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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X8493 X11,19,12,18 X17,5,18,20 X19,13,20,12 X9,16,10,17 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

q-9/2 - 2q-7/2 + 3q-5/2 - 4q-3/2 + 3q-1/2 - 5q1/2 + 3q3/2 - 2q5/2 + q7/2

A2 (sl(3)) Invariant:

- q-14 - q-10 - q-8 + q-6 + q-4 + 4q-2 + 4 + 3q2 + 2q4 - q6 - q10 - q12

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-4 - a-4z2 - a-3z-1 + 2a-3z - 2a-3z3 + 4a-2 - 7a-2z2 + 3a-2z4 - a-2z6 - 4a-1z-1 + 9a-1z - 13a-1z3 + 8a-1z5 - 2a-1z7 + 7 - 13z2 + 7z4 + z6 - z8 - 4az-1 + 11az - 19az3 + 16az5 - 4az7 + 4a2 - 11a2z2 + 8a2z4 + a2z6 - a2z8 - a3z-1 + 4a3z - 8a3z3 + 8a3z5 - 2a3z7 + a4 - 4a4z2 + 4a4z4 - a4z6

Khovanov Homology:

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

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