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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X16,11,17,12 X3,10,4,11 X2,15,3,16 X12,5,13,6 X6718 X9,14,10,15 X13,18,14,7 X17,4,18,5

Gauss Code:

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

Jones Polynomial:

- q-11/2 - q-7/2

A2 (sl(3)) Invariant:

- q-36 + q-24 + q-22 + 2q-20 + 2q-18 + 2q-16 + q-14 + q-12

HOMFLY-PT Polynomial:

- 2a7z-1 - 11a7z - 15a7z3 - 7a7z5 - a7z7 + 3a9z-1 + 9a9z + 6a9z3 + a9z5 - a11z-1 - a11z

Kauffman Polynomial:

- 2a7z-1 + 11a7z - 15a7z3 + 7a7z5 - a7z7 + 3a8 - 9a8z2 + 6a8z4 - a8z6 - 3a9z-1 + 12a9z - 15a9z3 + 7a9z5 - a9z7 + 3a10 - 9a10z2 + 6a10z4 - a10z6 - a11z-1 + a11z + a12

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -6             1 j = -8             1 j = -10         1     j = -12     1         j = -14     1 1       j = -16 1 1           j = -18 1 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, NonAlternating, 15]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, NonAlternating, 15]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 11, 17, 12], X[3, 10, 4, 11], X[2, 15, 3, 16], > X[12, 5, 13, 6], X[6, 7, 1, 8], X[9, 14, 10, 15], X[13, 18, 14, 7], > X[17, 4, 18, 5]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, -3, 9, 5, -6}, {6, -1, -7, 3, 2, -5, -8, 7, 4, -2, -9, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) -(7/2) -q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-36 -24 -22 2 2 2 -14 -12 -q + q + q + --- + --- + --- + q + q 20 18 16 q q q
In[8]:=	HOMFLYPT[Link[9, NonAlternating, 15]][a, z]
Out[8]=	7 9 11 -2 a 3 a a 7 9 11 7 3 9 3 7 5 ----- + ---- - --- - 11 a z + 9 a z - a z - 15 a z + 6 a z - 7 a z + z z z 9 5 7 7 > a z - a z
In[9]:=	Kauffman[Link[9, NonAlternating, 15]][a, z]
Out[9]=	7 9 11 8 10 12 2 a 3 a a 7 9 11 8 2 3 a + 3 a + a - ---- - ---- - --- + 11 a z + 12 a z + a z - 9 a z - z z z 10 2 7 3 9 3 8 4 10 4 7 5 9 5 > 9 a z - 15 a z - 15 a z + 6 a z + 6 a z + 7 a z + 7 a z - 8 6 10 6 7 7 9 7 > a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 1 1 1 1 1 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 18 6 16 6 18 5 16 5 14 4 12 4 14 3 q t q t q t q t q t q t q t 1 > ------ 10 2 q t

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L9n14 L9n16