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L11n150 L11n152

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DrawMorseLink

PD Presentation:

X8192 X18,9,19,10 X6718 X22,19,7,20 X12,5,13,6 X3,10,4,11 X4,15,5,16 X11,16,12,17 X20,13,21,14 X14,21,15,22 X17,2,18,3

Gauss Code:

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

Jones Polynomial:

q-25/2 - 2q-23/2 + 3q-21/2 - 4q-19/2 + 4q-17/2 - 4q-15/2 + 3q-13/2 - 3q-11/2 + q-9/2 - q-7/2

A2 (sl(3)) Invariant:

- q-42 - q-38 - q-36 + 2q-30 + 2q-26 + q-24 + q-22 + 2q-20 + q-18 + 2q-16 + q-12

HOMFLY-PT Polynomial:

- a7z-1 - 7a7z - 11a7z3 - 6a7z5 - a7z7 - a9z - 6a9z3 - 5a9z5 - a9z7 + 2a11z-1 + 6a11z + 5a11z3 + a11z5 - a13z-1 - a13z

Kauffman Polynomial:

- a7z-1 + 7a7z - 11a7z3 + 6a7z5 - a7z7 + a8 - 3a8z2 - 2a8z4 + 4a8z6 - a8z8 - 3a9z5 + 4a9z7 - a9z9 - 3a10 + 11a10z2 - 19a10z4 + 14a10z6 - 3a10z8 + 2a11z-1 - 9a11z + 12a11z3 - 7a11z5 + 4a11z7 - a11z9 - 5a12 + 19a12z2 - 20a12z4 + 10a12z6 - 2a12z8 + a13z-1 - a13z - a13z3 + 2a13z5 - a13z7 - 2a14 + 4a14z2 - 3a14z4 + a15z - 2a15z3 - a16z2

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -6                   1 j = -8                 1 1 j = -10               2     j = -12             1 1     j = -14           3 2       j = -16         1 1         j = -18       3 3           j = -20     1 2             j = -22   1 2               j = -24   1                 j = -26 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, 151]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 151]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 9, 19, 10], X[6, 7, 1, 8], X[22, 19, 7, 20], > X[12, 5, 13, 6], X[3, 10, 4, 11], X[4, 15, 5, 16], X[11, 16, 12, 17], > X[20, 13, 21, 14], X[14, 21, 15, 22], X[17, 2, 18, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -6, -7, 5, -3}, > {3, -1, 2, 6, -8, -5, 9, -10, 7, 8, -11, -2, 4, -9, 10, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) 2 3 4 4 4 3 3 -(9/2) q - ----- + ----- - ----- + ----- - ----- + ----- - ----- + q - 23/2 21/2 19/2 17/2 15/2 13/2 11/2 q q q q q q q -(7/2) > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 -38 -36 2 2 -24 -22 2 -18 2 -12 -q - q - q + --- + --- + q + q + --- + q + --- + q 30 26 20 16 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 151]][a, z]
Out[8]=	7 11 13 a 2 a a 7 9 11 13 7 3 9 3 -(--) + ----- - --- - 7 a z - a z + 6 a z - a z - 11 a z - 6 a z + z z z 11 3 7 5 9 5 11 5 7 7 9 7 > 5 a z - 6 a z - 5 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 151]][a, z]
Out[9]=	7 11 13 8 10 12 14 a 2 a a 7 11 13 a - 3 a - 5 a - 2 a - -- + ----- + --- + 7 a z - 9 a z - a z + z z z 15 8 2 10 2 12 2 14 2 16 2 7 3 > a z - 3 a z + 11 a z + 19 a z + 4 a z - a z - 11 a z + 11 3 13 3 15 3 8 4 10 4 12 4 > 12 a z - a z - 2 a z - 2 a z - 19 a z - 20 a z - 14 4 7 5 9 5 11 5 13 5 8 6 10 6 > 3 a z + 6 a z - 3 a z - 7 a z + 2 a z + 4 a z + 14 a z + 12 6 7 7 9 7 11 7 13 7 8 8 10 8 > 10 a z - a z + 4 a z + 4 a z - a z - a z - 3 a z - 12 8 9 9 11 9 > 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 1 1 2 1 2 3 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 26 9 24 8 22 8 22 7 20 7 20 6 18 6 q t q t q t q t q t q t q t 3 1 1 3 2 1 1 2 1 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + ---- 18 5 16 5 16 4 14 4 14 3 12 3 12 2 10 2 8 q t q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n151

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