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L11n281 L11n283

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X8493 X2,14,3,13 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-2 + 4q-1 - 5 + 9q - 7q2 + 9q3 - 7q4 + 4q5 - 2q6

A2 (sl(3)) Invariant:

- q-6 + 2q-4 + q-2 + 4 + 8q2 + 6q4 + 10q6 + 4q8 + 3q10 - q12 - 4q14 - q16 - 3q18 - q20

HOMFLY-PT Polynomial:

- a-6z-2 - a-6 + 4a-4z-2 + 4a-4 - a-4z4 - 5a-2z-2 - 5a-2 + a-2z2 + 3a-2z4 + a-2z6 + 2z-2 + 2 - z2 - z4

Kauffman Polynomial:

a-7z-1 - 3a-7z + 3a-7z3 - a-6z-2 + a-6 + 2a-6z4 + a-6z6 + 5a-5z-1 - 10a-5z + 7a-5z3 - a-5z5 + 2a-5z7 - 4a-4z-2 + 4a-4 + 3a-4z2 - 9a-4z4 + 5a-4z6 + a-4z8 + 9a-3z-1 - 12a-3z + 5a-3z3 - 8a-3z5 + 6a-3z7 - 5a-2z-2 + 5a-2 + 7a-2z2 - 20a-2z4 + 8a-2z6 + a-2z8 + 5a-1z-1 - 5a-1z - 6a-1z5 + 4a-1z7 - 2z-2 + 3 + 4z2 - 9z4 + 4z6 - az3 + az5

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 13                 2 j = 11               2   j = 9             5 2   j = 7           4 2     j = 5         3 5       j = 3       6 4         j = 1     3 7           j = -1   1 2             j = -3   3               j = -5 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, 282]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 282]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[8, 4, 9, 3], X[2, 14, 3, 13], > 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]=	-2 4 2 3 4 5 6 -5 - q + - + 9 q - 7 q + 9 q - 7 q + 4 q - 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 2 -2 2 4 6 8 10 12 14 16 4 - q + -- + q + 8 q + 6 q + 10 q + 4 q + 3 q - q - 4 q - q - 4 q 18 20 > 3 q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 282]][a, z]
Out[8]=	2 4 4 6 -6 4 5 2 1 4 5 2 z 4 z 3 z z 2 - a + -- - -- + -- - ----- + ----- - ----- - z + -- - z - -- + ---- + -- 4 2 2 6 2 4 2 2 2 2 4 2 2 a a z a z a z a z a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 282]][a, z]
Out[9]=	-6 4 5 2 1 4 5 1 5 9 5 3 + a + -- + -- - -- - ----- - ----- - ----- + ---- + ---- + ---- + --- - 4 2 2 6 2 4 2 2 2 7 5 3 a z a a z a z a z a z a z a z a z 2 2 3 3 3 3 z 10 z 12 z 5 z 2 3 z 7 z 3 z 7 z 5 z 3 > --- - ---- - ---- - --- + 4 z + ---- + ---- + ---- + ---- + ---- - a z - 7 5 3 a 4 2 7 5 3 a a a a a a a a 4 4 4 5 5 5 6 6 4 2 z 9 z 20 z z 8 z 6 z 5 6 z 5 z > 9 z + ---- - ---- - ----- - -- - ---- - ---- + a z + 4 z + -- + ---- + 6 4 2 5 3 a 6 4 a a a a a a a 6 7 7 7 8 8 8 z 2 z 6 z 4 z z z > ---- + ---- + ---- + ---- + -- + -- 2 5 3 a 4 2 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 3 1 2 3 q 3 5 5 2 7 q + 6 q + ----- + ----- + ---- + --- + --- + 4 q t + 3 q t + 5 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 13 5 > 4 q t + 2 q t + 5 q t + 2 q t + 2 q t + 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n282

L11n281 L11n283