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L11n227 L11n229

Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X17,9,18,22 X9,21,10,20 X6,13,7,14 X14,7,15,8 X8,15,1,16 X19,5,20,4 X5,19,6,18 X21,17,22,16

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 2q-11/2 - 3q-9/2 + 3q-7/2 - 3q-5/2 + 2q-3/2 - 2q-1/2 - q1/2 + q3/2 - q5/2 + q7/2

A2 (sl(3)) Invariant:

q-20 - q-18 + q-16 + q-14 + 2q-10 + 2q-6 + q-4 + 2q-2 + 3 - q6 - q8 - q10

HOMFLY-PT Polynomial:

a-1z-1 + 6a-1z + 5a-1z3 + a-1z5 - 3az-1 - 13az - 15az3 - 7az5 - az7 + 2a3z-1 + 4a3z + 2a3z3 + a5z + a5z3

Kauffman Polynomial:

- a-2 + 11a-2z2 - 15a-2z4 + 7a-2z6 - a-2z8 + a-1z-1 - 7a-1z + 14a-1z3 - 15a-1z5 + 7a-1z7 - a-1z9 - 3 + 20z2 - 31z4 + 15z6 - 2z8 + 3az-1 - 15az + 20az3 - 19az5 + 8az7 - az9 - 3a2 + 12a2z2 - 18a2z4 + 8a2z6 - a2z8 + 2a3z-1 - 6a3z + 6a3z3 - 2a3z5 + 4a4z4 - 2a4z6 + 3a5z3 + a5z5 - a5z7 - 3a6z2 + 6a6z4 - 2a6z6 - 2a7z + 3a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 8                       1 j = 6                         j = 4                   1 1   j = 2               1 1       j = 0             2   1       j = -2           2 3 1         j = -4         2 1             j = -6       1 2 1             j = -8     2 2                 j = -10   1 2                   j = -12   1                     j = -14 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, 228]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 228]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[17, 9, 18, 22], > X[9, 21, 10, 20], X[6, 13, 7, 14], X[14, 7, 15, 8], X[8, 15, 1, 16], > X[19, 5, 20, 4], X[5, 19, 6, 18], X[21, 17, 22, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, 9, -10, -6, 7, -8}, > {-5, -1, 2, -3, 6, -7, 8, 11, -4, 10, -9, 5, -11, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 2 3 3 3 2 2 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- - Sqrt[q] + q - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 7/2 > q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 -16 -14 2 2 -4 2 6 8 10 3 + q - q + q + q + --- + -- + q + -- - q - q - q 10 6 2 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 228]][a, z]
Out[8]=	3 3 1 3 a 2 a 6 z 3 5 5 z 3 3 3 --- - --- + ---- + --- - 13 a z + 4 a z + a z + ---- - 15 a z + 2 a z + a z z z a a 5 5 3 z 5 7 > a z + -- - 7 a z - a z a
In[9]:=	Kauffman[Link[11, NonAlternating, 228]][a, z]
Out[9]=	3 -2 2 1 3 a 2 a 7 z 3 7 2 -3 - a - 3 a + --- + --- + ---- - --- - 15 a z - 6 a z - 2 a z + 20 z + a z z z a 2 3 11 z 2 2 6 2 14 z 3 3 3 5 3 > ----- + 12 a z - 3 a z + ----- + 20 a z + 6 a z + 3 a z + 2 a a 4 5 7 3 4 15 z 2 4 4 4 6 4 15 z 5 > 3 a z - 31 z - ----- - 18 a z + 4 a z + 6 a z - ----- - 19 a z - 2 a a 6 3 5 5 5 7 5 6 7 z 2 6 4 6 6 6 > 2 a z + a z - a z + 15 z + ---- + 8 a z - 2 a z - 2 a z + 2 a 7 8 9 7 z 7 5 7 8 z 2 8 z 9 > ---- + 8 a z - a z - 2 z - -- - a z - -- - a z a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 2 2 2 1 2 2 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 2 1 1 2 t 2 2 2 2 4 3 4 4 8 5 > ----- + ---- + ---- + ---- + -- + q t + t + q t + q t + q t + q t 4 2 6 4 2 2 q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n228

L11n227 L11n229