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L11n161 L11n163

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DrawMorseLink

PD Presentation:

X8192 X10,3,11,4 X17,13,18,12 X14,5,15,6 X4,13,5,14 X11,19,12,18 X19,7,20,22 X15,21,16,20 X21,17,22,16 X2738 X6,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-11/2 - 2q-5/2 + 3q-3/2 - 4q-1/2 + 4q1/2 - 4q3/2 + 3q5/2 - 2q7/2 + q9/2

A2 (sl(3)) Invariant:

q-18 + q-16 + 2q-14 + 2q-12 + 2q-10 + 3q-8 + q-4 - q-2 - 1 - q4 + q6 - q14

HOMFLY-PT Polynomial:

2a-3z + a-3z3 + a-1z-1 - a-1z - 3a-1z3 - a-1z5 - 2az-1 - 5az - 4az3 - az5 + a5z-1 + a5z

Kauffman Polynomial:

- 3a-4z2 + 4a-4z4 - a-4z6 + 2a-3z - 7a-3z3 + 8a-3z5 - 2a-3z7 - 2a-2 + 5a-2z2 - 8a-2z4 + 8a-2z6 - 2a-2z8 + a-1z-1 - 2a-1z3 + 3a-1z7 - a-1z9 - 5 + 22z2 - 29z4 + 16z6 - 3z8 + 2az-1 - 8az + 13az3 - 13az5 + 6az7 - az9 - 3a2 + 10a2z2 - 15a2z4 + 7a2z6 - a2z8 + 3a3z - 6a3z3 + 2a3z5 + a4 - 4a4z2 + 2a4z4 - a5z-1 + 9a5z - 14a5z3 + 7a5z5 - a5z7

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 = 10                       1 j = 8                     1   j = 6                   2 1   j = 4                 2 1     j = 2               2 2       j = 0           1 3 2         j = -2           2 3           j = -4       1 2 2             j = -6         2               j = -8     1 1                 j = -10 1                       j = -12 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 162]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 162]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[17, 13, 18, 12], X[14, 5, 15, 6], > X[4, 13, 5, 14], X[11, 19, 12, 18], X[19, 7, 20, 22], X[15, 21, 16, 20], > X[21, 17, 22, 16], X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -5, 4, -11}, > {10, -1, 11, -2, -6, 3, 5, -4, -8, 9, -3, 6, -7, 8, -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 2 3 4 3/2 5/2 7/2 9/2 -q - ---- + ---- - ------- + 4 Sqrt[q] - 4 q + 3 q - 2 q + q 5/2 3/2 Sqrt[q] q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 2 2 2 3 -4 -2 4 6 14 -1 + q + q + --- + --- + --- + -- + q - q - q + q - q 14 12 10 8 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 162]][a, z]
Out[8]=	5 3 3 5 1 2 a a 2 z z 5 z 3 z 3 z 5 --- - --- + -- + --- - - - 5 a z + a z + -- - ---- - 4 a z - -- - a z a z z z 3 a 3 a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 162]][a, z]
Out[9]=	5 2 2 4 1 2 a a 2 z 3 5 2 -5 - -- - 3 a + a + --- + --- - -- + --- - 8 a z + 3 a z + 9 a z + 22 z - 2 a z z z 3 a a 2 2 3 3 3 z 5 z 2 2 4 2 7 z 2 z 3 3 3 > ---- + ---- + 10 a z - 4 a z - ---- - ---- + 13 a z - 6 a z - 4 2 3 a a a a 4 4 5 5 3 4 4 z 8 z 2 4 4 4 8 z 5 > 14 a z - 29 z + ---- - ---- - 15 a z + 2 a z + ---- - 13 a z + 4 2 3 a a a 6 6 7 7 3 5 5 5 6 z 8 z 2 6 2 z 3 z 7 > 2 a z + 7 a z + 16 z - -- + ---- + 7 a z - ---- + ---- + 6 a z - 4 2 3 a a a a 8 9 5 7 8 2 z 2 8 z 9 > a z - 3 z - ---- - a z - -- - a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 1 1 2 2 1 2 3 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + - + ---- + 2 12 6 10 6 8 4 8 3 4 3 6 2 4 2 t 4 q q t q t q t q t q t q t q t q t 2 2 2 2 4 2 4 3 6 3 6 4 8 4 > ---- + 2 t + 2 q t + 2 q t + 2 q t + q t + 2 q t + q t + q t + 2 q t 10 5 > q t

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