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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-29/2 - 2q-27/2 + 5q-25/2 - 8q-23/2 + 11q-21/2 - 12q-19/2 + 11q-17/2 - 11q-15/2 + 7q-13/2 - 5q-11/2 + 2q-9/2 - q-7/2

A2 (sl(3)) Invariant:

- q-44 - 2q-42 - q-40 - 3q-38 - q-34 + 3q-30 + 6q-26 + 2q-24 + 3q-22 + 2q-20 - q-18 + 2q-16 - q-14 + q-12

HOMFLY-PT Polynomial:

- 2a7z - 7a7z3 - 5a7z5 - a7z7 - 4a9z-1 - 15a9z - 20a9z3 - 11a9z5 - 2a9z7 + 7a11z-1 + 19a11z + 14a11z3 + 3a11z5 - 3a13z-1 - 4a13z - a13z3

Kauffman Polynomial:

2a7z - 7a7z3 + 5a7z5 - a7z7 + a8z2 - 8a8z4 + 8a8z6 - 2a8z8 + 4a9z-1 - 17a9z + 29a9z3 - 26a9z5 + 14a9z7 - 3a9z9 - 7a10 + 24a10z2 - 31a10z4 + 19a10z6 - 2a10z8 - a10z10 + 7a11z-1 - 30a11z + 53a11z3 - 44a11z5 + 24a11z7 - 6a11z9 - 7a12 + 24a12z2 - 30a12z4 + 20a12z6 - 4a12z8 - a12z10 + 3a13z-1 - 11a13z + 11a13z3 - 6a13z5 + 5a13z7 - 3a13z9 - a14z2 - 4a14z4 + 6a14z6 - 4a14z8 - 4a15z3 + 5a15z5 - 4a15z7 + 2a16z4 - 3a16z6 + 2a17z3 - 2a17z5 - a18 + 2a18z2 - a18z4

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 = -6                       1 j = -8                     2 1 j = -10                   3     j = -12                 4 2     j = -14               7 3       j = -16             5 5         j = -18           7 6           j = -20         4 5             j = -22       4 7               j = -24     1 4                 j = -26   1 4                   j = -28   1                     j = -30 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, 72]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 72]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[20, 13, 21, 14], X[16, 7, 17, 8], > X[18, 9, 19, 10], X[8, 17, 9, 18], X[10, 19, 11, 20], X[22, 15, 5, 16], > X[14, 21, 15, 22], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -6, 5, -7, 11, -2, 3, -9, 8, -4, 6, -5, > 7, -3, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(29/2) 2 5 8 11 12 11 11 7 q - ----- + ----- - ----- + ----- - ----- + ----- - ----- + ----- - 27/2 25/2 23/2 21/2 19/2 17/2 15/2 13/2 q q q q q q q q 5 2 -(7/2) > ----- + ---- - q 11/2 9/2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-44 2 -40 3 -34 3 6 2 3 2 -18 2 -q - --- - q - --- - q + --- + --- + --- + --- + --- - q + --- - 42 38 30 26 24 22 20 16 q q q q q q q q -14 -12 > q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 72]][a, z]
Out[8]=	9 11 13 -4 a 7 a 3 a 7 9 11 13 7 3 ----- + ----- - ----- - 2 a z - 15 a z + 19 a z - 4 a z - 7 a z - z z z 9 3 11 3 13 3 7 5 9 5 11 5 7 7 > 20 a z + 14 a z - a z - 5 a z - 11 a z + 3 a z - a z - 9 7 > 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 72]][a, z]
Out[9]=	9 11 13 10 12 18 4 a 7 a 3 a 7 9 11 -7 a - 7 a - a + ---- + ----- + ----- + 2 a z - 17 a z - 30 a z - z z z 13 8 2 10 2 12 2 14 2 18 2 7 3 > 11 a z + a z + 24 a z + 24 a z - a z + 2 a z - 7 a z + 9 3 11 3 13 3 15 3 17 3 8 4 > 29 a z + 53 a z + 11 a z - 4 a z + 2 a z - 8 a z - 10 4 12 4 14 4 16 4 18 4 7 5 9 5 > 31 a z - 30 a z - 4 a z + 2 a z - a z + 5 a z - 26 a z - 11 5 13 5 15 5 17 5 8 6 10 6 > 44 a z - 6 a z + 5 a z - 2 a z + 8 a z + 19 a z + 12 6 14 6 16 6 7 7 9 7 11 7 13 7 > 20 a z + 6 a z - 3 a z - a z + 14 a z + 24 a z + 5 a z - 15 7 8 8 10 8 12 8 14 8 9 9 11 9 > 4 a z - 2 a z - 2 a z - 4 a z - 4 a z - 3 a z - 6 a z - 13 9 10 10 12 10 > 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 1 1 4 1 4 4 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 30 11 28 10 26 10 26 9 24 9 24 8 22 8 q t q t q t q t q t q t q t 7 4 5 7 6 5 5 7 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 22 7 20 7 20 6 18 6 18 5 16 5 16 4 14 4 q t q t q t q t q t q t q t q t 3 4 2 3 2 > ------ + ------ + ------ + ------ + ---- 14 3 12 3 12 2 10 2 8 q t q t q t q t q t

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