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DrawMorseLink

PD Presentation:

X12,1,13,2 X16,8,17,7 X10,5,1,6 X6374 X4,9,5,10 X18,16,19,15 X22,19,11,20 X20,13,21,14 X14,21,15,22 X2,11,3,12 X8,18,9,17

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 3q-17/2 - 6q-15/2 + 12q-13/2 - 16q-11/2 + 19q-9/2 - 20q-7/2 + 16q-5/2 - 14q-3/2 + 8q-1/2 - 4q1/2 + q3/2

A2 (sl(3)) Invariant:

q-30 + q-28 - q-26 + q-24 - 2q-22 - 6q-20 - 4q-16 + 2q-14 + 4q-12 + 2q-10 + 8q-8 - q-6 + 4q-4 + q-2 - 2 + 2q2 - q4

HOMFLY-PT Polynomial:

az + 2az3 + az5 - 4a3z-1 - 9a3z - 8a3z3 - 4a3z5 - a3z7 + 8a5z-1 + 14a5z + 10a5z3 + 3a5z5 - 5a7z-1 - 7a7z - 3a7z3 + a9z-1 + a9z

Kauffman Polynomial:

- z2 + 2z4 - z6 + 2az - 8az3 + 10az5 - 4az7 - 2a2z2 - 3a2z4 + 12a2z6 - 6a2z8 - 4a3z-1 + 14a3z - 25a3z3 + 24a3z5 - 2a3z7 - 4a3z9 + 8a4 - 13a4z2 - 5a4z4 + 26a4z6 - 12a4z8 - a4z10 - 8a5z-1 + 21a5z - 33a5z3 + 27a5z5 - 7a5z9 + 14a6 - 29a6z2 + 12a6z4 + 13a6z6 - 10a6z8 - a6z10 - 5a7z-1 + 9a7z - 16a7z3 + 17a7z5 - 6a7z7 - 3a7z9 + 9a8 - 22a8z2 + 18a8z4 - 3a8z6 - 4a8z8 - a9z-1 - a9z + 2a9z3 + 3a9z5 - 4a9z7 + 2a10 - 5a10z2 + 6a10z4 - 3a10z6 - a11z + 2a11z3 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                       1 j = 2                     3   j = 0                   5 1   j = -2                 9 3     j = -4               9 7       j = -6             11 7         j = -8           8 9           j = -10         8 11             j = -12       4 8               j = -14     2 8                 j = -16   1 4                   j = -18   2                     j = -20 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, 343]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 343]]
Out[4]=	PD[X[12, 1, 13, 2], X[16, 8, 17, 7], X[10, 5, 1, 6], X[6, 3, 7, 4], > X[4, 9, 5, 10], X[18, 16, 19, 15], X[22, 19, 11, 20], X[20, 13, 21, 14], > X[14, 21, 15, 22], X[2, 11, 3, 12], X[8, 18, 9, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 4, -5, 3, -4, 2, -11, 5, -3}, > {10, -1, 8, -9, 6, -2, 11, -6, 7, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 3 6 12 16 19 20 16 14 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q 8 3/2 > ------- - 4 Sqrt[q] + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 -26 -24 2 6 4 2 4 2 8 -6 -2 + q + q - q + q - --- - --- - --- + --- + --- + --- + -- - q + 22 20 16 14 12 10 8 q q q q q q q 4 -2 2 4 > -- + q + 2 q - q 4 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 343]][a, z]
Out[8]=	3 5 7 9 -4 a 8 a 5 a a 3 5 7 9 3 ----- + ---- - ---- + -- + a z - 9 a z + 14 a z - 7 a z + a z + 2 a z - z z z z 3 3 5 3 7 3 5 3 5 5 5 3 7 > 8 a z + 10 a z - 3 a z + a z - 4 a z + 3 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 343]][a, z]
Out[9]=	3 5 7 9 4 6 8 10 4 a 8 a 5 a a 3 8 a + 14 a + 9 a + 2 a - ---- - ---- - ---- - -- + 2 a z + 14 a z + z z z z 5 7 9 11 2 2 2 4 2 6 2 > 21 a z + 9 a z - a z - a z - z - 2 a z - 13 a z - 29 a z - 8 2 10 2 3 3 3 5 3 7 3 9 3 > 22 a z - 5 a z - 8 a z - 25 a z - 33 a z - 16 a z + 2 a z + 11 3 4 2 4 4 4 6 4 8 4 10 4 > 2 a z + 2 z - 3 a z - 5 a z + 12 a z + 18 a z + 6 a z + 5 3 5 5 5 7 5 9 5 11 5 6 > 10 a z + 24 a z + 27 a z + 17 a z + 3 a z - a z - z + 2 6 4 6 6 6 8 6 10 6 7 3 7 > 12 a z + 26 a z + 13 a z - 3 a z - 3 a z - 4 a z - 2 a z - 7 7 9 7 2 8 4 8 6 8 8 8 3 9 > 6 a z - 4 a z - 6 a z - 12 a z - 10 a z - 4 a z - 4 a z - 5 9 7 9 4 10 6 10 > 7 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 9 1 2 1 4 2 8 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 7 16 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 8 8 11 8 9 11 7 9 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 5 t + 12 4 10 4 10 3 8 3 8 2 6 2 6 4 q t q t q t q t q t q t q t q t 3 t 2 2 2 4 3 > --- + t + 3 q t + q t 2 q

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