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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-21/2 + 2q-19/2 - 4q-17/2 + 6q-15/2 - 8q-13/2 + 8q-11/2 - 8q-9/2 + 7q-7/2 - 6q-5/2 + 3q-3/2 - 2q-1/2 + q1/2

A2 (sl(3)) Invariant:

q-30 + 2q-26 + q-22 + q-20 + 3q-16 - q-14 + 2q-12 + q-8 + q-6 - q-4 - 1

HOMFLY-PT Polynomial:

a3z-1 + 6a3z + 11a3z3 + 6a3z5 + a3z7 - 3a5z-1 - 19a5z - 33a5z3 - 24a5z5 - 8a5z7 - a5z9 + 2a7z-1 + 9a7z + 12a7z3 + 6a7z5 + a7z7

Kauffman Polynomial:

- a2 + 6a2z2 - 11a2z4 + 6a2z6 - a2z8 + a3z-1 - 6a3z + 19a3z3 - 24a3z5 + 12a3z7 - 2a3z9 - 3a4 + 17a4z2 - 25a4z4 + 9a4z6 + 2a4z8 - a4z10 + 3a5z-1 - 21a5z + 45a5z3 - 48a5z5 + 26a5z7 - 5a5z9 - 3a6 + 13a6z2 - 23a6z4 + 16a6z6 - a6z8 - a6z10 + 2a7z-1 - 10a7z + 13a7z3 - 9a7z5 + 9a7z7 - 3a7z9 - a8z2 - 2a8z4 + 9a8z6 - 4a8z8 + 3a9z - 10a9z3 + 12a9z5 - 5a9z7 - 2a10z2 + 5a10z4 - 4a10z6 - a11z + 2a11z3 - 3a11z5 + a12z2 - 2a12z4 + a13z - a13z3

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 = 2                       1 j = 0                     1   j = -2                   2 1   j = -4                 4 1     j = -6               4 3       j = -8             4 3         j = -10           4 4           j = -12         4 4             j = -14       2 4               j = -16     2 4                 j = -18     2                   j = -20 1 2                     j = -22 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, 304]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 304]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[16, 5, 17, 6], > X[22, 20, 9, 19], X[20, 14, 21, 13], X[14, 22, 15, 21], X[18, 7, 19, 8], > X[8, 9, 1, 10], X[4, 15, 5, 16], X[6, 17, 7, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -10, 4, -11, 8, -9}, > {9, -1, 2, -3, 6, -7, 10, -4, 11, -8, 5, -6, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 2 4 6 8 8 8 7 6 3 -q + ----- - ----- + ----- - ----- + ----- - ---- + ---- - ---- + ---- - 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q 2 > ------- + Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 2 -22 -20 3 -14 2 -8 -6 -4 -1 + q + --- + q + q + --- - q + --- + q + q - q 26 16 12 q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 304]][a, z]
Out[8]=	3 5 7 a 3 a 2 a 3 5 7 3 3 5 3 7 3 -- - ---- + ---- + 6 a z - 19 a z + 9 a z + 11 a z - 33 a z + 12 a z + z z z 3 5 5 5 7 5 3 7 5 7 7 7 5 9 > 6 a z - 24 a z + 6 a z + a z - 8 a z + a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 304]][a, z]
Out[9]=	3 5 7 2 4 6 a 3 a 2 a 3 5 7 9 -a - 3 a - 3 a + -- + ---- + ---- - 6 a z - 21 a z - 10 a z + 3 a z - z z z 11 13 2 2 4 2 6 2 8 2 10 2 12 2 > a z + a z + 6 a z + 17 a z + 13 a z - a z - 2 a z + a z + 3 3 5 3 7 3 9 3 11 3 13 3 2 4 > 19 a z + 45 a z + 13 a z - 10 a z + 2 a z - a z - 11 a z - 4 4 6 4 8 4 10 4 12 4 3 5 5 5 > 25 a z - 23 a z - 2 a z + 5 a z - 2 a z - 24 a z - 48 a z - 7 5 9 5 11 5 2 6 4 6 6 6 8 6 > 9 a z + 12 a z - 3 a z + 6 a z + 9 a z + 16 a z + 9 a z - 10 6 3 7 5 7 7 7 9 7 2 8 4 8 > 4 a z + 12 a z + 26 a z + 9 a z - 5 a z - a z + 2 a z - 6 8 8 8 3 9 5 9 7 9 4 10 6 10 > a z - 4 a z - 2 a z - 5 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 1 1 2 2 2 4 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 22 8 20 8 20 7 18 6 16 6 16 5 14 5 q q q t q t q t q t q t q t q t 4 4 4 4 4 4 3 4 t > ------ + ------ + ------ + ------ + ------ + ----- + ---- + ---- + -- + 14 4 12 4 12 3 10 3 10 2 8 2 8 6 4 q t q t q t q t q t q t q t q t q 2 2 t 2 t 2 3 > --- + t + -- + q t 2 2 q q

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