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L9n25 L9n27

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X11,17,12,16 X7,14,8,15 X13,8,14,9 X15,13,16,18 X17,5,18,12 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-20 + 2q-18 + 2q-16 + 4q-14 + 4q-12 + 4q-10 + 4q-8 + 2q-6 + 2q-4 + q-2 + 1 + q2 - q4

HOMFLY-PT Polynomial:

- z2 + a2z-2 + 3a2 + 3a2z2 + a2z4 - 2a4z-2 - 4a4 - 2a4z2 + a6z-2 + a6

Kauffman Polynomial:

a-1z - 1 + 3z2 + 3az - 5az3 + 2az5 + a2z-2 - 7a2 + 15a2z2 - 12a2z4 + 3a2z6 - 2a3z-1 + 7a3z - 6a3z3 - a3z5 + a3z7 + 2a4z-2 - 10a4 + 20a4z2 - 17a4z4 + 4a4z6 - 2a5z-1 + 5a5z - a5z3 - 3a5z5 + a5z7 + a6z-2 - 5a6 + 8a6z2 - 5a6z4 + a6z6

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9n26

L9n25 L9n27