a z + z z z z a 2 5 7 2 z 2 2 4 2 6 2 8 2 > 17 a z + 4 a z - 3 z + -- - 15 a z - 21 a z - 13 a z - 3 a z + 2 a 3 4 3 z 3 3 3 5 3 7 3 4 z 2 4 > ---- - 5 a z - 31 a z - 32 a z - 9 a z + 5 z - -- + 15 a z + a 2 a 5 4 4 6 4 8 4 3 z 5 3 5 5 5 > 16 a z + 10 a z + 3 a z - ---- + 7 a z + 33 a z + 32 a z + a 7 5 6 2 6 4 6 6 6 8 6 7 3 7 > 9 a z - 5 z - 2 a z + 7 a z + 3 a z - a z - 6 a z - 12 a z - 5 7 7 7 2 8 4 8 6 8 3 9 5 9 > 9 a z - 3 a z - 4 a z - 7 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 1 2 1 4 2 6 5 6 6 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 5 7 6 5 7 2 2 2 4 2 6 3 > ----- + ----- + ----- + ---- + ---- + 2 t + 4 q t + q t + 2 q t + q t 6 3 6 2 4 2 4 2 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a13

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