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DrawMorseLink

PD Presentation:

X8192 X12,3,13,4 X14,19,15,20 X16,5,17,6 X18,10,19,9 X4,15,5,16 X10,18,11,17 X20,13,7,14 X2738 X6,11,1,12

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 3q-17/2 - 7q-15/2 + 10q-13/2 - 13q-11/2 + 14q-9/2 - 13q-7/2 + 10q-5/2 - 7q-3/2 + 3q-1/2 - q1/2

A2 (sl(3)) Invariant:

q-30 + q-28 - q-26 + 3q-24 - q-20 + 3q-18 - 2q-16 + 2q-14 - q-12 + 3q-8 - 2q-6 + 3q-4 - 1 + q2

HOMFLY-PT Polynomial:

- az - az3 - a3z-1 - 3a3z + a3z5 + 2a5z-1 + 5a5z + 5a5z3 + 2a5z5 - 2a7z-1 - 5a7z - 3a7z3 + a9z-1 + a9z

Kauffman Polynomial:

- az + 2az3 - az5 - a2z2 + 5a2z4 - 3a2z6 - a3z-1 + 5a3z - 6a3z3 + 9a3z5 - 5a3z7 + 2a4z2 - 5a4z4 + 8a4z6 - 5a4z8 - 2a5z-1 + 12a5z - 26a5z3 + 22a5z5 - 6a5z7 - 2a5z9 - a6 + 6a6z2 - 20a6z4 + 22a6z6 - 10a6z8 - 2a7z-1 + 12a7z - 26a7z3 + 22a7z5 - 6a7z7 - 2a7z9 + 2a8z2 - 5a8z4 + 8a8z6 - 5a8z8 - a9z-1 + 5a9z - 6a9z3 + 9a9z5 - 5a9z7 - a10z2 + 5a10z4 - 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 j = 2                     1 j = 0                   2   j = -2                 5 1   j = -4               6 3     j = -6             7 4       j = -8           7 6         j = -10         6 7           j = -12       4 7             j = -14     3 6               j = -16   1 5                 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[10, Alternating, 81]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 81]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[14, 19, 15, 20], X[16, 5, 17, 6], > X[18, 10, 19, 9], X[4, 15, 5, 16], X[10, 18, 11, 17], X[20, 13, 7, 14], > X[2, 7, 3, 8], X[6, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -6, 4, -10}, > {9, -1, 5, -7, 10, -2, 8, -3, 6, -4, 7, -5, 3, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 3 7 10 13 14 13 10 7 -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 3 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 -26 3 -20 3 2 2 -12 3 2 3 -1 + q + q - q + --- - q + --- - --- + --- - q + -- - -- + -- + 24 18 16 14 8 6 4 q q q q q q q 2 > q
In[8]:=	HOMFLYPT[Link[10, Alternating, 81]][a, z]
Out[8]=	3 5 7 9 a 2 a 2 a a 3 5 7 9 3 -(--) + ---- - ---- + -- - a z - 3 a z + 5 a z - 5 a z + a z - a z + z z z z 5 3 7 3 3 5 5 5 > 5 a z - 3 a z + a z + 2 a z
In[9]:=	Kauffman[Link[10, Alternating, 81]][a, z]
Out[9]=	3 5 7 9 6 a 2 a 2 a a 3 5 7 9 -a - -- - ---- - ---- - -- - a z + 5 a z + 12 a z + 12 a z + 5 a z - z z z z 11 2 2 4 2 6 2 8 2 10 2 3 3 3 > a z - a z + 2 a z + 6 a z + 2 a z - a z + 2 a z - 6 a z - 5 3 7 3 9 3 11 3 2 4 4 4 6 4 > 26 a z - 26 a z - 6 a z + 2 a z + 5 a z - 5 a z - 20 a z - 8 4 10 4 5 3 5 5 5 7 5 9 5 > 5 a z + 5 a z - a z + 9 a z + 22 a z + 22 a z + 9 a z - 11 5 2 6 4 6 6 6 8 6 10 6 3 7 > a z - 3 a z + 8 a z + 22 a z + 8 a z - 3 a z - 5 a z - 5 7 7 7 9 7 4 8 6 8 8 8 5 9 > 6 a z - 6 a z - 5 a z - 5 a z - 10 a z - 5 a z - 2 a z - 7 9 > 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 1 2 1 5 3 6 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 7 6 7 7 6 7 4 6 t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 2 t + -- + 12 4 10 4 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q t q t q 2 2 > q t

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