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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-23/2 - 2q-21/2 + 5q-19/2 - 8q-17/2 + 10q-15/2 - 12q-13/2 + 11q-11/2 - 11q-9/2 + 7q-7/2 - 5q-5/2 + 3q-3/2 - q-1/2

A2 (sl(3)) Invariant:

- q-36 - 2q-34 - q-32 - 2q-30 - q-28 + 2q-26 + q-24 + 5q-22 + 2q-20 + 2q-18 + 3q-16 - q-14 + 3q-12 - q-10 - q-4 + q-2

HOMFLY-PT Polynomial:

- a3z - 3a3z3 - a3z5 + a5z + 4a5z3 + 4a5z5 + a5z7 - 3a7z-1 - 10a7z - 11a7z3 - 3a7z5 + 5a9z-1 + 9a9z + 3a9z3 - 2a11z-1 - a11z

Kauffman Polynomial:

a3z - 4a3z3 + 4a3z5 - a3z7 + 3a4z2 - 15a4z4 + 13a4z6 - 3a4z8 + 2a5z3 - 10a5z5 + 11a5z7 - 3a5z9 + 5a6z2 - 17a6z4 + 16a6z6 - 2a6z8 - a6z10 - 3a7z-1 + 9a7z - 4a7z3 - 8a7z5 + 14a7z7 - 5a7z9 + 5a8 - 9a8z2 + 4a8z4 + 6a8z6 - 2a8z8 - a8z10 - 5a9z-1 + 15a9z - 20a9z3 + 14a9z5 - 2a9z7 - 2a9z9 + 5a10 - 13a10z2 + 9a10z4 - 3a10z8 - 2a11z-1 + 5a11z - 8a11z3 + 6a11z5 - 4a11z7 + 2a12z4 - 3a12z6 + 2a13z3 - 2a13z5 - a14 + 2a14z2 - 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                   3 1   j = -6                 5 3     j = -8               6 2       j = -10             5 5         j = -12           7 6           j = -14         4 6             j = -16       4 6               j = -18     1 4                 j = -20   1 4                   j = -22   1                     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, 123]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 123]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[16, 12, 17, 11], X[12, 16, 13, 15], > X[22, 17, 5, 18], X[18, 7, 19, 8], X[20, 9, 21, 10], X[8, 19, 9, 20], > X[10, 21, 11, 22], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 6, -8, 7, -9, 3, -4, 11, -2, 4, -3, 5, -6, > 8, -7, 9, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) 2 5 8 10 12 11 11 7 5 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]=	-36 2 -32 2 -28 2 -24 5 2 2 3 -14 -q - --- - q - --- - q + --- + q + --- + --- + --- + --- - q + 34 30 26 22 20 18 16 q q q q q q q 3 -10 -4 -2 > --- - q - q + q 12 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 123]][a, z]
Out[8]=	7 9 11 -3 a 5 a 2 a 3 5 7 9 11 3 3 ----- + ---- - ----- - a z + a z - 10 a z + 9 a z - a z - 3 a z + z z z 5 3 7 3 9 3 3 5 5 5 7 5 5 7 > 4 a z - 11 a z + 3 a z - a z + 4 a z - 3 a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 123]][a, z]
Out[9]=	7 9 11 8 10 14 3 a 5 a 2 a 3 7 9 11 5 a + 5 a - a - ---- - ---- - ----- + a z + 9 a z + 15 a z + 5 a z + z z z 4 2 6 2 8 2 10 2 14 2 3 3 5 3 > 3 a z + 5 a z - 9 a z - 13 a z + 2 a z - 4 a z + 2 a z - 7 3 9 3 11 3 13 3 4 4 6 4 8 4 > 4 a z - 20 a z - 8 a z + 2 a z - 15 a z - 17 a z + 4 a z + 10 4 12 4 14 4 3 5 5 5 7 5 9 5 > 9 a z + 2 a z - a z + 4 a z - 10 a z - 8 a z + 14 a z + 11 5 13 5 4 6 6 6 8 6 12 6 3 7 > 6 a z - 2 a z + 13 a z + 16 a z + 6 a z - 3 a z - a z + 5 7 7 7 9 7 11 7 4 8 6 8 8 8 > 11 a z + 14 a z - 2 a z - 4 a z - 3 a z - 2 a z - 2 a z - 10 8 5 9 7 9 9 9 6 10 8 10 > 3 a z - 3 a z - 5 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 1 1 1 4 1 4 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 6 4 6 7 6 5 5 6 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 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 2 5 t 2 t 2 > ---- + ---- + -- + --- + t 8 6 4 2 q t q t q q

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