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L11n405 L11n407

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X11,22,12,19 X10,4,11,3 X5,21,6,20 X21,5,22,18 X19,12,20,13 X14,9,15,10 X2,14,3,13 X8,15,9,16

Gauss Code:

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

Jones Polynomial:

- q-7 + q-6 - q-5 + 2q-4 + q-1 + 2q - q2 + q3

A2 (sl(3)) Invariant:

- q-22 - q-20 - q-18 + 2q-14 + 2q-12 + 3q-10 + q-8 + q-6 + 2q-4 + 3q-2 + 5 + 4q2 + 3q4 + 2q6 + q8 + q10

HOMFLY-PT Polynomial:

a-2z-2 + 2a-2 + a-2z2 - 2z-2 - 4 - 4z2 - z4 + a2z-2 + 4a4 + 4a4z2 + a4z4 - 2a6 - a6z2

Kauffman Polynomial:

a-2z-2 - 4a-2 + 6a-2z2 - 5a-2z4 + a-2z6 - 2a-1z-1 + 4a-1z - 4a-1z5 + a-1z7 + 2z-2 - 7 + 12z2 - 2z4 - 4z6 + z8 - 2az-1 + 8az - 10az3 + 10az5 - 6az7 + az9 + a2z-2 - 12a2z2 + 22a2z4 - 13a2z6 + 2a2z8 - 2a3z3 + 8a3z5 - 6a3z7 + a3z9 + 8a4 - 28a4z2 + 30a4z4 - 14a4z6 + 2a4z8 - 8a5z + 18a5z3 - 12a5z5 + 2a5z7 + 4a6 - 10a6z2 + 11a6z4 - 6a6z6 + a6z8 - 4a7z + 10a7z3 - 6a7z5 + a7z7

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                       1 j = 5                         j = 3                   2 1   j = 1               3 1       j = -1             2 5 2       j = -3           1 2 2         j = -5         1 2 1           j = -7       2 2 2             j = -9       1                 j = -11   1 2 1                 j = -13                         j = -15 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, 406]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 406]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[11, 22, 12, 19], > X[10, 4, 11, 3], X[5, 21, 6, 20], X[21, 5, 22, 18], X[19, 12, 20, 13], > X[14, 9, 15, 10], X[2, 14, 3, 13], X[8, 15, 9, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -3}, {-8, 6, -7, 4}, > {-6, -1, 2, -11, 9, -5, -4, 8, 10, -9, 11, -2, 3, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-7 -6 -5 2 1 2 3 -q + q - q + -- + - + 2 q - q + q 4 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 -18 2 2 3 -8 -6 2 3 2 4 5 - q - q - q + --- + --- + --- + q + q + -- + -- + 4 q + 3 q + 14 12 10 4 2 q q q q q 6 8 10 > 2 q + q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 406]][a, z]
Out[8]=	2 2 2 4 6 2 1 a 2 z 4 2 6 2 4 -4 + -- + 4 a - 2 a - -- + ----- + -- - 4 z + -- + 4 a z - a z - z + 2 2 2 2 2 2 a z a z z a 4 4 > a z
In[9]:=	Kauffman[Link[11, NonAlternating, 406]][a, z]
Out[9]=	2 4 4 6 2 1 a 2 2 a 4 z 5 -7 - -- + 8 a + 4 a + -- + ----- + -- - --- - --- + --- + 8 a z - 8 a z - 2 2 2 2 2 a z z a a z a z z 2 7 2 6 z 2 2 4 2 6 2 3 > 4 a z + 12 z + ---- - 12 a z - 28 a z - 10 a z - 10 a z - 2 a 4 3 3 5 3 7 3 4 5 z 2 4 4 4 > 2 a z + 18 a z + 10 a z - 2 z - ---- + 22 a z + 30 a z + 2 a 5 6 6 4 4 z 5 3 5 5 5 7 5 6 z > 11 a z - ---- + 10 a z + 8 a z - 12 a z - 6 a z - 4 z + -- - a 2 a 7 2 6 4 6 6 6 z 7 3 7 5 7 7 7 > 13 a z - 14 a z - 6 a z + -- - 6 a z - 6 a z + 2 a z + a z + a 8 2 8 4 8 6 8 9 3 9 > z + 2 a z + 2 a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 5 1 1 2 1 1 2 2 -- + - + 3 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 7 11 6 11 5 11 4 9 4 7 4 7 3 q q t q t q t q t q t q t q t 1 2 2 1 1 2 2 2 t 3 2 > ----- + ----- + ----- + ----- + ---- + ---- + --- + --- + q t + 2 q t + 5 3 7 2 5 2 3 2 5 3 q t q q t q t q t q t q t q t 3 3 7 4 > q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n406

L11n405 L11n407