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L11n412 L11n414

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DrawMorseLink

PD Presentation:

X8192 X5,15,6,14 X10,3,11,4 X13,5,14,4 X2738 X6,9,1,10 X11,18,12,19 X17,12,18,7 X20,16,21,15 X22,20,13,19 X16,22,17,21

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-16 + 2q-14 + 4q-12 + 5q-10 + 7q-8 + 7q-6 + 3q-4 + 3q-2 - 1 - q2 - q4 - q6 + q8 - q10 - q16

HOMFLY-PT Polynomial:

- a-4 - a-4z2 - a-2z-2 + a-2 + 2a-2z2 + a-2z4 + 4z-2 + 6 + 3z2 + z4 - 5a2z-2 - 9a2 - 6a2z2 - a2z4 + 2a4z-2 + 3a4 + a4z2

Kauffman Polynomial:

a-5z - 3a-5z3 + a-5z5 - a-4 + 3a-4z2 - 6a-4z4 + 2a-4z6 + a-3z-1 - 2a-3z + 4a-3z3 - 6a-3z5 + 2a-3z7 - a-2z-2 + 3a-2z2 - a-2z4 - 2a-2z6 + a-2z8 + 5a-1z-1 - 19a-1z + 26a-1z3 - 14a-1z5 + 3a-1z7 - 4z-2 + 13 - 16z2 + 15z4 - 6z6 + z8 + 9az-1 - 35az + 41az3 - 16az5 + 2az7 - 5a2z-2 + 22a2 - 39a2z2 + 31a2z4 - 10a2z6 + a2z8 + 5a3z-1 - 19a3z + 22a3z3 - 9a3z5 + a3z7 - 2a4z-2 + 11a4 - 23a4z2 + 21a4z4 - 8a4z6 + 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                   2 1   j = 5                 2 1     j = 3               2 2       j = 1           1 3 2         j = -1           2 4           j = -3       1 2 1             j = -5         3               j = -7     1                   j = -9 1                       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, 413]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 413]]
Out[4]=	PD[X[8, 1, 9, 2], X[5, 15, 6, 14], X[10, 3, 11, 4], X[13, 5, 14, 4], > X[2, 7, 3, 8], X[6, 9, 1, 10], X[11, 18, 12, 19], X[17, 12, 18, 7], > X[20, 16, 21, 15], X[22, 20, 13, 19], X[16, 22, 17, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -5, 3, 4, -2, -6}, {5, -1, 6, -3, -7, 8}, > {-4, 2, 9, -11, -8, 7, 10, -9, 11, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-5 -3 2 2 2 3 4 5 4 + q + q + -- - - - 4 q + 4 q - 3 q + 2 q - q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 4 5 7 7 3 3 2 4 6 8 10 16 -1 + q + --- + --- + --- + -- + -- + -- + -- - q - 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, 413]][a, z]
Out[8]=	2 4 2 2 -4 -2 2 4 4 1 5 a 2 a 2 z 2 z 6 - a + a - 9 a + 3 a + -- - ----- - ---- + ---- + 3 z - -- + ---- - 2 2 2 2 2 4 2 z a z z z a a 4 2 2 4 2 4 z 2 4 > 6 a z + a z + z + -- - a z 2 a
In[9]:=	Kauffman[Link[11, NonAlternating, 413]][a, z]
Out[9]=	2 4 3 -4 2 4 4 1 5 a 2 a 1 5 9 a 5 a 13 - a + 22 a + 11 a - -- - ----- - ---- - ---- + ---- + --- + --- + ---- + 2 2 2 2 2 3 a z z z z a z z z a z 2 2 z 2 z 19 z 3 2 3 z 3 z 2 2 > -- - --- - ---- - 35 a z - 19 a z - 16 z + ---- + ---- - 39 a z - 5 3 a 4 2 a a a a 3 3 3 4 4 4 2 3 z 4 z 26 z 3 3 3 4 6 z z > 23 a z - ---- + ---- + ----- + 41 a z + 22 a z + 15 z - ---- - -- + 5 3 a 4 2 a a a a 5 5 5 6 2 4 4 4 z 6 z 14 z 5 3 5 6 2 z > 31 a z + 21 a z + -- - ---- - ----- - 16 a z - 9 a z - 6 z + ---- - 5 3 a 4 a a a 6 7 7 8 2 z 2 6 4 6 2 z 3 z 7 3 7 8 z > ---- - 10 a z - 8 a z + ---- + ---- + 2 a z + a z + z + -- + 2 3 a 2 a a a 2 8 4 8 > a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 1 1 1 3 2 1 2 q - + 3 q + ------ + ----- + ----- + ----- + ----- + ----- + ---- + --- + - + q 11 6 9 6 7 4 3 3 5 2 3 2 3 q t t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 9 4 > 2 q t + 2 q t + 2 q t + 2 q t + q t + 2 q t + q t + q t + 11 5 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n413

L11n412 L11n414