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L11n136 L11n138

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DrawMorseLink

PD Presentation:

X8192 X18,11,19,12 X10,4,11,3 X2,17,3,18 X12,5,13,6 X6718 X16,10,17,9 X13,20,14,21 X15,22,16,7 X4,20,5,19 X21,14,22,15

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 4q-17/2 - 7q-15/2 + 10q-13/2 - 12q-11/2 + 12q-9/2 - 11q-7/2 + 7q-5/2 - 5q-3/2 + q-1/2

A2 (sl(3)) Invariant:

q-28 - 2q-26 + q-24 - 2q-22 - 2q-20 + 2q-18 - 2q-16 + 4q-14 + 3q-10 + 4q-8 + 3q-4 - q-2

HOMFLY-PT Polynomial:

- 2a3z-1 - 3a3z + a3z3 + a3z5 + 3a5z-1 + 2a5z - 4a5z3 - 4a5z5 - a5z7 - a7z-1 + 2a7z3 + a7z5

Kauffman Polynomial:

- a2z4 - 2a3z-1 + 3a3z + 4a3z3 - 5a3z5 + 3a4 - 4a4z2 + 4a4z4 - 3a4z6 - a4z8 - 3a5z-1 + 4a5z + 2a5z3 - a5z5 - 2a5z7 - a5z9 + 3a6 - 9a6z2 + 8a6z4 + 2a6z6 - 5a6z8 - a7z-1 + a7z - 7a7z3 + 16a7z5 - 8a7z7 - a7z9 + a8 - 7a8z2 + 10a8z4 + a8z6 - 4a8z8 - 4a9z3 + 11a9z5 - 6a9z7 - 2a10z2 + 7a10z4 - 4a10z6 + a11z3 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 j = 0                   1 j = -2                 4   j = -4               4 2   j = -6             7 3     j = -8           6 5       j = -10         6 6         j = -12       4 6           j = -14     3 6             j = -16   1 4               j = -18   3                 j = -20 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, 137]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 137]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 11, 19, 12], X[10, 4, 11, 3], X[2, 17, 3, 18], > X[12, 5, 13, 6], X[6, 7, 1, 8], X[16, 10, 17, 9], X[13, 20, 14, 21], > X[15, 22, 16, 7], X[4, 20, 5, 19], X[21, 14, 22, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -10, 5, -6}, > {6, -1, 7, -3, 2, -5, -8, 11, -9, -7, 4, -2, 10, 8, -11, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 4 7 10 12 12 11 7 5 1 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 2 -24 2 2 2 2 4 3 4 3 -2 q - --- + q - --- - --- + --- - --- + --- + --- + -- + -- - q 26 22 20 18 16 14 10 8 4 q q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 137]][a, z]
Out[8]=	3 5 7 -2 a 3 a a 3 5 3 3 5 3 7 3 3 5 ----- + ---- - -- - 3 a z + 2 a z + a z - 4 a z + 2 a z + a z - z z z 5 5 7 5 5 7 > 4 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 137]][a, z]
Out[9]=	3 5 7 4 6 8 2 a 3 a a 3 5 7 4 2 3 a + 3 a + a - ---- - ---- - -- + 3 a z + 4 a z + a z - 4 a z - z z z 6 2 8 2 10 2 3 3 5 3 7 3 9 3 > 9 a z - 7 a z - 2 a z + 4 a z + 2 a z - 7 a z - 4 a z + 11 3 2 4 4 4 6 4 8 4 10 4 3 5 > a z - a z + 4 a z + 8 a z + 10 a z + 7 a z - 5 a z - 5 5 7 5 9 5 11 5 4 6 6 6 8 6 > a z + 16 a z + 11 a z - a z - 3 a z + 2 a z + a z - 10 6 5 7 7 7 9 7 4 8 6 8 8 8 > 4 a z - 2 a z - 8 a z - 6 a z - a z - 5 a z - 4 a z - 5 9 7 9 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 1 3 1 4 3 6 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 7 16 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 6 6 6 6 5 7 3 4 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + t 12 4 10 4 10 3 8 3 8 2 6 2 6 4 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 L11n137

L11n136 L11n138