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L11n349 L11n351

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X14,4,15,3 X11,21,12,20 X7,18,8,19 X9,16,10,17 X17,8,18,9 X19,13,20,22 X13,10,14,11 X21,5,22,12 X2536 X4,16,1,15

Gauss Code:

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

Jones Polynomial:

q-5 - q-4 + 2q-3 - q-2 + q-1 + 2 - q + 2q2 - 2q3 + 2q4 - q5

A2 (sl(3)) Invariant:

q-16 + q-14 + 2q-12 + 3q-10 + 3q-8 + 4q-6 + 2q-4 + 4q-2 + 3 + 2q2 + 2q4 + q8 - q16

HOMFLY-PT Polynomial:

- a-4 - a-4z2 + 2a-2 + 3a-2z2 + a-2z4 + z-2 + 2 - 2a2z-2 - 5a2 - 4a2z2 - a2z4 + a4z-2 + 2a4 + a4z2

Kauffman Polynomial:

a-5z - 3a-5z3 + a-5z5 - a-4 + 4a-4z2 - 7a-4z4 + 2a-4z6 + a-3z - 3a-3z5 + a-3z7 + 2a-2z2 - 4a-2z4 + a-2z6 - 3a-1z + 10a-1z3 - 7a-1z5 + a-1z7 - z-2 + 9 - 23z2 + 31z4 - 15z6 + 2z8 + 2az-1 - 8az + 6az3 + 6az5 - 6az7 + az9 - 2a2z-2 + 13a2 - 36a2z2 + 44a2z4 - 21a2z6 + 3a2z8 + 2a3z-1 - 5a3z - a3z3 + 9a3z5 - 6a3z7 + a3z9 - a4z-2 + 6a4 - 15a4z2 + 16a4z4 - 7a4z6 + a4z8

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 = 11                       1 j = 9                     1   j = 7                   1 1   j = 5               1 2 1     j = 3             1 1 1       j = 1           1 4 2         j = -1         1 2 4           j = -3       1 1               j = -5     1 1 1               j = -7   1 2                   j = -9                         j = -11 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, 350]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 350]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[11, 21, 12, 20], X[7, 18, 8, 19], > X[9, 16, 10, 17], X[17, 8, 18, 9], X[19, 13, 20, 22], X[13, 10, 14, 11], > X[21, 5, 22, 12], X[2, 5, 3, 6], X[4, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -4, 6, -5, 8, -3, 9}, > {-8, -2, 11, 5, -6, 4, -7, 3, -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-5 -4 2 -2 1 2 3 4 5 2 + q - q + -- - q + - - q + 2 q - 2 q + 2 q - q 3 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 2 3 3 4 2 4 2 4 8 16 3 + q + q + --- + --- + -- + -- + -- + -- + 2 q + 2 q + q - q 12 10 8 6 4 2 q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 350]][a, z]
Out[8]=	2 4 2 2 -4 2 2 4 -2 2 a a z 3 z 2 2 4 2 2 - a + -- - 5 a + 2 a + z - ---- + -- - -- + ---- - 4 a z + a z + 2 2 2 4 2 a z z a a 4 z 2 4 > -- - a z 2 a
In[9]:=	Kauffman[Link[11, NonAlternating, 350]][a, z]
Out[9]=	2 4 3 -4 2 4 -2 2 a a 2 a 2 a z z 3 z 9 - a + 13 a + 6 a - z - ---- - -- + --- + ---- + -- + -- - --- - 8 a z - 2 2 z z 5 3 a z z a a 2 2 3 3 3 2 4 z 2 z 2 2 4 2 3 z 10 z > 5 a z - 23 z + ---- + ---- - 36 a z - 15 a z - ---- + ----- + 4 2 5 a a a a 4 4 5 5 3 3 3 4 7 z 4 z 2 4 4 4 z 3 z > 6 a z - a z + 31 z - ---- - ---- + 44 a z + 16 a z + -- - ---- - 4 2 5 3 a a a a 5 6 6 7 7 z 5 3 5 6 2 z z 2 6 4 6 z > ---- + 6 a z + 9 a z - 15 z + ---- + -- - 21 a z - 7 a z + -- + a 4 2 3 a a a 7 z 7 3 7 8 2 8 4 8 9 3 9 > -- - 6 a z - 6 a z + 2 z + 3 a z + a z + a z + a z a
In[10]:=	Kh[L][q, t]
Out[10]=	4 3 1 1 2 1 1 1 1 1 - + 4 q + q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ----- + q 11 6 7 5 7 4 5 4 5 3 3 3 5 2 3 2 q t q t q t q t q t q t q t q t 1 2 q 3 5 3 2 5 2 5 3 7 3 > ---- + --- + - + 2 q t + q t + q t + q t + 2 q t + q t + q t + 2 q t t q t 7 4 9 4 11 5 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n350

L11n349 L11n351