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L9n22 L9n24

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X11,16,12,17 X9,11,10,18 X17,5,18,10 X4,15,1,16

Gauss Code:

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

Jones Polynomial:

- q-7 + 2q-6 - q-5 + 3q-4 - q-3 + 2q-2 - q-1 + 1

A2 (sl(3)) Invariant:

- q-26 + q-20 + 2q-18 + 3q-16 + 5q-14 + 4q-12 + 5q-10 + 3q-8 + 2q-6 + q-4 + q-2 + 1

HOMFLY-PT Polynomial:

a2z-2 + 3a2 + 4a2z2 + a2z4 - 2a4z-2 - 5a4 - 7a4z2 - 5a4z4 - a4z6 + a6z-2 + 3a6 + 4a6z2 + a6z4 - a8

Kauffman Polynomial:

a2z-2 - 4a2 + 7a2z2 - 5a2z4 + a2z6 - 2a3z-1 + 3a3z + 2a3z3 - 4a3z5 + a3z7 + 2a4z-2 - 9a4 + 18a4z2 - 14a4z4 + 3a4z6 - 2a5z-1 + 5a5z - a5z3 - 3a5z5 + a5z7 + a6z-2 - 8a6 + 13a6z2 - 9a6z4 + 2a6z6 + 3a7z - 3a7z3 + a7z5 - 2a8 + 2a8z2 + a9z

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 1               1 j = -1                 j = -3           2 1   j = -5       1 1 1     j = -7       3 1       j = -9   1 1 2         j = -11   2 1           j = -13   1             j = -15 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]=	3
In[3]:=	Show[DrawMorseLink[Link[9, NonAlternating, 23]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, NonAlternating, 23]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[3, 8, 4, 9], X[13, 2, 14, 3], > X[14, 7, 15, 8], X[11, 16, 12, 17], X[9, 11, 10, 18], X[17, 5, 18, 10], > X[4, 15, 1, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -9}, {-2, -1, 5, 3, -7, 8}, {-6, 2, -4, -5, 9, 6, -8, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-7 2 -5 3 -3 2 1 1 - q + -- - q + -- - q + -- - - 6 4 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -20 2 3 5 4 5 3 2 -4 -2 1 - q + q + --- + --- + --- + --- + --- + -- + -- + q + q 18 16 14 12 10 8 6 q q q q q q q
In[8]:=	HOMFLYPT[Link[9, NonAlternating, 23]][a, z]
Out[8]=	2 4 6 2 4 6 8 a 2 a a 2 2 4 2 6 2 3 a - 5 a + 3 a - a + -- - ---- + -- + 4 a z - 7 a z + 4 a z + 2 2 2 z z z 2 4 4 4 6 4 4 6 > a z - 5 a z + a z - a z
In[9]:=	Kauffman[Link[9, NonAlternating, 23]][a, z]
Out[9]=	2 4 6 3 5 2 4 6 8 a 2 a a 2 a 2 a 3 5 -4 a - 9 a - 8 a - 2 a + -- + ---- + -- - ---- - ---- + 3 a z + 5 a z + 2 2 2 z z z z z 7 9 2 2 4 2 6 2 8 2 3 3 5 3 > 3 a z + a z + 7 a z + 18 a z + 13 a z + 2 a z + 2 a z - a z - 7 3 2 4 4 4 6 4 3 5 5 5 7 5 > 3 a z - 5 a z - 14 a z - 9 a z - 4 a z - 3 a z + a z + 2 6 4 6 6 6 3 7 5 7 > a z + 3 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-5 2 1 1 2 1 1 1 2 3 q + -- + ------ + ------ + ------ + ----- + ------ + ----- + ----- + ----- + 3 15 5 13 4 11 4 9 4 11 3 9 3 9 2 7 2 q q t q t q t q t q t q t q t q t 1 1 1 t 2 > ----- + ---- + ---- + -- + q t 5 2 7 5 3 q t q t q t q

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