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DrawMorseLink

PD Presentation:

X8192 X16,11,17,12 X10,4,11,3 X2,15,3,16 X12,5,13,6 X6718 X14,10,15,9 X18,14,7,13 X4,18,5,17

Gauss Code:

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

Jones Polynomial:

q-11/2 - 4q-9/2 + 7q-7/2 - 10q-5/2 + 11q-3/2 - 12q-1/2 + 9q1/2 - 7q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-16 + 2q-14 - q-12 + 2q-10 + 2q-8 - q-6 + 4q-4 - q-2 + 4 - q4 + q6 - 2q8 + q10

HOMFLY-PT Polynomial:

- 2a-1z3 - a-1z5 - az-1 - az + 4az3 + 4az5 + az7 + a3z-1 - 2a3z3 - a3z5

Kauffman Polynomial:

a-3z3 - a-3z5 - 2a-2z2 + 7a-2z4 - 4a-2z6 - 6a-1z3 + 12a-1z5 - 6a-1z7 - 6z2 + 14z4 - 3z6 - 3z8 + az-1 - az - 12az3 + 25az5 - 13az7 - a2 - 6a2z2 + 16a2z4 - 6a2z6 - 3a2z8 + a3z-1 - a3z - 2a3z3 + 8a3z5 - 7a3z7 - 2a4z2 + 8a4z4 - 7a4z6 + 3a5z3 - 4a5z5 - a6z4

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                   1 j = 6                 3   j = 4               4 1   j = 2             5 3     j = 0           7 4       j = -2         5 6         j = -4       5 6           j = -6     3 6             j = -8   1 4               j = -10   3                 j = -12 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, 20]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 20]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 11, 17, 12], X[10, 4, 11, 3], X[2, 15, 3, 16], > X[12, 5, 13, 6], X[6, 7, 1, 8], X[14, 10, 15, 9], X[18, 14, 7, 13], > X[4, 18, 5, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -9, 5, -6}, {6, -1, 7, -3, 2, -5, 8, -7, 4, -2, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 4 7 10 11 12 3/2 5/2 q - ---- + ---- - ---- + ---- - ------- + 9 Sqrt[q] - 7 q + 4 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 7/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 -12 2 2 -6 4 -2 4 6 8 10 4 - q + --- - q + --- + -- - q + -- - q - q + q - 2 q + q 14 10 8 4 q q q q
In[8]:=	HOMFLYPT[Link[9, Alternating, 20]][a, z]
Out[8]=	3 3 5 a a 2 z 3 3 3 z 5 3 5 7 -(-) + -- - a z - ---- + 4 a z - 2 a z - -- + 4 a z - a z + a z z z a a
In[9]:=	Kauffman[Link[9, Alternating, 20]][a, z]
Out[9]=	3 2 3 3 2 a a 3 2 2 z 2 2 4 2 z 6 z -a + - + -- - a z - a z - 6 z - ---- - 6 a z - 2 a z + -- - ---- - z z 2 3 a a a 4 3 3 3 5 3 4 7 z 2 4 4 4 6 4 > 12 a z - 2 a z + 3 a z + 14 z + ---- + 16 a z + 8 a z - a z - 2 a 5 5 6 z 12 z 5 3 5 5 5 6 4 z 2 6 > -- + ----- + 25 a z + 8 a z - 4 a z - 3 z - ---- - 6 a z - 3 a 2 a a 7 4 6 6 z 7 3 7 8 2 8 > 7 a z - ---- - 13 a z - 7 a z - 3 z - 3 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	6 1 3 1 4 3 6 5 6 7 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 4 q q t q t q t q t q t q t q t q t 5 2 2 2 4 2 4 3 6 3 8 4 > ---- + 4 t + 5 q t + 3 q t + 4 q t + q t + 3 q t + q t 2 q t

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