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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X9,14,10,15 X3849 X5,11,6,10 X11,5,12,20 X13,19,14,18 X19,13,20,12 X15,2,16,3

Gauss Code:

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

Jones Polynomial:

- 2q-11/2 + 2q-9/2 - 3q-7/2 + 3q-5/2 - 3q-3/2 + 3q-1/2 - 2q1/2 + q3/2 - q5/2

A2 (sl(3)) Invariant:

q-22 + q-20 + 2q-18 + 2q-16 + q-14 + q-12 - q-10 - q-6 - q-4 + q2 + q4 + q6 + q8

HOMFLY-PT Polynomial:

- a-1z-1 - 3a-1z - a-1z3 + 2az-1 + 4az + 4az3 + az5 - a3z-1 + 3a3z3 + a3z5 - a5z-1 - 3a5z - a5z3 + a7z-1

Kauffman Polynomial:

- a-1z-1 + 6a-1z - 11a-1z3 + 6a-1z5 - a-1z7 - 1 + 3z2 - 7z4 + 5z6 - z8 - 2az-1 + 13az - 23az3 + 15az5 - 3az7 - 3a2 + 4a2z2 - 2a2z4 + 3a2z6 - a2z8 - a3z-1 + 7a3z - 8a3z3 + 7a3z5 - 2a3z7 - 2a4 + a4z2 + 4a4z4 - 2a4z6 + a5z-1 - 3a5z + 4a5z3 - 2a5z5 - a6 - a6z4 + a7z-1 - 3a7z

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 6                 1 j = 4                   j = 2             2 1   j = 0           1       j = -2         2 2       j = -4       2 2         j = -6     1 1           j = -8   1 2             j = -10 1 1               j = -12 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, 7]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 7]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[9, 14, 10, 15], > X[3, 8, 4, 9], X[5, 11, 6, 10], X[11, 5, 12, 20], X[13, 19, 14, 18], > X[19, 13, 20, 12], X[15, 2, 16, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 10, -5, -3}, {-6, -1, 2, 5, -4, 6, -7, 9, -8, 4, -10, -2, 3, 8, > -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-2 2 3 3 3 3 3/2 5/2 ----- + ---- - ---- + ---- - ---- + ------- - 2 Sqrt[q] + q - q 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 2 2 -14 -12 -10 -6 -4 2 4 6 8 q + q + --- + --- + q + q - q - q - q + q + q + q + q 18 16 q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 7]][a, z]
Out[8]=	3 5 7 3 1 2 a a a a 3 z 5 z 3 3 3 -(---) + --- - -- - -- + -- - --- + 4 a z - 3 a z - -- + 4 a z + 3 a z - a z z z z z a a 5 3 5 3 5 > a z + a z + a z
In[9]:=	Kauffman[Link[10, NonAlternating, 7]][a, z]
Out[9]=	3 5 7 2 4 6 1 2 a a a a 6 z 3 -1 - 3 a - 2 a - a - --- - --- - -- + -- + -- + --- + 13 a z + 7 a z - a z z z z z a 3 5 7 2 2 2 4 2 11 z 3 3 3 > 3 a z - 3 a z + 3 z + 4 a z + a z - ----- - 23 a z - 8 a z + a 5 5 3 4 2 4 4 4 6 4 6 z 5 3 5 > 4 a z - 7 z - 2 a z + 4 a z - a z + ---- + 15 a z + 7 a z - a 7 5 5 6 2 6 4 6 z 7 3 7 8 2 8 > 2 a z + 5 z + 3 a z - 2 a z - -- - 3 a z - 2 a z - z - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 2 1 1 1 2 1 1 2 -- + -- + ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + t + 4 2 12 4 10 4 10 3 8 3 8 2 6 2 6 4 q q q t q t q t q t q t q t q t q t 2 t 2 2 2 3 6 4 > --- + 2 q t + q t + q t 2 q

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