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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,17,12,16 X15,9,16,18 X17,13,18,12 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-6 - q-5 + 4q-4 - 2q-3 + 4q-2 - 3q-1 + 3 - 2q

A2 (sl(3)) Invariant:

q-20 + 3q-18 + 4q-16 + 7q-14 + 7q-12 + 6q-10 + 5q-8 + q-6 - 2q-2 - 2 - q2 - 2q4

HOMFLY-PT Polynomial:

- z-2 - 3 - 2z2 + 4a2z-2 + 10a2 + 8a2z2 + 2a2z4 - 5a4z-2 - 8a4 - 3a4z2 + 2a6z-2 + a6

Kauffman Polynomial:

- a-1z-1 + 3a-1z + z-2 - 2 + 2z2 + z4 - 5az-1 + 13az - 12az3 + 4az5 + 4a2z-2 - 10a2 + 16a2z2 - 14a2z4 + 4a2z6 - 9a3z-1 + 21a3z - 17a3z3 + 2a3z5 + a3z7 + 5a4z-2 - 14a4 + 23a4z2 - 20a4z4 + 5a4z6 - 5a5z-1 + 11a5z - 5a5z3 - 2a5z5 + a5z7 + 2a6z-2 - 7a6 + 9a6z2 - 5a6z4 + a6z6

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 j = 3               2 j = 1             1   j = -1           3 3   j = -3         1 1 1   j = -5       1 3       j = -7     3 1         j = -9   1 4           j = -11                 j = -13 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]=	3
In[3]:=	Show[DrawMorseLink[Link[9, NonAlternating, 22]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, NonAlternating, 22]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 7, 15, 8], X[8, 13, 5, 14], > X[11, 17, 12, 16], X[15, 9, 16, 18], X[17, 13, 18, 12], X[2, 5, 3, 6], > X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9}, {8, -1, 3, -4}, {9, -2, -5, 7, 4, -3, -6, 5, -7, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-6 -5 4 2 4 3 3 + q - q + -- - -- + -- - - - 2 q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 3 4 7 7 6 5 -6 2 2 4 -2 + q + --- + --- + --- + --- + --- + -- + q - -- - q - 2 q 18 16 14 12 10 8 2 q q q q q q q
In[8]:=	HOMFLYPT[Link[9, NonAlternating, 22]][a, z]
Out[8]=	2 4 6 2 4 6 -2 4 a 5 a 2 a 2 2 2 4 2 -3 + 10 a - 8 a + a - z + ---- - ---- + ---- - 2 z + 8 a z - 3 a z + 2 2 2 z z z 2 4 > 2 a z
In[9]:=	Kauffman[Link[9, NonAlternating, 22]][a, z]
Out[9]=	2 4 6 3 2 4 6 -2 4 a 5 a 2 a 1 5 a 9 a -2 - 10 a - 14 a - 7 a + z + ---- + ---- + ---- - --- - --- - ---- - 2 2 2 a z z z z z z 5 5 a 3 z 3 5 2 2 2 4 2 > ---- + --- + 13 a z + 21 a z + 11 a z + 2 z + 16 a z + 23 a z + z a 6 2 3 3 3 5 3 4 2 4 4 4 > 9 a z - 12 a z - 17 a z - 5 a z + z - 14 a z - 20 a z - 6 4 5 3 5 5 5 2 6 4 6 6 6 3 7 > 5 a z + 4 a z + 2 a z - 2 a z + 4 a z + 5 a z + a z + a z + 5 7 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	-3 3 1 1 4 3 1 1 3 1 q + - + q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 9 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t q t 1 3 3 > ---- + --- + 2 q t 3 q t q t

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