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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-19/2 - 2q-17/2 + 3q-15/2 - 4q-13/2 + 4q-11/2 - 4q-9/2 + 3q-7/2 - 3q-5/2 + q-3/2 - q-1/2

A2 (sl(3)) Invariant:

- q-28 - q-24 + 2q-16 + 2q-12 + q-10 + 2q-8 + 2q-6 + q-4 + q-2

HOMFLY-PT Polynomial:

- 2a3z-1 - 7a3z - 5a3z3 - a3z5 + 3a5z-1 + 10a5z + 12a5z3 + 6a5z5 + a5z7 - a7z-1 - 4a7z - 4a7z3 - a7z5

Kauffman Polynomial:

- 2a3z-1 + 9a3z - 12a3z3 + 6a3z5 - a3z7 + 3a4 - 5a4z2 - 2a4z4 + 4a4z6 - a4z8 - 3a5z-1 + 14a5z - 27a5z3 + 19a5z5 - 4a5z7 + 3a6 - 13a6z2 + 11a6z4 - a6z8 - a7z-1 + 3a7z - 6a7z3 + 9a7z5 - 3a7z7 + a8 - 5a8z2 + 10a8z4 - 4a8z6 - 2a9z + 7a9z3 - 4a9z5 + 2a10z2 - 3a10z4 - 2a11z3 - a12z2

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 0                   1 j = -2                     j = -4               3 1   j = -6             1 1     j = -8           3 2       j = -10         2 2         j = -12       2 2           j = -14     1 2             j = -16   1 2               j = -18   1                 j = -20 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, 29]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 29]]
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, 11, 17, 12], X[14, 6, 15, 5], X[4, 16, 5, 15], X[18, 13, 7, 14], > X[12, 17, 13, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 5, -9, 8, -6, 7, -5, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 2 3 4 4 4 3 3 -(3/2) q - ----- + ----- - ----- + ----- - ---- + ---- - ---- + q - 17/2 15/2 13/2 11/2 9/2 7/2 5/2 q q q q q q q 1 > ------- Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -24 2 2 -10 2 2 -4 -2 -q - q + --- + --- + q + -- + -- + q + q 16 12 8 6 q q q q
In[8]:=	HOMFLYPT[Link[9, Alternating, 29]][a, z]
Out[8]=	3 5 7 -2 a 3 a a 3 5 7 3 3 5 3 7 3 ----- + ---- - -- - 7 a z + 10 a z - 4 a z - 5 a z + 12 a z - 4 a z - z z z 3 5 5 5 7 5 5 7 > a z + 6 a z - a z + a z
In[9]:=	Kauffman[Link[9, Alternating, 29]][a, z]
Out[9]=	3 5 7 4 6 8 2 a 3 a a 3 5 7 9 3 a + 3 a + a - ---- - ---- - -- + 9 a z + 14 a z + 3 a z - 2 a z - z z z 4 2 6 2 8 2 10 2 12 2 3 3 5 3 > 5 a z - 13 a z - 5 a z + 2 a z - a z - 12 a z - 27 a z - 7 3 9 3 11 3 4 4 6 4 8 4 10 4 > 6 a z + 7 a z - 2 a z - 2 a z + 11 a z + 10 a z - 3 a z + 3 5 5 5 7 5 9 5 4 6 8 6 3 7 > 6 a z + 19 a z + 9 a z - 4 a z + 4 a z - 4 a z - a z - 5 7 7 7 4 8 6 8 > 4 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 3 1 1 1 2 1 2 2 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 20 7 18 6 16 6 16 5 14 5 14 4 12 4 q q t q t q t q t q t q t q t 2 2 2 3 2 1 t 2 > ------ + ------ + ------ + ----- + ---- + ---- + -- + t 12 3 10 3 10 2 8 2 8 6 4 q t q t q t q t q t q t q

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