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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X20,10,7,9 X2738 X16,11,17,12 X12,5,13,6 X4,18,5,17 X14,20,15,19 X18,14,19,13 X6,15,1,16

Gauss Code:

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

Jones Polynomial:

q-9/2 - 4q-7/2 + 7q-5/2 - 12q-3/2 + 14q-1/2 - 16q1/2 + 15q3/2 - 12q5/2 + 8q7/2 - 4q9/2 + q11/2

A2 (sl(3)) Invariant:

- q-14 + q-12 + 2q-10 + 5q-6 + q-2 + 2 - 3q2 + 3q4 - 3q6 + 2q8 + q10 - 2q12 + 2q14 - q16

HOMFLY-PT Polynomial:

a-3z + 2a-3z3 + a-3z5 - 3a-1z - 6a-1z3 - 4a-1z5 - a-1z7 - az-1 + 2az + 5az3 + 2az5 + a3z-1 - a3z - a3z3

Kauffman Polynomial:

- a-6z4 + 2a-5z3 - 4a-5z5 - 3a-4z2 + 8a-4z4 - 8a-4z6 + 2a-3z - 7a-3z3 + 13a-3z5 - 10a-3z7 - 6a-2z2 + 11a-2z4 + 2a-2z6 - 7a-2z8 + 6a-1z - 27a-1z3 + 40a-1z5 - 15a-1z7 - 2a-1z9 - 4z2 - 2z4 + 21z6 - 12z8 + az-1 + 5az - 27az3 + 34az5 - 9az7 - 2az9 - a2 - 2a2z2 - 2a2z4 + 10a2z6 - 5a2z8 + a3z-1 + a3z - 9a3z3 + 11a3z5 - 4a3z7 - a4z2 + 2a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12                     1 j = 10                   3   j = 8                 5 1   j = 6               7 3     j = 4             8 5       j = 2           8 7         j = 0         7 9           j = -2       5 7             j = -4     3 8               j = -6   1 4                 j = -8   3                   j = -10 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[10, Alternating, 52]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 52]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[20, 10, 7, 9], X[2, 7, 3, 8], > X[16, 11, 17, 12], X[12, 5, 13, 6], X[4, 18, 5, 17], X[14, 20, 15, 19], > X[18, 14, 19, 13], X[6, 15, 1, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -7, 6, -10}, > {4, -1, 3, -2, 5, -6, 9, -8, 10, -5, 7, -9, 8, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 4 7 12 14 3/2 5/2 q - ---- + ---- - ---- + ------- - 16 Sqrt[q] + 15 q - 12 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 > 8 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 2 5 -2 2 4 6 8 10 12 2 - q + q + --- + -- + q - 3 q + 3 q - 3 q + 2 q + q - 2 q + 10 6 q q 14 16 > 2 q - q
In[8]:=	HOMFLYPT[Link[10, Alternating, 52]][a, z]
Out[8]=	3 3 3 5 a a z 3 z 3 2 z 6 z 3 3 3 z -(-) + -- + -- - --- + 2 a z - a z + ---- - ---- + 5 a z - a z + -- - z z 3 a 3 a 3 a a a 5 7 4 z 5 z > ---- + 2 a z - -- a a
In[9]:=	Kauffman[Link[10, Alternating, 52]][a, z]
Out[9]=	3 2 2 2 a a 2 z 6 z 3 2 3 z 6 z 2 2 -a + - + -- + --- + --- + 5 a z + a z - 4 z - ---- - ---- - 2 a z - z z 3 a 4 2 a a a 3 3 3 4 4 4 2 2 z 7 z 27 z 3 3 3 4 z 8 z > a z + ---- - ---- - ----- - 27 a z - 9 a z - 2 z - -- + ---- + 5 3 a 6 4 a a a a 4 5 5 5 11 z 2 4 4 4 4 z 13 z 40 z 5 3 5 > ----- - 2 a z + 2 a z - ---- + ----- + ----- + 34 a z + 11 a z + 2 5 3 a a a a 6 6 7 7 6 8 z 2 z 2 6 4 6 10 z 15 z 7 3 7 > 21 z - ---- + ---- + 10 a z - a z - ----- - ----- - 9 a z - 4 a z - 4 2 3 a a a a 8 9 8 7 z 2 8 2 z 9 > 12 z - ---- - 5 a z - ---- - 2 a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 4 3 8 5 7 7 9 + 8 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + ---- + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t q t q t 2 4 4 2 6 2 6 3 8 3 8 4 > 7 q t + 8 q t + 5 q t + 7 q t + 3 q t + 5 q t + q t + 10 4 12 5 > 3 q t + q t

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