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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X6,9,7,10 X20,7,21,8 X8,19,1,20 X18,13,19,14 X16,6,17,5 X4,18,5,17 X22,15,9,16 X14,21,15,22

Gauss Code:

{{1, -2, 3, -9, 8, -4, 5, -6}, {4, -1, 2, -3, 7, -11, 10, -8, 9, -7, 6, -5, 11, -10}}

Jones Polynomial:

q-23/2 - 3q-21/2 + 7q-19/2 - 12q-17/2 + 15q-15/2 - 17q-13/2 + 17q-11/2 - 15q-9/2 + 10q-7/2 - 7q-5/2 + 3q-3/2 - q-1/2

A2 (sl(3)) Invariant:

- q-34 + q-32 - 2q-30 - q-28 + 2q-26 - 2q-24 + 4q-22 - q-20 + q-18 + 3q-16 - q-14 + 5q-12 - q-10 + q-8 + q-6 - q-4 + q-2

HOMFLY-PT Polynomial:

- 2a3z - 3a3z3 - a3z5 - 2a5z-1 - 5a5z + 3a5z5 + a5z7 + 3a7z-1 + 8a7z + 7a7z3 + 4a7z5 + a7z7 - a9z-1 - 3a9z - 3a9z3 - a9z5

Kauffman Polynomial:

2a3z - 5a3z3 + 4a3z5 - a3z7 + 2a4z2 - 11a4z4 + 11a4z6 - 3a4z8 + 2a5z-1 - 7a5z + 7a5z3 - 12a5z5 + 13a5z7 - 4a5z9 - 3a6 + 11a6z2 - 21a6z4 + 18a6z6 - a6z8 - 2a6z10 + 3a7z-1 - 12a7z + 23a7z3 - 31a7z5 + 30a7z7 - 10a7z9 - 3a8 + 17a8z2 - 29a8z4 + 29a8z6 - 7a8z8 - 2a8z10 + a9z-1 - 4a9z3 + 3a9z5 + 7a9z7 - 6a9z9 - a10 + 4a10z2 - 12a10z4 + 16a10z6 - 9a10z8 + 3a11z - 13a11z3 + 15a11z5 - 9a11z7 - 3a12z2 + 6a12z4 - 6a12z6 + 2a13z3 - 3a13z5 + a14z2 - a14z4

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 0                       1 j = -2                     2   j = -4                   5 1   j = -6                 6 3     j = -8               9 4       j = -10             8 6         j = -12           9 9           j = -14         7 9             j = -16       5 8               j = -18     2 7                 j = -20   1 5                   j = -22   2                     j = -24 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[11, Alternating, 308]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 308]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[6, 9, 7, 10], > X[20, 7, 21, 8], X[8, 19, 1, 20], X[18, 13, 19, 14], X[16, 6, 17, 5], > X[4, 18, 5, 17], X[22, 15, 9, 16], X[14, 21, 15, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -9, 8, -4, 5, -6}, > {4, -1, 2, -3, 7, -11, 10, -8, 9, -7, 6, -5, 11, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) 3 7 12 15 17 17 15 10 7 q - ----- + ----- - ----- + ----- - ----- + ----- - ---- + ---- - ---- + 21/2 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 q q q q q q q q q 3 1 > ---- - ------- 3/2 Sqrt[q] q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 -32 2 -28 2 2 4 -20 -18 3 -14 5 -q + q - --- - q + --- - --- + --- - q + q + --- - q + --- - 30 26 24 22 16 12 q q q q q q -10 -8 -6 -4 -2 > q + q + q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 308]][a, z]
Out[8]=	5 7 9 -2 a 3 a a 3 5 7 9 3 3 7 3 ----- + ---- - -- - 2 a z - 5 a z + 8 a z - 3 a z - 3 a z + 7 a z - z z z 9 3 3 5 5 5 7 5 9 5 5 7 7 7 > 3 a z - a z + 3 a z + 4 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 308]][a, z]
Out[9]=	5 7 9 6 8 10 2 a 3 a a 3 5 7 11 -3 a - 3 a - a + ---- + ---- + -- + 2 a z - 7 a z - 12 a z + 3 a z + z z z 4 2 6 2 8 2 10 2 12 2 14 2 3 3 > 2 a z + 11 a z + 17 a z + 4 a z - 3 a z + a z - 5 a z + 5 3 7 3 9 3 11 3 13 3 4 4 6 4 > 7 a z + 23 a z - 4 a z - 13 a z + 2 a z - 11 a z - 21 a z - 8 4 10 4 12 4 14 4 3 5 5 5 7 5 > 29 a z - 12 a z + 6 a z - a z + 4 a z - 12 a z - 31 a z + 9 5 11 5 13 5 4 6 6 6 8 6 > 3 a z + 15 a z - 3 a z + 11 a z + 18 a z + 29 a z + 10 6 12 6 3 7 5 7 7 7 9 7 11 7 > 16 a z - 6 a z - a z + 13 a z + 30 a z + 7 a z - 9 a z - 4 8 6 8 8 8 10 8 5 9 7 9 9 9 > 3 a z - a z - 7 a z - 9 a z - 4 a z - 10 a z - 6 a z - 6 10 8 10 > 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 1 2 1 5 2 7 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 24 9 22 8 20 8 20 7 18 7 18 6 16 6 q q q t q t q t q t q t q t q t 8 7 9 9 9 8 6 9 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 16 5 14 5 14 4 12 4 12 3 10 3 10 2 8 2 q t q t q t q t q t q t q t q t 4 6 t 2 t 2 > ---- + ---- + -- + --- + t 8 6 4 2 q t q t q q

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