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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X5,12,6,13 X8493 X9,16,10,17 X17,20,18,5 X11,19,12,18 X19,11,20,10 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-20 - q-18 + q-16 + 2q-14 - q-12 + q-10 - 2q-8 - q-6 + 4 + q2 + 2q4 + 2q6

HOMFLY-PT Polynomial:

- 2a-1z-1 - 2a-1z + 4az-1 + 6az + 3az3 - 3a3z-1 - 5a3z - 3a3z3 - a3z5 + a5z-1 + a5z + a5z3

Kauffman Polynomial:

- 2a-1z-1 + 5a-1z - 3a-1z3 + 2 - 2z2 - z6 - 4az-1 + 13az - 14az3 + 5az5 - 2az7 + 3a2 - 10a2z2 + 7a2z4 - 2a2z6 - a2z8 - 3a3z-1 + 9a3z - 12a3z3 + 11a3z5 - 5a3z7 + 3a4 - 11a4z2 + 14a4z4 - 4a4z6 - a4z8 - a5z-1 + a5z3 + 5a5z5 - 3a5z7 + a6 - 3a6z2 + 7a6z4 - 3a6z6 - a7z + 2a7z3 - a7z5

Khovanov Homology:

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

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