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L11n277 L11n279

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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X18,10,19,9 X17,11,18,22 X11,21,12,20 X21,17,22,16 X4,15,1,16 X10,20,5,19

Gauss Code:

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

Jones Polynomial:

q-4 + q-3 + q-2 + q-1 + q - q2 + q3 - q4

A2 (sl(3)) Invariant:

2q-14 + 3q-12 + 5q-10 + 7q-8 + 7q-6 + 6q-4 + 3q-2 + 1 - q2 - 2q4 - q6 - q8 - q10 - q12

HOMFLY-PT Polynomial:

- a-2z-2 - 3a-2 - 4a-2z2 - a-2z4 + 4z-2 + 10 + 11z2 + 6z4 + z6 - 5a2z-2 - 9a2 - 6a2z2 - a2z4 + 2a4z-2 + 2a4

Kauffman Polynomial:

a-3z-1 - 5a-3z + 10a-3z3 - 6a-3z5 + a-3z7 - a-2z-2 + 3a-2 - 6a-2z2 + 10a-2z4 - 6a-2z6 + a-2z8 + 5a-1z-1 - 18a-1z + 25a-1z3 - 13a-1z5 + 2a-1z7 - 4z-2 + 12 - 18z2 + 16z4 - 7z6 + z8 + 9az-1 - 24az + 21az3 - 8az5 + az7 - 5a2z-2 + 15a2 - 17a2z2 + 7a2z4 - a2z6 + 5a3z-1 - 11a3z + 6a3z3 - a3z5 - 2a4z-2 + 7a4 - 5a4z2 + a4z4

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 9                   1 j = 7                     j = 5               1 1   j = 3           1 1       j = 1             1       j = -1       1 3 1         j = -3     1   1           j = -5     3 1             j = -7 1   1               j = -9 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 278]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 278]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[3, 8, 4, 9], X[13, 2, 14, 3], > X[14, 7, 15, 8], X[18, 10, 19, 9], X[17, 11, 18, 22], X[11, 21, 12, 20], > X[21, 17, 22, 16], X[4, 15, 1, 16], X[10, 20, 5, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -10}, {-2, -1, 5, 3, 6, -11}, > {-8, 2, -4, -5, 10, 9, -7, -6, 11, 8, -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-4 -3 -2 1 2 3 4 q + q + q + - + q - q + q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 3 5 7 7 6 3 2 4 6 8 10 12 1 + --- + --- + --- + -- + -- + -- + -- - q - 2 q - q - q - q - q 14 12 10 8 6 4 2 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 278]][a, z]
Out[8]=	2 4 2 3 2 4 4 1 5 a 2 a 2 4 z 2 2 10 - -- - 9 a + 2 a + -- - ----- - ---- + ---- + 11 z - ---- - 6 a z + 2 2 2 2 2 2 2 a z a z z z a 4 4 z 2 4 6 > 6 z - -- - a z + z 2 a
In[9]:=	Kauffman[Link[11, NonAlternating, 278]][a, z]
Out[9]=	2 4 3 3 2 4 4 1 5 a 2 a 1 5 9 a 5 a 12 + -- + 15 a + 7 a - -- - ----- - ---- - ---- + ---- + --- + --- + ---- - 2 2 2 2 2 2 3 a z z z a z a z z z a z 2 3 5 z 18 z 3 2 6 z 2 2 4 2 10 z > --- - ---- - 24 a z - 11 a z - 18 z - ---- - 17 a z - 5 a z + ----- + 3 a 2 3 a a a 3 4 5 25 z 3 3 3 4 10 z 2 4 4 4 6 z > ----- + 21 a z + 6 a z + 16 z + ----- + 7 a z + a z - ---- - a 2 3 a a 5 6 7 7 8 13 z 5 3 5 6 6 z 2 6 z 2 z 7 8 z > ----- - 8 a z - a z - 7 z - ---- - a z + -- + ---- + a z + z + -- a 2 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	-3 3 1 1 1 3 1 1 1 t 3 q + - + ----- + ----- + ----- + ----- + ----- + ---- + --- + - + q t + q 9 4 7 4 7 2 5 2 3 2 5 q t q q t q t q t q t q t q t 2 3 2 5 3 5 4 9 5 > q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n278

L11n277 L11n279