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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X7,14,8,15 X20,15,17,16 X18,11,19,12 X12,17,13,18 X16,19,5,20 X13,8,14,9 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

2q-10 - 4q-9 + 7q-8 - 8q-7 + 9q-6 - 7q-5 + 7q-4 - 3q-3 + q-2

A2 (sl(3)) Invariant:

2q-32 + 3q-30 + 3q-26 + 2q-24 + 2q-22 + 5q-20 + 2q-18 + 5q-16 + q-14 + q-12 + 2q-10 - 2q-8 + q-6

HOMFLY-PT Polynomial:

a4z2 + a4z4 + a6z-2 + 6a6 + 8a6z2 + 3a6z4 - 2a8z-2 - 8a8 - 5a8z2 + a10z-2 + 2a10

Kauffman Polynomial:

- a4z2 + a4z4 - 2a5z3 + 3a5z5 + a6z-2 - 6a6 + 10a6z2 - 11a6z4 + 6a6z6 - 2a7z-1 + 8a7z - 8a7z3 - a7z5 + 4a7z7 + 2a8z-2 - 9a8 + 16a8z2 - 18a8z4 + 8a8z6 + a8z8 - 2a9z-1 + 8a9z - 8a9z3 - 3a9z5 + 5a9z7 + a10z-2 - 2a10 - 3a10z4 + 2a10z6 + a10z8 - 2a11z3 + a11z5 + a11z7 + 2a12 - 5a12z2 + 3a12z4

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -3                 1 j = -5               3 1 j = -7             4     j = -9           3 3     j = -11         6 4       j = -13       4 5         j = -15     3 4           j = -17   1 4             j = -19 1 3               j = -21 2

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]=	3
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 89]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 89]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[7, 14, 8, 15], X[20, 15, 17, 16], > X[18, 11, 19, 12], X[12, 17, 13, 18], X[16, 19, 5, 20], X[13, 8, 14, 9], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {6, -5, 7, -4}, > {9, -1, -3, 8, 10, -2, 5, -6, -8, 3, 4, -7}]
In[6]:=	Jones[L][q]
Out[6]=	2 4 7 8 9 7 7 3 -2 --- - -- + -- - -- + -- - -- + -- - -- + q 10 9 8 7 6 5 4 3 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 3 3 2 2 5 2 5 -14 -12 2 2 -6 --- + --- + --- + --- + --- + --- + --- + --- + q + q + --- - -- + q 32 30 26 24 22 20 18 16 10 8 q q q q q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 89]][a, z]
Out[8]=	6 8 10 6 8 10 a 2 a a 4 2 6 2 8 2 4 4 6 a - 8 a + 2 a + -- - ---- + --- + a z + 8 a z - 5 a z + a z + 2 2 2 z z z 6 4 > 3 a z
In[9]:=	Kauffman[Link[10, NonAlternating, 89]][a, z]
Out[9]=	6 8 10 7 9 6 8 10 12 a 2 a a 2 a 2 a 7 -6 a - 9 a - 2 a + 2 a + -- + ---- + --- - ---- - ---- + 8 a z + 2 2 2 z z z z z 9 4 2 6 2 8 2 12 2 5 3 7 3 > 8 a z - a z + 10 a z + 16 a z - 5 a z - 2 a z - 8 a z - 9 3 11 3 4 4 6 4 8 4 10 4 12 4 > 8 a z - 2 a z + a z - 11 a z - 18 a z - 3 a z + 3 a z + 5 5 7 5 9 5 11 5 6 6 8 6 10 6 > 3 a z - a z - 3 a z + a z + 6 a z + 8 a z + 2 a z + 7 7 9 7 11 7 8 8 10 8 > 4 a z + 5 a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-5 -3 2 1 3 1 4 3 4 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 21 8 19 8 19 7 17 7 17 6 15 6 15 5 q t q t q t q t q t q t q t 4 5 6 4 3 3 4 3 > ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- 13 5 13 4 11 4 11 3 9 3 9 2 7 2 5 q t q t q t q t q t q t q t q t

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