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L11n162 L11n164

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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 X22,19,7,20 X20,15,21,16 X16,21,17,22 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-21/2 - 3q-19/2 + 5q-17/2 - 8q-15/2 + 9q-13/2 - 10q-11/2 + 9q-9/2 - 7q-7/2 + 4q-5/2 - 2q-3/2

A2 (sl(3)) Invariant:

- q-32 + q-30 - q-26 + 3q-24 + 2q-20 + 2q-18 + 2q-14 - 2q-12 + q-10 + q-8 - q-6 + 2q-4

HOMFLY-PT Polynomial:

- 3a3z - 2a3z3 - a5z-1 - a5z + a5z3 + a5z5 + a7z-1 + 2a7z + 2a7z3 + a7z5 - a9z - a9z3

Kauffman Polynomial:

3a3z - 3a3z3 + a4z2 - 2a4z4 - a4z6 + a5z-1 - 2a5z - 2a5z7 - a6 + a6z4 - 2a6z8 + a7z-1 - 2a7z - 2a7z3 + 7a7z5 - 3a7z7 - a7z9 - a8z2 - a8z4 + 9a8z6 - 5a8z8 + 5a9z - 14a9z3 + 17a9z5 - 4a9z7 - a9z9 - 2a10z2 - a10z4 + 7a10z6 - 3a10z8 + 2a11z - 9a11z3 + 10a11z5 - 3a11z7 - 2a12z2 + 3a12z4 - a12z6

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -2                   2 j = -4                 3 1 j = -6               4 1   j = -8             5 3     j = -10           5 4       j = -12         4 5         j = -14       4 5           j = -16     2 5             j = -18   1 3               j = -20   2                 j = -22 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, 163]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 163]]
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[22, 19, 7, 20], X[20, 15, 21, 16], > X[16, 21, 17, 22], 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]=	-(21/2) 3 5 8 9 10 9 7 4 2 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -30 -26 3 2 2 2 2 -10 -8 -6 2 -q + q - q + --- + --- + --- + --- - --- + q + q - q + -- 24 20 18 14 12 4 q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 163]][a, z]
Out[8]=	5 7 a a 3 5 7 9 3 3 5 3 7 3 -(--) + -- - 3 a z - a z + 2 a z - a z - 2 a z + a z + 2 a z - z z 9 3 5 5 7 5 > a z + a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 163]][a, z]
Out[9]=	5 7 6 a a 3 5 7 9 11 4 2 8 2 -a + -- + -- + 3 a z - 2 a z - 2 a z + 5 a z + 2 a z + a z - a z - z z 10 2 12 2 3 3 7 3 9 3 11 3 4 4 > 2 a z - 2 a z - 3 a z - 2 a z - 14 a z - 9 a z - 2 a z + 6 4 8 4 10 4 12 4 7 5 9 5 11 5 > a z - a z - a z + 3 a z + 7 a z + 17 a z + 10 a z - 4 6 8 6 10 6 12 6 5 7 7 7 9 7 > a z + 9 a z + 7 a z - a z - 2 a z - 3 a z - 4 a z - 11 7 6 8 8 8 10 8 7 9 9 9 > 3 a z - 2 a z - 5 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 2 1 2 1 3 2 5 4 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q t q t q t q t q t q t q t 5 4 5 5 4 5 3 4 1 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q t q t q t q t q t q t q t q t q t 3 > ---- 4 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n163

L11n162 L11n164