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L11n173 L11n175

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DrawMorseLink

PD Presentation:

X8192 X20,9,21,10 X14,5,15,6 X11,18,12,19 X3,10,4,11 X7,13,8,12 X16,13,17,14 X17,7,18,22 X6,15,1,16 X4,21,5,22 X19,2,20,3

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-30 + 2q-26 + q-24 + q-22 - 2q-18 + q-16 - q-14 + 2q-12 + q-10 + q-8 + q-6 + q-2

HOMFLY-PT Polynomial:

- a3z-1 - 4a3z - 4a3z3 - a3z5 + 2a5z-1 + 4a5z + 7a5z3 + 5a5z5 + a5z7 - 2a7z-1 - 4a7z - 4a7z3 - a7z5 + a9z-1 + a9z

Kauffman Polynomial:

- a3z-1 + 5a3z - 8a3z3 + 5a3z5 - a3z7 + 2a4z2 - 10a4z4 + 9a4z6 - 2a4z8 - 2a5z-1 + 9a5z - 19a5z3 + 11a5z5 + a5z7 - a5z9 - a6 + 8a6z2 - 19a6z4 + 17a6z6 - 4a6z8 - 2a7z-1 + 9a7z - 15a7z3 + 9a7z5 + a7z7 - a7z9 + 6a8z2 - 10a8z4 + 8a8z6 - 2a8z8 - a9z-1 + 3a9z - 4a9z3 + 3a9z5 - a9z7 - a10z4 - 2a11z

Khovanov Homology:

trqj 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             3 1   j = -6           2 2     j = -8         3 2       j = -10       2 2         j = -12     2 3           j = -14   1 2             j = -16 1 2               j = -18 2

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, 174]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 174]]
Out[4]=	PD[X[8, 1, 9, 2], X[20, 9, 21, 10], X[14, 5, 15, 6], X[11, 18, 12, 19], > X[3, 10, 4, 11], X[7, 13, 8, 12], X[16, 13, 17, 14], X[17, 7, 18, 22], > X[6, 15, 1, 16], X[4, 21, 5, 22], X[19, 2, 20, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -5, -10, 3, -9}, > {-6, -1, 2, 5, -4, 6, 7, -3, 9, -7, -8, 4, -11, -2, 10, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-2 3 4 5 5 4 4 2 1 ----- + ----- - ----- + ----- - ---- + ---- - ---- + ---- - ------- 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]=	-30 2 -24 -22 2 -16 -14 2 -10 -8 -6 -2 q + --- + q + q - --- + q - q + --- + q + q + q + q 26 18 12 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 174]][a, z]
Out[8]=	3 5 7 9 a 2 a 2 a a 3 5 7 9 3 3 -(--) + ---- - ---- + -- - 4 a z + 4 a z - 4 a z + a z - 4 a z + z z z z 5 3 7 3 3 5 5 5 7 5 5 7 > 7 a z - 4 a z - a z + 5 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 174]][a, z]
Out[9]=	3 5 7 9 6 a 2 a 2 a a 3 5 7 9 11 -a - -- - ---- - ---- - -- + 5 a z + 9 a z + 9 a z + 3 a z - 2 a z + z z z z 4 2 6 2 8 2 3 3 5 3 7 3 9 3 > 2 a z + 8 a z + 6 a z - 8 a z - 19 a z - 15 a z - 4 a z - 4 4 6 4 8 4 10 4 3 5 5 5 7 5 > 10 a z - 19 a z - 10 a z - a z + 5 a z + 11 a z + 9 a z + 9 5 4 6 6 6 8 6 3 7 5 7 7 7 9 7 > 3 a z + 9 a z + 17 a z + 8 a z - a z + a z + a z - a z - 4 8 6 8 8 8 5 9 7 9 > 2 a z - 4 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 2 1 2 1 2 2 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 18 6 16 6 16 5 14 5 14 4 12 4 12 3 q q q t q t q t q t q t q t q t 2 2 3 2 2 t t 2 > ------ + ------ + ----- + ---- + ---- + -- + -- + t 10 3 10 2 8 2 8 6 4 2 q t q t q t q t q t q q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n174

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