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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X7,16,8,17 X17,22,18,5 X11,18,12,19 X21,12,22,13 X13,20,14,21 X19,14,20,15 X15,8,16,9 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-25/2 - q-23/2 + 2q-21/2 - 2q-19/2 + q-17/2 - q-15/2 - q-13/2 - q-9/2 + q-7/2 - q-5/2

A2 (sl(3)) Invariant:

- q-40 - 2q-38 - q-36 - 2q-34 - q-32 + q-30 + q-28 + 3q-26 + 2q-24 + 3q-22 + 2q-20 + q-18 + 2q-16 + q-8

HOMFLY-PT Polynomial:

- 2a5z - 4a5z3 - a5z5 - 2a7z-1 - 5a7z - 5a7z3 - a7z5 + 2a9z-1 + 3a9z + a11z-1 + 2a11z - a13z-1

Kauffman Polynomial:

- 2a5z + 4a5z3 - a5z5 - 2a6z2 + 4a6z4 - a6z6 - 2a7z-1 + 8a7z - 10a7z3 + 6a7z5 - a7z7 + 2a8 - 8a8z2 + 7a8z4 - a8z6 - 2a9z-1 + 13a9z - 24a9z3 + 14a9z5 - 2a9z7 - 4a10 + 13a10z2 - 22a10z4 + 13a10z6 - 2a10z8 + a11z-1 + 2a11z - 5a11z3 - 3a11z5 + 5a11z7 - a11z9 - 9a12 + 33a12z2 - 41a12z4 + 20a12z6 - 3a12z8 + a13z-1 - a13z + 5a13z3 - 10a13z5 + 6a13z7 - a13z9 - 4a14 + 14a14z2 - 16a14z4 + 7a14z6 - a14z8

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                     1 1 j = -8                 1 1     j = -10               1 1 1     j = -12             2 4 1       j = -14           1   1         j = -16         1 3 2           j = -18       2 1               j = -20       1 1               j = -22   1 2                   j = -24                         j = -26 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, NonAlternating, 71]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 71]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[7, 16, 8, 17], X[17, 22, 18, 5], > X[11, 18, 12, 19], X[21, 12, 22, 13], X[13, 20, 14, 21], X[19, 14, 20, 15], > X[15, 8, 16, 9], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, -5, 6, -7, 8, -9, 3, -4, 5, > -8, 7, -6, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) -(23/2) 2 2 -(17/2) -(15/2) -(13/2) q - q + ----- - ----- + q - q - q - 21/2 19/2 q q -(9/2) -(7/2) -(5/2) > q + q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-40 2 -36 2 -32 -30 -28 3 2 3 2 -18 -q - --- - q - --- - q + q + q + --- + --- + --- + --- + q + 38 34 26 24 22 20 q q q q q q 2 -8 > --- + q 16 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 71]][a, z]
Out[8]=	7 9 11 13 -2 a 2 a a a 5 7 9 11 5 3 ----- + ---- + --- - --- - 2 a z - 5 a z + 3 a z + 2 a z - 4 a z - z z z z 7 3 5 5 7 5 > 5 a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 71]][a, z]
Out[9]=	7 9 11 13 8 10 12 14 2 a 2 a a a 5 7 2 a - 4 a - 9 a - 4 a - ---- - ---- + --- + --- - 2 a z + 8 a z + z z z z 9 11 13 6 2 8 2 10 2 12 2 > 13 a z + 2 a z - a z - 2 a z - 8 a z + 13 a z + 33 a z + 14 2 5 3 7 3 9 3 11 3 13 3 6 4 > 14 a z + 4 a z - 10 a z - 24 a z - 5 a z + 5 a z + 4 a z + 8 4 10 4 12 4 14 4 5 5 7 5 9 5 > 7 a z - 22 a z - 41 a z - 16 a z - a z + 6 a z + 14 a z - 11 5 13 5 6 6 8 6 10 6 12 6 14 6 > 3 a z - 10 a z - a z - a z + 13 a z + 20 a z + 7 a z - 7 7 9 7 11 7 13 7 10 8 12 8 14 8 > a z - 2 a z + 5 a z + 6 a z - 2 a z - 3 a z - a z - 11 9 13 9 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 1 2 1 2 1 1 q + q + ------- + ------- + ------ + ------ + ------ + ------ + ------ + 26 11 22 10 22 9 20 8 18 8 20 7 18 7 q t q t q t q t q t q t q t 1 3 1 2 2 1 4 1 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 16 7 16 6 14 6 16 5 12 5 14 4 12 4 10 4 q t q t q t q t q t q t q t q t 1 1 1 1 1 1 > ------ + ------ + ----- + ------ + ----- + ---- 12 3 10 3 8 3 10 2 8 2 6 q t q t q t q t q t q t

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