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L11n424 L11n426

Knotscape

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DrawMorseLink

PD Presentation:

X8192 X16,8,17,7 X5,14,6,15 X3,10,4,11 X13,4,14,5 X17,2,18,3 X9,19,10,18 X21,7,22,12 X11,13,12,22 X20,16,21,15 X6,19,1,20

Gauss Code:

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

Jones Polynomial:

q-6 - 2q-5 + 4q-4 - 4q-3 + 6q-2 - 5q-1 + 5 - 3q + 2q2

A2 (sl(3)) Invariant:

q-18 + 2q-14 + 2q-12 + 2q-10 + 5q-8 + 3q-6 + 4q-4 + 2q-2 + 1 + 2q2 + 2q6 + q8

HOMFLY-PT Polynomial:

a-2 + z-2 + 2 + 2z2 + z4 - 2a2z-2 - 6a2 - 9a2z2 - 5a2z4 - a2z6 + a4z-2 + 3a4 + 3a4z2 + a4z4

Kauffman Polynomial:

- 2a-2 + 2a-2z2 + a-1z5 - z-2 + 3 - 3z2 + 7z4 - 4z6 + z8 + 2az-1 - 9az + 10az3 - 3az7 + az9 - 2a2z-2 + 11a2 - 27a2z2 + 33a2z4 - 19a2z6 + 4a2z8 + 2a3z-1 - 9a3z + 14a3z3 - 8a3z5 - a3z7 + a3z9 - a4z-2 + 7a4 - 19a4z2 + 22a4z4 - 14a4z6 + 3a4z8 + 4a5z3 - 7a5z5 + 2a5z7 + 3a6z2 - 4a6z4 + a6z6

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 5                 2 j = 3               2 1 j = 1             3 1   j = -1           4 4     j = -3         2 1       j = -5       2 4         j = -7     2 2           j = -9     2             j = -11 1 2               j = -13 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, 425]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 425]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 8, 17, 7], X[5, 14, 6, 15], X[3, 10, 4, 11], > X[13, 4, 14, 5], X[17, 2, 18, 3], X[9, 19, 10, 18], X[21, 7, 22, 12], > X[11, 13, 12, 22], X[20, 16, 21, 15], X[6, 19, 1, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 6, -4, 5, -3, -11}, {2, -1, -7, 4, -9, 8}, > {-5, 3, 10, -2, -6, 7, 11, -10, -8, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-6 2 4 4 6 5 2 5 + q - -- + -- - -- + -- - - - 3 q + 2 q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 2 2 5 3 4 2 2 6 8 1 + q + --- + --- + --- + -- + -- + -- + -- + 2 q + 2 q + q 14 12 10 8 6 4 2 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 425]][a, z]
Out[8]=	2 4 -2 2 4 -2 2 a a 2 2 2 4 2 4 2 + a - 6 a + 3 a + z - ---- + -- + 2 z - 9 a z + 3 a z + z - 2 2 z z 2 4 4 4 2 6 > 5 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 425]][a, z]
Out[9]=	2 4 3 2 2 4 -2 2 a a 2 a 2 a 3 2 3 - -- + 11 a + 7 a - z - ---- - -- + --- + ---- - 9 a z - 9 a z - 3 z + 2 2 2 z z a z z 2 2 z 2 2 4 2 6 2 3 3 3 5 3 > ---- - 27 a z - 19 a z + 3 a z + 10 a z + 14 a z + 4 a z + 2 a 5 4 2 4 4 4 6 4 z 3 5 5 5 6 > 7 z + 33 a z + 22 a z - 4 a z + -- - 8 a z - 7 a z - 4 z - a 2 6 4 6 6 6 7 3 7 5 7 8 2 8 > 19 a z - 14 a z + a z - 3 a z - a z + 2 a z + z + 4 a z + 4 8 9 3 9 > 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 1 2 2 2 2 2 4 - + 3 q + ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + q 13 6 11 6 11 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 2 1 4 3 3 2 5 2 > ----- + ---- + --- + q t + 2 q t + q t + 2 q t 3 2 3 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n425

L11n424 L11n426