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DrawMorseLink

PD Presentation:

X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X16,11,17,12 X18,15,9,16 X12,17,13,18 X2536 X9,1,10,4

Gauss Code:

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

Jones Polynomial:

- q-5 + q-4 - q-3 + q-2 + 2 + q + q3

A2 (sl(3)) Invariant:

- q-16 - q-14 - 2q-12 - 2q-10 - q-8 + 3q-4 + 5q-2 + 7 + 7q2 + 5q4 + 4q6 + 2q8 + q10

HOMFLY-PT Polynomial:

2a-2z-2 + 3a-2 + a-2z2 - 5z-2 - 9 - 6z2 - z4 + 4a2z-2 + 8a2 + 5a2z2 + a2z4 - a4z-2 - 2a4 - a4z2

Kauffman Polynomial:

2a-2z-2 - 7a-2 + 10a-2z2 - 6a-2z4 + a-2z6 - 5a-1z-1 + 11a-1z - 7a-1z3 + a-1z5 + 5z-2 - 14 + 18z2 - 8z4 + z6 - 9az-1 + 21az - 13az3 + 2az5 + 4a2z-2 - 10a2 + 11a2z2 - 6a2z4 + a2z6 - 5a3z-1 + 13a3z - 10a3z3 + 2a3z5 + a4z-2 - 2a4 + 3a4z2 - 4a4z4 + a4z6 - a5z-1 + 3a5z - 4a5z3 + a5z5

Khovanov Homology:

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

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