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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X7,16,8,17 X11,20,12,21 X17,22,18,5 X21,18,22,19 X19,10,20,11 X9,14,10,15 X15,8,16,9 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-30 - q-28 - q-26 - q-24 - q-20 + 2q-18 + q-16 + q-14 + 2q-12 + 3q-8 + q-6 + 2q-4 + q-2 + q2

HOMFLY-PT Polynomial:

- az-1 - 2az - az3 + a3z-1 + 2a3z + 3a3z3 + a3z5 - a5z-1 - 2a5z - a5z3 + 2a7z-1 + 3a7z + a7z3 - a9z-1 - a9z

Kauffman Polynomial:

az-1 - 3az + 3az3 - az5 - a2 - a2z2 + 5a2z4 - 2a2z6 + a3z-1 - 5a3z + 9a3z3 - a3z5 - a3z7 - a4z2 + 6a4z4 - 3a4z6 + a5z-1 - 7a5z + 10a5z3 - 4a5z5 - 4a6 + 15a6z2 - 20a6z4 + 8a6z6 - a6z8 + 2a7z-1 - 8a7z + 14a7z3 - 18a7z5 + 8a7z7 - a7z9 - 7a8 + 27a8z2 - 36a8z4 + 16a8z6 - 2a8z8 + a9z-1 - 3a9z + 10a9z3 - 14a9z5 + 7a9z7 - a9z9 - 3a10 + 12a10z2 - 15a10z4 + 7a10z6 - a10z8

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 = 2                       1 j = 0                     1   j = -2                   3 1   j = -4                 2 3     j = -6             1 3 1       j = -8             1 2         j = -10         1 3 3           j = -12       1   1             j = -14       1 2               j = -16   1 1                   j = -18                         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, NonAlternating, 91]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 91]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[7, 16, 8, 17], X[11, 20, 12, 21], > X[17, 22, 18, 5], X[21, 18, 22, 19], X[19, 10, 20, 11], X[9, 14, 10, 15], > X[15, 8, 16, 9], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -3, 9, -8, 7, -4, -2, 11, 8, -9, 3, -5, 6, > -7, 4, -6, 5}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) -(17/2) -(15/2) 2 3 4 3 4 2 q - q + q - ----- + ---- - ---- + ---- - ---- + ------- - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q > Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 -26 -24 -20 2 -16 -14 2 3 -6 2 -q - q - q - q - q + --- + q + q + --- + -- + q + -- + 18 12 8 4 q q q q -2 2 > q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 91]][a, z]
Out[8]=	3 5 7 9 a a a 2 a a 3 5 7 9 3 -(-) + -- - -- + ---- - -- - 2 a z + 2 a z - 2 a z + 3 a z - a z - a z + z z z z z 3 3 5 3 7 3 3 5 > 3 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 91]][a, z]
Out[9]=	3 5 7 9 2 6 8 10 a a a 2 a a 3 5 -a - 4 a - 7 a - 3 a + - + -- + -- + ---- + -- - 3 a z - 5 a z - 7 a z - z z z z z 7 9 2 2 4 2 6 2 8 2 10 2 > 8 a z - 3 a z - a z - a z + 15 a z + 27 a z + 12 a z + 3 3 3 5 3 7 3 9 3 2 4 4 4 > 3 a z + 9 a z + 10 a z + 14 a z + 10 a z + 5 a z + 6 a z - 6 4 8 4 10 4 5 3 5 5 5 7 5 > 20 a z - 36 a z - 15 a z - a z - a z - 4 a z - 18 a z - 9 5 2 6 4 6 6 6 8 6 10 6 3 7 > 14 a z - 2 a z - 3 a z + 8 a z + 16 a z + 7 a z - a z + 7 7 9 7 6 8 8 8 10 8 7 9 9 9 > 8 a z + 7 a z - a z - 2 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 1 1 1 1 1 2 1 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 9 16 8 16 7 14 6 12 6 14 5 10 5 q q q t q t q t q t q t q t q t 1 3 3 1 1 2 3 1 2 > ------ + ------ + ------ + ----- + ----- + ----- + ----- + ---- + ---- + 12 4 10 4 10 3 8 3 6 3 8 2 6 2 6 4 q t q t q t q t q t q t q t q t q t t 2 2 > t + -- + q t 2 q

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