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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X6718 X16,13,17,14 X14,6,15,5 X4,16,5,15 X18,11,7,12 X12,17,13,18

Gauss Code:

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

Jones Polynomial:

q-17/2 - 2q-15/2 + 3q-13/2 - 5q-11/2 + 5q-9/2 - 5q-7/2 + 4q-5/2 - 3q-3/2 + q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-26 + 2q-18 + q-14 + 2q-8 + 2q-4 + q-2 + 1 + q2

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

az-1 - 4az + 4az3 - az5 - a2 + a2z2 + 2a2z4 - a2z6 + a3z-1 - 3a3z + 3a3z3 + a3z5 - a3z7 + 5a4z2 - 6a4z4 + 3a4z6 - a4z8 + 4a5z - 12a5z3 + 9a5z5 - 3a5z7 + a6z2 - 4a6z4 + 2a6z6 - a6z8 + 2a7z - 7a7z3 + 5a7z5 - 2a7z7 - a8z2 + 3a8z4 - 2a8z6 - a9z + 4a9z3 - 2a9z5 + 2a10z2 - a10z4

Khovanov Homology:

trqj r = -7 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         3 3         j = -10       2 2           j = -12     1 3             j = -14   1 2               j = -16   1                 j = -18 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, 30]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 30]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[6, 7, 1, 8], > X[16, 13, 17, 14], X[14, 6, 15, 5], X[4, 16, 5, 15], X[18, 11, 7, 12], > X[12, 17, 13, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 8, -9, 5, -6, 7, -5, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 2 3 5 5 5 4 3 1 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 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]=	-26 2 -14 2 2 -2 2 1 - q + --- + q + -- + -- + q + q 18 8 4 q q q
In[8]:=	HOMFLYPT[Link[9, Alternating, 30]][a, z]
Out[8]=	3 a a 3 5 7 3 3 3 5 3 -(-) + -- - 3 a z + 2 a z + 2 a z - 2 a z - a z + 3 a z + 3 a z - z z 7 3 3 5 5 5 > a z + a z + a z
In[9]:=	Kauffman[Link[9, Alternating, 30]][a, z]
Out[9]=	3 2 a a 3 5 7 9 2 2 4 2 -a + - + -- - 4 a z - 3 a z + 4 a z + 2 a z - a z + a z + 5 a z + z z 6 2 8 2 10 2 3 3 3 5 3 7 3 > a z - a z + 2 a z + 4 a z + 3 a z - 12 a z - 7 a z + 9 3 2 4 4 4 6 4 8 4 10 4 5 3 5 > 4 a z + 2 a z - 6 a z - 4 a z + 3 a z - a z - a z + a z + 5 5 7 5 9 5 2 6 4 6 6 6 8 6 3 7 > 9 a z + 5 a z - 2 a z - a z + 3 a z + 2 a z - 2 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 1 1 1 2 1 3 2 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 18 7 16 6 14 6 14 5 12 5 12 4 10 4 q q t q t q t q t q t q t q t 2 3 3 3 2 2 t 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + -- + q t 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q

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