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DrawMorseLink

PD Presentation:

X8192 X16,6,17,5 X10,20,11,19 X18,10,19,9 X20,12,13,11 X2,13,3,14 X14,3,15,4 X4758 X12,16,7,15 X6,18,1,17

Gauss Code:

{{1, -6, 7, -8, 2, -10}, {8, -1, 4, -3, 5, -9}, {6, -7, 9, -2, 10, -4, 3, -5}}

Jones Polynomial:

q-3 - 3q-2 + 6q-1 - 8 + 12q - 11q2 + 12q3 - 9q4 + 6q5 - 3q6 + q7

A2 (sl(3)) Invariant:

q-8 - q-6 + 2q-4 + 2 + 5q2 + 2q4 + 8q6 + 2q8 + 4q10 + q12 - q14 + 2q16 - q18 + q20

HOMFLY-PT Polynomial:

a-4z-2 + 2a-4 + 5a-4z2 + 4a-4z4 + a-4z6 - 2a-2z-2 - 5a-2 - 12a-2z2 - 13a-2z4 - 6a-2z6 - a-2z8 + z-2 + 3 + 5z2 + 4z4 + z6

Kauffman Polynomial:

- a-8z2 + a-8z4 - 3a-7z3 + 3a-7z5 - a-6 + 3a-6z2 - 6a-6z4 + 5a-6z6 + 6a-5z3 - 9a-5z5 + 6a-5z7 - a-4z-2 + 2a-4 - 3a-4z2 + 7a-4z4 - 8a-4z6 + 5a-4z8 + 2a-3z-1 - 5a-3z + 12a-3z3 - 13a-3z5 + 3a-3z7 + 2a-3z9 - 2a-2z-2 + 6a-2 - 18a-2z2 + 29a-2z4 - 26a-2z6 + 9a-2z8 + 2a-1z-1 - 5a-1z + 9a-1z3 - 10a-1z5 + 2a-1z9 - z-2 + 4 - 9z2 + 12z4 - 12z6 + 4z8 + 6az3 - 9az5 + 3az7 + 2a2z2 - 3a2z4 + a2z6

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 15                     1 j = 13                   2   j = 11                 4 1   j = 9               6 3     j = 7             6 3       j = 5           6 7         j = 3         6 5           j = 1       3 7             j = -1     3 5               j = -3   1 4                 j = -5   2                   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[10, Alternating, 162]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 162]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 6, 17, 5], X[10, 20, 11, 19], X[18, 10, 19, 9], > X[20, 12, 13, 11], X[2, 13, 3, 14], X[14, 3, 15, 4], X[4, 7, 5, 8], > X[12, 16, 7, 15], X[6, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -6, 7, -8, 2, -10}, {8, -1, 4, -3, 5, -9}, > {6, -7, 9, -2, 10, -4, 3, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-3 3 6 2 3 4 5 6 7 -8 + q - -- + - + 12 q - 11 q + 12 q - 9 q + 6 q - 3 q + q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 2 2 4 6 8 10 12 14 16 2 + q - q + -- + 5 q + 2 q + 8 q + 2 q + 4 q + q - q + 2 q - 4 q 18 20 > q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 162]][a, z]
Out[8]=	2 2 4 4 2 5 -2 1 2 2 5 z 12 z 4 4 z 13 z 3 + -- - -- + z + ----- - ----- + 5 z + ---- - ----- + 4 z + ---- - ----- + 4 2 4 2 2 2 4 2 4 2 a a a z a z a a a a 6 6 8 6 z 6 z z > z + -- - ---- - -- 4 2 2 a a a
In[9]:=	Kauffman[Link[10, Alternating, 162]][a, z]
Out[9]=	2 -6 2 6 -2 1 2 2 2 5 z 5 z 2 z 4 - a + -- + -- - z - ----- - ----- + ---- + --- - --- - --- - 9 z - -- + 4 2 4 2 2 2 3 a z 3 a 8 a a a z a z a z a a 2 2 2 3 3 3 3 3 z 3 z 18 z 2 2 3 z 6 z 12 z 9 z 3 > ---- - ---- - ----- + 2 a z - ---- + ---- + ----- + ---- + 6 a z + 6 4 2 7 5 3 a a a a a a a 4 4 4 4 5 5 5 5 4 z 6 z 7 z 29 z 2 4 3 z 9 z 13 z 10 z > 12 z + -- - ---- + ---- + ----- - 3 a z + ---- - ---- - ----- - ----- - 8 6 4 2 7 5 3 a a a a a a a a 6 6 6 7 7 5 6 5 z 8 z 26 z 2 6 6 z 3 z 7 > 9 a z - 12 z + ---- - ---- - ----- + a z + ---- + ---- + 3 a z + 6 4 2 5 3 a a a a a 8 8 9 9 8 5 z 9 z 2 z 2 z > 4 z + ---- + ---- + ---- + ---- 4 2 3 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 2 1 4 3 5 3 q 3 7 q + 6 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 5 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 > 6 q t + 7 q t + 6 q t + 3 q t + 6 q t + 3 q t + 4 q t + 11 5 13 5 15 6 > q t + 2 q t + q t

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