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L11n204 L11n206

Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X7,16,8,17 X18,12,19,11 X2,19,3,20 X3,12,4,13 X13,21,14,20 X14,5,15,6 X6,9,7,10 X22,16,9,15 X17,8,18,1 X21,4,22,5

Gauss Code:

{{1, -4, -5, 11, 7, -8, -2, 10}, {8, -1, 3, 5, -6, -7, 9, 2, -10, -3, 4, 6, -11, -9}}

Jones Polynomial:

- q-17/2 - q-7/2

A2 (sl(3)) Invariant:

q-26 + 2q-24 + 2q-22 + q-20 + q-18 + q-16 + q-14 + q-12 - q-6

HOMFLY-PT Polynomial:

a3z-1 + a3z - 3a5z-1 - 9a5z - 6a5z3 - a5z5 + 2a7z-1 + 4a7z + a7z3

Kauffman Polynomial:

- a2 + a3z-1 - a3z - 3a4 + 9a4z2 - 6a4z4 + a4z6 + 3a5z-1 - 12a5z + 15a5z3 - 7a5z5 + a5z7 - 3a6 + 9a6z2 - 6a6z4 + a6z6 + 2a7z-1 - 4a7z + a7z3 + 7a9z - 14a9z3 + 7a9z5 - a9z7

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -2               1 1 j = -4               1 1 j = -6           1 2     j = -8             1     j = -10       1 2 1       j = -12                   j = -14     1 1           j = -16 1                 j = -18 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[11, NonAlternating, 205]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 205]]
Out[4]=	PD[X[10, 1, 11, 2], X[7, 16, 8, 17], X[18, 12, 19, 11], X[2, 19, 3, 20], > X[3, 12, 4, 13], X[13, 21, 14, 20], X[14, 5, 15, 6], X[6, 9, 7, 10], > X[22, 16, 9, 15], X[17, 8, 18, 1], X[21, 4, 22, 5]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, -5, 11, 7, -8, -2, 10}, > {8, -1, 3, 5, -6, -7, 9, 2, -10, -3, 4, 6, -11, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) -(7/2) -q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 2 2 -20 -18 -16 -14 -12 -6 q + --- + --- + q + q + q + q + q - q 24 22 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 205]][a, z]
Out[8]=	3 5 7 a 3 a 2 a 3 5 7 5 3 7 3 5 5 -- - ---- + ---- + a z - 9 a z + 4 a z - 6 a z + a z - a z z z z
In[9]:=	Kauffman[Link[11, NonAlternating, 205]][a, z]
Out[9]=	3 5 7 2 4 6 a 3 a 2 a 3 5 7 9 -a - 3 a - 3 a + -- + ---- + ---- - a z - 12 a z - 4 a z + 7 a z + z z z 4 2 6 2 5 3 7 3 9 3 4 4 6 4 > 9 a z + 9 a z + 15 a z + a z - 14 a z - 6 a z - 6 a z - 5 5 9 5 4 6 6 6 5 7 9 7 > 7 a z + 7 a z + a z + a z + a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 -2 1 1 1 1 1 2 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 18 8 16 8 14 6 14 5 10 5 10 4 10 3 q t q t q t q t q t q t q t 1 1 2 1 1 > ----- + ----- + ----- + ---- + ---- 6 3 8 2 6 2 4 2 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n205

L11n204 L11n206