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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X22,13,5,14 X14,7,15,8 X18,9,19,10 X20,17,21,18 X16,21,17,22 X8,15,9,16 X10,19,11,20 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

q-27/2 - 3q-25/2 + 7q-23/2 - 11q-21/2 + 16q-19/2 - 19q-17/2 + 18q-15/2 - 17q-13/2 + 12q-11/2 - 8q-9/2 + 3q-7/2 - q-5/2

A2 (sl(3)) Invariant:

- q-42 - q-40 + q-38 - 3q-36 - q-34 + q-32 - 4q-30 + 3q-28 + q-26 + 3q-24 + 5q-22 - q-20 + 5q-18 - q-16 + 3q-12 - 2q-10 + q-8

HOMFLY-PT Polynomial:

- a5z - 2a5z3 - a5z5 - 2a7z-1 - 6a7z - 8a7z3 - 3a7z5 + 2a9z-1 + a9z - 3a9z3 - 2a9z5 + a11z-1 + 5a11z + 3a11z3 - a13z-1 - a13z

Kauffman Polynomial:

- a5z + 2a5z3 - a5z5 - a6z2 + 4a6z4 - 3a6z6 - 2a7z-1 + 7a7z - 11a7z3 + 11a7z5 - 6a7z7 + 3a8 - 7a8z2 + 4a8z4 + 5a8z6 - 6a8z8 - 2a9z-1 + 7a9z - 11a9z3 + 11a9z5 - 2a9z7 - 4a9z9 + 3a10z2 - 14a10z4 + 22a10z6 - 11a10z8 - a10z10 + a11z-1 - 2a11z - a11z3 + 4a11z5 + 6a11z7 - 7a11z9 - 3a12 + 9a12z2 - 17a12z4 + 22a12z6 - 9a12z8 - a12z10 + a13z-1 - 9a13z3 + 13a13z5 - a13z7 - 3a13z9 - 3a14z2 + 7a14z6 - 4a14z8 + a15z - 6a15z3 + 8a15z5 - 3a15z7 + a16 - 3a16z2 + 3a16z4 - a16z6

Khovanov Homology:

trqj r = -11 r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -4                       1 j = -6                     3 1 j = -8                   5     j = -10                 7 3     j = -12               10 5       j = -14             9 8         j = -16           10 9           j = -18         6 9             j = -20       5 10               j = -22     2 6                 j = -24   1 5                   j = -26   2                     j = -28 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, 61]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 61]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[22, 13, 5, 14], X[14, 7, 15, 8], > X[18, 9, 19, 10], X[20, 17, 21, 18], X[16, 21, 17, 22], X[8, 15, 9, 16], > X[10, 19, 11, 20], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -8, 5, -9, 11, -2, 3, -4, 8, -7, 6, -5, > 9, -6, 7, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(27/2) 3 7 11 16 19 18 17 12 q - ----- + ----- - ----- + ----- - ----- + ----- - ----- + ----- - 25/2 23/2 21/2 19/2 17/2 15/2 13/2 11/2 q q q q q q q q 8 3 -(5/2) > ---- + ---- - q 9/2 7/2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 -40 -38 3 -34 -32 4 3 -26 3 5 -20 -q - q + q - --- - q + q - --- + --- + q + --- + --- - q + 36 30 28 24 22 q q q q q 5 -16 3 2 -8 > --- - q + --- - --- + q 18 12 10 q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 61]][a, z]
Out[8]=	7 9 11 13 -2 a 2 a a a 5 7 9 11 13 5 3 ----- + ---- + --- - --- - a z - 6 a z + a z + 5 a z - a z - 2 a z - z z z z 7 3 9 3 11 3 5 5 7 5 9 5 > 8 a z - 3 a z + 3 a z - a z - 3 a z - 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 61]][a, z]
Out[9]=	7 9 11 13 8 12 16 2 a 2 a a a 5 7 9 3 a - 3 a + a - ---- - ---- + --- + --- - a z + 7 a z + 7 a z - z z z z 11 15 6 2 8 2 10 2 12 2 14 2 > 2 a z + a z - a z - 7 a z + 3 a z + 9 a z - 3 a z - 16 2 5 3 7 3 9 3 11 3 13 3 15 3 > 3 a z + 2 a z - 11 a z - 11 a z - a z - 9 a z - 6 a z + 6 4 8 4 10 4 12 4 16 4 5 5 7 5 > 4 a z + 4 a z - 14 a z - 17 a z + 3 a z - a z + 11 a z + 9 5 11 5 13 5 15 5 6 6 8 6 > 11 a z + 4 a z + 13 a z + 8 a z - 3 a z + 5 a z + 10 6 12 6 14 6 16 6 7 7 9 7 11 7 > 22 a z + 22 a z + 7 a z - a z - 6 a z - 2 a z + 6 a z - 13 7 15 7 8 8 10 8 12 8 14 8 9 9 > a z - 3 a z - 6 a z - 11 a z - 9 a z - 4 a z - 4 a z - 11 9 13 9 10 10 12 10 > 7 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 2 1 5 2 6 5 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 28 11 26 10 24 10 24 9 22 9 22 8 20 8 q t q t q t q t q t q t q t 10 6 9 10 9 9 8 10 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 20 7 18 7 18 6 16 6 16 5 14 5 14 4 12 4 q t q t q t q t q t q t q t q t 5 7 3 5 3 > ------ + ------ + ------ + ----- + ---- 12 3 10 3 10 2 8 2 6 q t q t q t q t q t

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