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DrawMorseLink

PD Presentation:

X6172 X18,7,19,8 X4,19,1,20 X5,12,6,13 X3849 X13,17,14,16 X9,15,10,14 X15,11,16,10 X11,20,12,5 X17,2,18,3

Gauss Code:

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

Jones Polynomial:

q-13/2 - 3q-11/2 + 4q-9/2 - 5q-7/2 + 4q-5/2 - 5q-3/2 + 3q-1/2 - 2q1/2 + q3/2

A2 (sl(3)) Invariant:

- q-24 - q-20 + q-18 + q-16 + q-14 + 3q-12 + q-10 + 4q-8 + q-6 + q-4 - 1 - q4

HOMFLY-PT Polynomial:

az-1 + 4az + 4az3 + az5 - 4a3z-1 - 13a3z - 13a3z3 - 6a3z5 - a3z7 + 4a5z-1 + 8a5z + 5a5z3 + a5z5 - a7z-1 - a7z

Kauffman Polynomial:

1 - 4z2 + 4z4 - z6 - az-1 + 5az - 9az3 + 8az5 - 2az7 + 4a2 - 13a2z2 + 11a2z4 - a2z8 - 4a3z-1 + 17a3z - 26a3z3 + 20a3z5 - 5a3z7 + 7a4 - 18a4z2 + 13a4z4 - a4z6 - a4z8 - 4a5z-1 + 14a5z - 20a5z3 + 12a5z5 - 3a5z7 + 4a6 - 10a6z2 + 6a6z4 - 2a6z6 - a7z-1 + 2a7z - 3a7z3 + a8 - a8z2

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n17

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