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L11n233 L11n235

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DrawMorseLink

PD Presentation:

X10,1,11,2 X5,20,6,21 X3,15,4,14 X15,5,16,4 X19,22,20,9 X7,17,8,16 X11,18,12,19 X17,12,18,13 X2,9,3,10 X13,1,14,8 X21,6,22,7

Gauss Code:

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

Jones Polynomial:

2q-11/2 - 5q-9/2 + 8q-7/2 - 12q-5/2 + 12q-3/2 - 12q-1/2 + 10q1/2 - 7q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-18 - q-16 + 2q-14 - 2q-12 + 2q-10 + 2q-8 + 4q-4 - 2q-2 + 3 - q2 + 3q6 - q8 + q10

HOMFLY-PT Polynomial:

- a-1z-1 - 4a-1z - 3a-1z3 - a-1z5 + az-1 + 7az + 10az3 + 5az5 + az7 - 2a3z - 2a3z3 - a3z5 - a5z

Kauffman Polynomial:

- a-3z + 2a-3z3 - a-3z5 - 2a-2z2 + 5a-2z4 - 3a-2z6 - a-1z-1 + 4a-1z - 6a-1z3 + 9a-1z5 - 5a-1z7 + 1 - z2 + 3z4 + 3z6 - 4z8 - az-1 + 8az - 19az3 + 22az5 - 9az7 - az9 - 2a2z2 + a2z4 + 6a2z6 - 6a2z8 - 5a3z3 + 9a3z5 - 5a3z7 - a3z9 + a4z2 - 2a4z8 - 3a5z + 6a5z3 - 3a5z5 - a5z7 + 4a6z2 - 3a6z4

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                   1 j = 6                 2   j = 4               5 1   j = 2             5 2     j = 0           7 5       j = -2         7 7         j = -4       5 5           j = -6     3 7             j = -8   2 5               j = -10   3                 j = -12 2

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, 234]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 234]]
Out[4]=	PD[X[10, 1, 11, 2], X[5, 20, 6, 21], X[3, 15, 4, 14], X[15, 5, 16, 4], > X[19, 22, 20, 9], X[7, 17, 8, 16], X[11, 18, 12, 19], X[17, 12, 18, 13], > X[2, 9, 3, 10], X[13, 1, 14, 8], X[21, 6, 22, 7]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, -3, 4, -2, 11, -6, 10}, > {9, -1, -7, 8, -10, 3, -4, 6, -8, 7, -5, 2, -11, 5}]
In[6]:=	Jones[L][q]
Out[6]=	2 5 8 12 12 12 3/2 5/2 ----- - ---- + ---- - ---- + ---- - ------- + 10 Sqrt[q] - 7 q + 3 q - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 7/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 2 2 2 2 4 2 2 6 8 10 3 - q - q + --- - --- + --- + -- + -- - -- - q + 3 q - q + q 14 12 10 8 4 2 q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 234]][a, z]
Out[8]=	3 5 1 a 4 z 3 5 3 z 3 3 3 z -(---) + - - --- + 7 a z - 2 a z - a z - ---- + 10 a z - 2 a z - -- + a z z a a a 5 3 5 7 > 5 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 234]][a, z]
Out[9]=	2 1 a z 4 z 5 2 2 z 2 2 4 2 1 - --- - - - -- + --- + 8 a z - 3 a z - z - ---- - 2 a z + a z + a z z 3 a 2 a a 3 3 4 6 2 2 z 6 z 3 3 3 5 3 4 5 z 2 4 > 4 a z + ---- - ---- - 19 a z - 5 a z + 6 a z + 3 z + ---- + a z - 3 a 2 a a 5 5 6 6 4 z 9 z 5 3 5 5 5 6 3 z 2 6 > 3 a z - -- + ---- + 22 a z + 9 a z - 3 a z + 3 z - ---- + 6 a z - 3 a 2 a a 7 5 z 7 3 7 5 7 8 2 8 4 8 9 3 9 > ---- - 9 a z - 5 a z - a z - 4 z - 6 a z - 2 a z - a z - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	7 2 3 2 5 3 7 5 5 7 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 4 q q t q t q t q t q t q t q t q t 7 2 2 2 4 2 4 3 6 3 8 4 > ---- + 5 t + 5 q t + 2 q t + 5 q t + q t + 2 q t + q t 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n234

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