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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X18,7,19,8 X4,19,1,20 X11,14,12,15 X3849 X5,13,6,12 X13,5,14,20 X9,16,10,17 X15,10,16,11 X17,2,18,3

Gauss Code:

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

Jones Polynomial:

- 2q-15/2 + 3q-13/2 - 4q-11/2 + 5q-9/2 - 5q-7/2 + 4q-5/2 - 3q-3/2 + q-1/2 - q1/2

A2 (sl(3)) Invariant:

q-28 + q-26 + 2q-24 + q-22 - q-20 - 2q-16 + 2q-8 + 2q-4 + q-2 + 1 + q2

HOMFLY-PT Polynomial:

- az-1 - 3az - az3 + a3z-1 + 2a3z + 3a3z3 + a3z5 + a5z-1 + 2a5z + 3a5z3 + a5z5 - 2a7z-1 - 3a7z - a7z3 + a9z-1

Kauffman Polynomial:

az-1 - 4az + 4az3 - az5 - a2 + a2z2 + 2a2z4 - a2z6 + a3z-1 - 3a3z + 3a3z3 + a3z5 - a3z7 - 2a4 + 6a4z2 - 6a4z4 + 3a4z6 - a4z8 - a5z-1 + 9a5z - 16a5z3 + 10a5z5 - 3a5z7 - 3a6 + 7a6z2 - 8a6z4 + 3a6z6 - a6z8 - 2a7z-1 + 13a7z - 18a7z3 + 8a7z5 - 2a7z7 - a8 + 2a8z2 - a8z6 - a9z-1 + 5a9z - 3a9z3

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                 1 j = 0                   j = -2             3 1   j = -4           2 1     j = -6         3 2       j = -8       2 2         j = -10     2 3           j = -12   2 3             j = -14   1               j = -16 2

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, NonAlternating, 19]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 19]]
Out[4]=	PD[X[6, 1, 7, 2], X[18, 7, 19, 8], X[4, 19, 1, 20], X[11, 14, 12, 15], > X[3, 8, 4, 9], X[5, 13, 6, 12], X[13, 5, 14, 20], X[9, 16, 10, 17], > X[15, 10, 16, 11], X[17, 2, 18, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 10, -5, -3}, {-6, -1, 2, 5, -8, 9, -4, 6, -7, 4, -9, 8, -10, -2, > 3, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-2 3 4 5 5 4 3 1 ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - Sqrt[q] 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -26 2 -22 -20 2 2 2 -2 2 1 + q + q + --- + q - q - --- + -- + -- + q + q 24 16 8 4 q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 19]][a, z]
Out[8]=	3 5 7 9 a a a 2 a a 3 5 7 3 -(-) + -- + -- - ---- + -- - 3 a z + 2 a z + 2 a z - 3 a z - a z + z z z z z 3 3 5 3 7 3 3 5 5 5 > 3 a z + 3 a z - a z + a z + a z
In[9]:=	Kauffman[Link[10, NonAlternating, 19]][a, z]
Out[9]=	3 5 7 9 2 4 6 8 a a a 2 a a 3 5 -a - 2 a - 3 a - a + - + -- - -- - ---- - -- - 4 a z - 3 a z + 9 a z + z z z z z 7 9 2 2 4 2 6 2 8 2 3 3 3 > 13 a z + 5 a z + a z + 6 a z + 7 a z + 2 a z + 4 a z + 3 a z - 5 3 7 3 9 3 2 4 4 4 6 4 5 > 16 a z - 18 a z - 3 a z + 2 a z - 6 a z - 8 a z - a z + 3 5 5 5 7 5 2 6 4 6 6 6 8 6 3 7 > a z + 10 a z + 8 a z - a z + 3 a z + 3 a z - a z - a z - 5 7 7 7 4 8 6 8 > 3 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 3 2 1 2 3 2 3 2 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 2 16 6 14 5 12 5 12 4 10 4 10 3 8 3 q q t q t q t q t q t q t q t 2 3 2 2 t 2 2 > ----- + ----- + ---- + ---- + -- + q t 8 2 6 2 6 4 2 q t q t q t q t q

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