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L11a161 L11a163

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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X6718 X16,11,17,12 X14,6,15,5 X4,16,5,15 X20,13,21,14 X22,17,7,18 X18,21,19,22 X12,19,13,20

Gauss Code:

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

Jones Polynomial:

q-23/2 - 3q-21/2 + 6q-19/2 - 9q-17/2 + 12q-15/2 - 13q-13/2 + 12q-11/2 - 11q-9/2 + 7q-7/2 - 5q-5/2 + 2q-3/2 - q-1/2

A2 (sl(3)) Invariant:

- q-34 + q-32 - q-30 - q-28 - 3q-24 + 2q-22 - q-20 + 2q-18 + 3q-16 + 4q-12 + 2q-8 + q-6 + q-2

HOMFLY-PT Polynomial:

- a3z-1 - 4a3z - 4a3z3 - a3z5 + a5z + 4a5z3 + 4a5z5 + a5z7 + 2a7z-1 + 4a7z + 5a7z3 + 4a7z5 + a7z7 - a9z-1 - 2a9z - 3a9z3 - a9z5

Kauffman Polynomial:

- a3z-1 + 5a3z - 8a3z3 + 5a3z5 - a3z7 + a4 + a4z2 - 8a4z4 + 8a4z6 - 2a4z8 + 2a5z - 7a5z3 + 2a5z5 + 5a5z7 - 2a5z9 - 3a6 + 12a6z2 - 22a6z4 + 17a6z6 - 2a6z8 - a6z10 + 2a7z-1 - 7a7z + 16a7z3 - 23a7z5 + 20a7z7 - 6a7z9 - 5a8 + 22a8z2 - 36a8z4 + 27a8z6 - 6a8z8 - a8z10 + a9z-1 - 3a9z + 6a9z3 - 8a9z5 + 8a9z7 - 4a9z9 - 2a10 + 8a10z2 - 15a10z4 + 13a10z6 - 6a10z8 + a11z - 6a11z3 + 9a11z5 - 6a11z7 - 2a12z2 + 6a12z4 - 5a12z6 + 3a13z3 - 3a13z5 + a14z2 - a14z4

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 0                       1 j = -2                     1   j = -4                   4 1   j = -6                 4 2     j = -8               7 3       j = -10             6 5         j = -12           7 6           j = -14         5 6             j = -16       4 7               j = -18     2 5                 j = -20   1 4                   j = -22   2                     j = -24 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, Alternating, 162]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 162]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[6, 7, 1, 8], > X[16, 11, 17, 12], X[14, 6, 15, 5], X[4, 16, 5, 15], X[20, 13, 21, 14], > X[22, 17, 7, 18], X[18, 21, 19, 22], X[12, 19, 13, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -7, 6, -4}, > {4, -1, 2, -3, 5, -11, 8, -6, 7, -5, 9, -10, 11, -8, 10, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) 3 6 9 12 13 12 11 7 5 q - ----- + ----- - ----- + ----- - ----- + ----- - ---- + ---- - ---- + 21/2 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 q q q q q q q q q 2 1 > ---- - ------- 3/2 Sqrt[q] q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 -32 -30 -28 3 2 -20 2 3 4 2 -6 -2 -q + q - q - q - --- + --- - q + --- + --- + --- + -- + q + q 24 22 18 16 12 8 q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 162]][a, z]
Out[8]=	3 7 9 a 2 a a 3 5 7 9 3 3 5 3 -(--) + ---- - -- - 4 a z + a z + 4 a z - 2 a z - 4 a z + 4 a z + z z z 7 3 9 3 3 5 5 5 7 5 9 5 5 7 7 7 > 5 a z - 3 a z - a z + 4 a z + 4 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 162]][a, z]
Out[9]=	3 7 9 4 6 8 10 a 2 a a 3 5 7 9 a - 3 a - 5 a - 2 a - -- + ---- + -- + 5 a z + 2 a z - 7 a z - 3 a z + z z z 11 4 2 6 2 8 2 10 2 12 2 14 2 > a z + a z + 12 a z + 22 a z + 8 a z - 2 a z + a z - 3 3 5 3 7 3 9 3 11 3 13 3 4 4 > 8 a z - 7 a z + 16 a z + 6 a z - 6 a z + 3 a z - 8 a z - 6 4 8 4 10 4 12 4 14 4 3 5 5 5 > 22 a z - 36 a z - 15 a z + 6 a z - a z + 5 a z + 2 a z - 7 5 9 5 11 5 13 5 4 6 6 6 8 6 > 23 a z - 8 a z + 9 a z - 3 a z + 8 a z + 17 a z + 27 a z + 10 6 12 6 3 7 5 7 7 7 9 7 11 7 > 13 a z - 5 a z - a z + 5 a z + 20 a z + 8 a z - 6 a z - 4 8 6 8 8 8 10 8 5 9 7 9 9 9 > 2 a z - 2 a z - 6 a z - 6 a z - 2 a z - 6 a z - 4 a z - 6 10 8 10 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 1 2 1 4 2 5 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 24 9 22 8 20 8 20 7 18 7 18 6 16 6 q q q t q t q t q t q t q t q t 7 5 6 7 6 6 5 7 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 16 5 14 5 14 4 12 4 12 3 10 3 10 2 8 2 q t q t q t q t q t q t q t q t 3 4 t t 2 > ---- + ---- + -- + -- + t 8 6 4 2 q t q t q q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a162

L11a161 L11a163