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L11n226 L11n228

Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X12,4,13,3 X22,12,9,11 X2,9,3,10 X17,21,18,20 X6,14,7,13 X14,8,15,7 X8,16,1,15 X19,4,20,5 X5,18,6,19 X21,17,22,16

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 2q-3/2 - 4q-1/2 + 5q1/2 - 7q3/2 + 6q5/2 - 6q7/2 + 5q9/2 - 3q11/2 + q13/2

A2 (sl(3)) Invariant:

q-8 + q-6 + q-2 + 2q2 + 2q4 + 2q6 + 3q8 - 2q10 - 2q14 + q18 + q22 - q24

HOMFLY-PT Polynomial:

a-7z - 3a-5z - 4a-5z3 - a-5z5 + 6a-3z + 8a-3z3 + 5a-3z5 + a-3z7 - a-1z-1 - 7a-1z - 8a-1z3 - 2a-1z5 + az-1 + 3az + az3

Kauffman Polynomial:

- a-8z2 + a-7z - 3a-7z3 + a-6z2 - a-6z4 - a-6z6 + 2a-5z - 7a-5z3 + 8a-5z5 - 3a-5z7 + 2a-4z2 - 7a-4z4 + 9a-4z6 - 3a-4z8 + 6a-3z - 18a-3z3 + 17a-3z5 - 2a-3z7 - a-3z9 + 3a-2z2 - 17a-2z4 + 19a-2z6 - 5a-2z8 - a-1z-1 + 10a-1z - 22a-1z3 + 14a-1z5 - a-1z9 + 1 + 3z2 - 11z4 + 9z6 - 2z8 - az-1 + 5az - 8az3 + 5az5 - 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 = 14                   1 j = 12                 2   j = 10               3 1   j = 8             3 2     j = 6           3 3       j = 4         4 3         j = 2       3 5           j = 0     1 2             j = -2   1 3               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[11, NonAlternating, 227]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 227]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[22, 12, 9, 11], X[2, 9, 3, 10], > X[17, 21, 18, 20], X[6, 14, 7, 13], X[14, 8, 15, 7], X[8, 16, 1, 15], > X[19, 4, 20, 5], X[5, 18, 6, 19], X[21, 17, 22, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, 9, -10, -6, 7, -8}, > {4, -1, 3, -2, 6, -7, 8, 11, -5, 10, -9, 5, -11, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 2 4 3/2 5/2 7/2 9/2 -q + ---- - ------- + 5 Sqrt[q] - 7 q + 6 q - 6 q + 5 q - 3/2 Sqrt[q] q 11/2 13/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 -2 2 4 6 8 10 14 18 22 24 q + q + q + 2 q + 2 q + 2 q + 3 q - 2 q - 2 q + q + q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 227]][a, z]
Out[8]=	3 3 3 5 1 a z 3 z 6 z 7 z 4 z 8 z 8 z 3 z -(---) + - + -- - --- + --- - --- + 3 a z - ---- + ---- - ---- + a z - -- + a z z 7 5 3 a 5 3 a 5 a a a a a a 5 5 7 5 z 2 z z > ---- - ---- + -- 3 a 3 a a
In[9]:=	Kauffman[Link[11, NonAlternating, 227]][a, z]
Out[9]=	2 2 2 2 1 a z 2 z 6 z 10 z 2 z z 2 z 3 z 1 - --- - - + -- + --- + --- + ---- + 5 a z + 3 z - -- + -- + ---- + ---- - a z z 7 5 3 a 8 6 4 2 a a a a a a a 3 3 3 3 4 4 4 5 3 z 7 z 18 z 22 z 3 4 z 7 z 17 z 8 z > ---- - ---- - ----- - ----- - 8 a z - 11 z - -- - ---- - ----- + ---- + 7 5 3 a 6 4 2 5 a a a a a a a 5 5 6 6 6 7 7 17 z 14 z 5 6 z 9 z 19 z 3 z 2 z 7 > ----- + ----- + 5 a z + 9 z - -- + ---- + ----- - ---- - ---- - a z - 3 a 6 4 2 5 3 a a a a a a 8 8 9 9 8 3 z 5 z z z > 2 z - ---- - ---- - -- - -- 4 2 3 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 1 -2 3 2 3 q 4 5 q + 4 q + ----- + ----- + ----- + t + ----- + - + ---- + 3 q t + 6 4 4 3 2 3 2 2 t t q t q t q t q t 6 6 2 8 2 8 3 10 3 10 4 12 4 14 5 > 3 q t + 3 q t + 3 q t + 2 q t + 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n227

L11n226 L11n228