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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X14,8,15,7 X18,14,7,13 X6,17,1,18 X16,11,17,12 X12,6,13,5 X4,16,5,15

Gauss Code:

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

Jones Polynomial:

q-11/2 - 3q-9/2 + 5q-7/2 - 8q-5/2 + 8q-3/2 - 9q-1/2 + 7q1/2 - 5q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-16 + q-14 - q-12 + 2q-10 + 3q-8 + q-6 + 4q-4 - q-2 + 2 - q2 - q4 + q6 - q8 + q10

HOMFLY-PT Polynomial:

- 2a-1z - 3a-1z3 - a-1z5 - az-1 + 3az + 8az3 + 5az5 + az7 + a3z-1 - 2a3z - 3a3z3 - a3z5

Kauffman Polynomial:

2a-3z3 - a-3z5 - 3a-2z2 + 7a-2z4 - 3a-2z6 + 3a-1z - 8a-1z3 + 10a-1z5 - 4a-1z7 - 4z2 + 6z4 - 2z8 + az-1 + 4az - 17az3 + 19az5 - 8az7 - a2 - 2a2z2 + 4a2z4 - a2z6 - 2a2z8 + a3z-1 - 3a3z3 + 5a3z5 - 4a3z7 + 4a4z4 - 4a4z6 - a5z + 4a5z3 - 3a5z5 + a6z2 - a6z4

Khovanov Homology:

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

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