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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-15/2 + 2q-13/2 - 3q-11/2 + 4q-9/2 - 5q-7/2 + 4q-5/2 - 4q-3/2 + 2q-1/2 - 2q1/2 + q3/2

A2 (sl(3)) Invariant:

q-22 + q-18 + q-12 + 3q-8 + q-6 + 2q-4 + q-2 - q4

HOMFLY-PT Polynomial:

3az + 4az3 + az5 - a3z-1 - 9a3z - 12a3z3 - 6a3z5 - a3z7 + a5z-1 + 4a5z + 4a5z3 + a5z5

Kauffman Polynomial:

- 3z2 + 4z4 - z6 + 4az - 11az3 + 9az5 - 2az7 - 4a2z2 + 3a2z4 + 2a2z6 - a2z8 - a3z-1 + 10a3z - 20a3z3 + 16a3z5 - 4a3z7 + a4 - 4a4z2 + 3a4z4 + a4z6 - a4z8 - a5z-1 + 5a5z - 7a5z3 + 5a5z5 - 2a5z7 - a6z2 + 2a6z4 - 2a6z6 + a7z3 - 2a7z5 + 2a8z2 - 2a8z4 + a9z - a9z3

Khovanov Homology:

trqj 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                 1   j = 0               1 1   j = -2             3 1     j = -4           2 2       j = -6         3 2         j = -8       1 2           j = -10     2 3             j = -12   1 2               j = -14   1                 j = -16 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[9, Alternating, 39]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 39]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[14, 5, 15, 6], > X[6, 9, 7, 10], X[16, 8, 17, 7], X[18, 16, 9, 15], X[8, 18, 1, 17], > X[4, 13, 5, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -9, 4, -5, 6, -8}, {5, -1, 2, -3, 9, -4, 7, -6, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 2 3 4 5 4 4 2 -q + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 2 Sqrt[q] + 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -18 -12 3 -6 2 -2 4 q + q + q + -- + q + -- + q - q 8 4 q q
In[8]:=	HOMFLYPT[Link[9, Alternating, 39]][a, z]
Out[8]=	3 5 a a 3 5 3 3 3 5 3 5 -(--) + -- + 3 a z - 9 a z + 4 a z + 4 a z - 12 a z + 4 a z + a z - z z 3 5 5 5 3 7 > 6 a z + a z - a z
In[9]:=	Kauffman[Link[9, Alternating, 39]][a, z]
Out[9]=	3 5 4 a a 3 5 9 2 2 2 4 2 a - -- - -- + 4 a z + 10 a z + 5 a z + a z - 3 z - 4 a z - 4 a z - z z 6 2 8 2 3 3 3 5 3 7 3 9 3 4 > a z + 2 a z - 11 a z - 20 a z - 7 a z + a z - a z + 4 z + 2 4 4 4 6 4 8 4 5 3 5 5 5 > 3 a z + 3 a z + 2 a z - 2 a z + 9 a z + 16 a z + 5 a z - 7 5 6 2 6 4 6 6 6 7 3 7 5 7 > 2 a z - z + 2 a z + a z - 2 a z - 2 a z - 4 a z - 2 a z - 2 8 4 8 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 1 1 1 2 2 3 1 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 4 2 16 6 14 5 12 5 12 4 10 4 10 3 8 3 8 2 q q q t q t q t q t q t q t q t q t 3 2 2 t 2 2 2 4 3 > ----- + ---- + ---- + t + -- + t + q t + q t 6 2 6 4 2 q t q t q t q

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