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DrawMorseLink

PD Presentation:

X8192 X16,7,17,8 X3,10,4,11 X2,15,3,16 X14,10,15,9 X11,19,12,18 X12,5,13,6 X6,21,1,22 X20,14,21,13 X22,17,7,18 X19,4,20,5

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-26 + q-22 + q-20 + 2q-16 + q-14 + 2q-12 + q-10 - q-4 + 1

HOMFLY-PT Polynomial:

- az + a3z + a3z3 - a5z-1 - 5a5z - 4a5z3 - a5z5 + a7z-1 + 2a7z + a7z3

Kauffman Polynomial:

- az - a2z2 + 2a3z3 - a3z5 + 2a4z2 - 4a4z4 + 4a4z6 - a4z8 + a5z-1 - 3a5z + 4a5z3 - 4a5z5 + 4a5z7 - a5z9 - a6 + 7a6z2 - 17a6z4 + 14a6z6 - 3a6z8 + a7z-1 - a7z - 5a7z3 + 2a7z5 + 3a7z7 - a7z9 + 4a8z2 - 13a8z4 + 10a8z6 - 2a8z8 + 3a9z - 7a9z3 + 5a9z5 - a9z7

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = 0                 1 j = -2               2 1 j = -4             1     j = -6           2 2     j = -8         2 1       j = -10       1 2         j = -12     2 2           j = -14   1 2             j = -16   1               j = -18 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, 158]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 158]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 7, 17, 8], X[3, 10, 4, 11], X[2, 15, 3, 16], > X[14, 10, 15, 9], X[11, 19, 12, 18], X[12, 5, 13, 6], X[6, 21, 1, 22], > X[20, 14, 21, 13], X[22, 17, 7, 18], X[19, 4, 20, 5]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, -3, 11, 7, -8}, > {2, -1, 5, 3, -6, -7, 9, -5, 4, -2, 10, 6, -11, -9, 8, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 2 3 3 4 3 3 2 1 -q + ----- - ----- + ----- - ---- + ---- - ---- + ---- - ------- 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -22 -20 2 -14 2 -10 -4 1 + q + q + q + --- + q + --- + q - q 16 12 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 158]][a, z]
Out[8]=	5 7 a a 3 5 7 3 3 5 3 7 3 5 5 -(--) + -- - a z + a z - 5 a z + 2 a z + a z - 4 a z + a z - a z z z
In[9]:=	Kauffman[Link[11, NonAlternating, 158]][a, z]
Out[9]=	5 7 6 a a 5 7 9 2 2 4 2 6 2 -a + -- + -- - a z - 3 a z - a z + 3 a z - a z + 2 a z + 7 a z + z z 8 2 3 3 5 3 7 3 9 3 4 4 6 4 > 4 a z + 2 a z + 4 a z - 5 a z - 7 a z - 4 a z - 17 a z - 8 4 3 5 5 5 7 5 9 5 4 6 6 6 > 13 a z - a z - 4 a z + 2 a z + 5 a z + 4 a z + 14 a z + 8 6 5 7 7 7 9 7 4 8 6 8 8 8 5 9 > 10 a z + 4 a z + 3 a z - a z - a z - 3 a z - 2 a z - a z - 7 9 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	-2 1 1 1 2 2 2 1 1 + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 18 8 16 7 14 7 14 6 12 6 12 5 10 5 q t q t q t q t q t q t q t 2 2 1 2 2 1 2 > ------ + ----- + ----- + ----- + ----- + ----- + ---- 10 4 8 4 8 3 6 3 6 2 4 2 2 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 L11n158

L11n157 L11n159