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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9/2 - 3q-7/2 + 6q-5/2 - 9q-3/2 + 10q-1/2 - 13q1/2 + 11q3/2 - 9q5/2 + 6q7/2 - 3q9/2 + q11/2

A2 (sl(3)) Invariant:

- q-14 - 2q-8 + 2q-6 + 3q-2 + 5 + q2 + 4q4 - 2q6 + q8 - 2q12 + q14 - q16

HOMFLY-PT Polynomial:

a-3z-1 + 3a-3z + 3a-3z3 + a-3z5 - 4a-1z-1 - 9a-1z - 10a-1z3 - 5a-1z5 - a-1z7 + 4az-1 + 8az + 7az3 + 2az5 - a3z-1 - 2a3z - a3z3

Kauffman Polynomial:

a-6z2 - a-6z4 + 3a-5z3 - 3a-5z5 + a-4 - 3a-4z2 + 6a-4z4 - 5a-4z6 - a-3z-1 + 6a-3z - 11a-3z3 + 10a-3z5 - 6a-3z7 + 4a-2 - 12a-2z2 + 9a-2z4 + a-2z6 - 4a-2z8 - 4a-1z-1 + 17a-1z - 38a-1z3 + 35a-1z5 - 11a-1z7 - a-1z9 + 7 - 16z2 + 7z4 + 11z6 - 7z8 - 4az-1 + 15az - 32az3 + 31az5 - 8az7 - az9 + 4a2 - 11a2z2 + 8a2z4 + 4a2z6 - 3a2z8 - a3z-1 + 4a3z - 8a3z3 + 9a3z5 - 3a3z7 + a4 - 3a4z2 + 3a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12                     1 j = 10                   2   j = 8                 4 1   j = 6               5 2     j = 4             6 4       j = 2           7 5         j = 0         5 8           j = -2       4 5             j = -4     2 5               j = -6   1 4                 j = -8   2                   j = -10 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, 29]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 29]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 8, 15, 7], X[20, 12, 5, 11], > X[8, 20, 9, 19], X[16, 9, 17, 10], X[18, 15, 19, 16], X[10, 17, 11, 18], > X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 3, -5, 6, -8, 4, -2, 10, -3, 7, -6, 8, -7, > 5, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 3 6 9 10 3/2 5/2 q - ---- + ---- - ---- + ------- - 13 Sqrt[q] + 11 q - 9 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 > 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 2 2 3 2 4 6 8 12 14 16 5 - q - -- + -- + -- + q + 4 q - 2 q + q - 2 q + q - q 8 6 2 q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 29]][a, z]
Out[8]=	3 3 3 1 4 4 a a 3 z 9 z 3 3 z 10 z 3 ---- - --- + --- - -- + --- - --- + 8 a z - 2 a z + ---- - ----- + 7 a z - 3 a z z z 3 a 3 a a z a a 5 5 7 3 3 z 5 z 5 z > a z + -- - ---- + 2 a z - -- 3 a a a
In[9]:=	Kauffman[Link[10, Alternating, 29]][a, z]
Out[9]=	3 -4 4 2 4 1 4 4 a a 6 z 17 z 7 + a + -- + 4 a + a - ---- - --- - --- - -- + --- + ---- + 15 a z + 2 3 a z z z 3 a a a z a 2 2 2 3 3 3 2 z 3 z 12 z 2 2 4 2 3 z 11 z > 4 a z - 16 z + -- - ---- - ----- - 11 a z - 3 a z + ---- - ----- - 6 4 2 5 3 a a a a a 3 4 4 4 38 z 3 3 3 4 z 6 z 9 z 2 4 4 4 > ----- - 32 a z - 8 a z + 7 z - -- + ---- + ---- + 8 a z + 3 a z - a 6 4 2 a a a 5 5 5 6 6 3 z 10 z 35 z 5 3 5 6 5 z z 2 6 > ---- + ----- + ----- + 31 a z + 9 a z + 11 z - ---- + -- + 4 a z - 5 3 a 4 2 a a a a 7 7 8 9 4 6 6 z 11 z 7 3 7 8 4 z 2 8 z 9 > a z - ---- - ----- - 8 a z - 3 a z - 7 z - ---- - 3 a z - -- - a z 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 4 2 5 4 5 5 8 + 7 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + ---- + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t q t q t 2 4 4 2 6 2 6 3 8 3 8 4 > 5 q t + 6 q t + 4 q t + 5 q t + 2 q t + 4 q t + q t + 10 4 12 5 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a29

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