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L9n11 L9n13

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5/2 - q-1/2 - q1/2 + q3/2 - q5/2 + q7/2 - q9/2 + q11/2

A2 (sl(3)) Invariant:

q-8 + 2q-6 + 3q-4 + 3q-2 + 4 + 2q2 - q6 - 2q8 - q10 - q12 - q18

HOMFLY-PT Polynomial:

a-5z + 2a-3z-1 + 2a-3z - 5a-1z-1 - 9a-1z - 6a-1z3 - a-1z5 + 3az-1 + 4az + az3

Kauffman Polynomial:

- a-6 + 3a-6z2 - a-6z4 - a-5z + 3a-5z3 - a-5z5 + 2a-4z2 - a-4z4 - 2a-3z-1 + 3a-3z - a-3z3 + 5a-2 - 13a-2z2 + 7a-2z4 - a-2z6 - 5a-1z-1 + 16a-1z - 19a-1z3 + 8a-1z5 - a-1z7 + 5 - 12z2 + 7z4 - z6 - 3az-1 + 12az - 15az3 + 7az5 - az7

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9n12

L9n11 L9n13