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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X12,8,13,7 X18,13,15,14 X16,10,17,9 X8,16,9,15 X14,17,5,18 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

q-3 - 2q-2 + 5q-1 - 6 + 9q - 7q2 + 8q3 - 6q4 + 3q5 - q6

A2 (sl(3)) Invariant:

q-10 + q-8 + q-6 + 4q-4 + 2q-2 + 4 + 5q2 + 3q4 + 6q6 + q8 + 2q10 - q12 - 2q14 + q16 - q18

HOMFLY-PT Polynomial:

- 2a-4 - 2a-4z2 - a-4z4 + a-2z-2 + 6a-2 + 7a-2z2 + 4a-2z4 + a-2z6 - 2z-2 - 6 - 6z2 - 2z4 + a2z-2 + 2a2 + a2z2

Kauffman Polynomial:

a-7z3 + 3a-6z4 + 2a-5z - 5a-5z3 + 6a-5z5 - 4a-4 + 10a-4z2 - 14a-4z4 + 8a-4z6 + 6a-3z - 7a-3z3 - 4a-3z5 + 5a-3z7 + a-2z-2 - 12a-2 + 30a-2z2 - 35a-2z4 + 11a-2z6 + a-2z8 - 2a-1z-1 + 6a-1z + 3a-1z3 - 16a-1z5 + 7a-1z7 + 2z-2 - 11 + 26z2 - 22z4 + 4z6 + z8 - 2az-1 + 2az + 4az3 - 6az5 + 2az7 + a2z-2 - 4a2 + 6a2z2 - 4a2z4 + a2z6

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 13                   1 j = 11                 2   j = 9               4 1   j = 7             4 2     j = 5           3 4       j = 3         6 4         j = 1       4 7           j = -1     1 2             j = -3   1 4               j = -5   1                 j = -7 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, Alternating, 54]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 54]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[18, 13, 15, 14], > X[16, 10, 17, 9], X[8, 16, 9, 15], X[14, 17, 5, 18], X[2, 5, 3, 6], > X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9}, {6, -5, 7, -4}, {8, -1, 3, -6, 5, -2, 9, -3, 4, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-3 2 5 2 3 4 5 6 -6 + q - -- + - + 9 q - 7 q + 8 q - 6 q + 3 q - q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -8 -6 4 2 2 4 6 8 10 12 4 + q + q + q + -- + -- + 5 q + 3 q + 6 q + q + 2 q - q - 4 2 q q 14 16 18 > 2 q + q - q
In[8]:=	HOMFLYPT[Link[9, Alternating, 54]][a, z]
Out[8]=	2 2 2 2 6 2 2 1 a 2 2 z 7 z 2 2 4 -6 - -- + -- + 2 a - -- + ----- + -- - 6 z - ---- + ---- + a z - 2 z - 4 2 2 2 2 2 4 2 a a z a z z a a 4 4 6 z 4 z z > -- + ---- + -- 4 2 2 a a a
In[9]:=	Kauffman[Link[9, Alternating, 54]][a, z]
Out[9]=	2 4 12 2 2 1 a 2 2 a 2 z 6 z 6 z -11 - -- - -- - 4 a + -- + ----- + -- - --- - --- + --- + --- + --- + 2 a z + 4 2 2 2 2 2 a z z 5 3 a a a z a z z a a 2 2 3 3 3 3 2 10 z 30 z 2 2 z 5 z 7 z 3 z 3 > 26 z + ----- + ----- + 6 a z + -- - ---- - ---- + ---- + 4 a z - 4 2 7 5 3 a a a a a a 4 4 4 5 5 5 4 3 z 14 z 35 z 2 4 6 z 4 z 16 z 5 > 22 z + ---- - ----- - ----- - 4 a z + ---- - ---- - ----- - 6 a z + 6 4 2 5 3 a a a a a a 6 6 7 7 8 6 8 z 11 z 2 6 5 z 7 z 7 8 z > 4 z + ---- + ----- + a z + ---- + ---- + 2 a z + z + -- 4 2 3 a 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 4 1 2 4 q 3 7 q + 6 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 4 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 13 5 > 3 q t + 4 q t + 4 q t + 2 q t + 4 q t + q t + 2 q t + q t

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