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L11n175 L11n177

Knotscape

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DrawMorseLink

PD Presentation:

X8192 X20,9,21,10 X14,5,15,6 X11,18,12,19 X3,10,4,11 X12,7,13,8 X16,13,17,14 X22,17,7,18 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:

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

A2 (sl(3)) Invariant:

- q-42 + q-40 + q-38 - 2q-32 - q-30 - 2q-28 + q-26 + 2q-24 + 2q-22 + 3q-20 + 2q-18 + 2q-16 + q-12

HOMFLY-PT Polynomial:

- 2a7z-1 - 8a7z - 11a7z3 - 6a7z5 - a7z7 + 3a9z-1 + 2a9z - 5a9z3 - 5a9z5 - a9z7 - a11z-1 + 4a11z + 5a11z3 + a11z5 - a13z

Kauffman Polynomial:

- 2a7z-1 + 8a7z - 11a7z3 + 6a7z5 - a7z7 + 3a8 - 5a8z2 - a8z4 + 4a8z6 - a8z8 - 3a9z-1 + 7a9z - 6a9z3 - a9z5 + 4a9z7 - a9z9 + 3a10 + 3a10z2 - 18a10z4 + 15a10z6 - 3a10z8 - a11z-1 - a11z3 - 2a11z5 + 4a11z7 - a11z9 + a12 + 8a12z2 - 17a12z4 + 11a12z6 - 2a12z8 + a13z - 6a13z3 + 5a13z5 - a13z7

Khovanov Homology:

trqj 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 1     j = -14         3 2       j = -16     1 2 1         j = -18     2 3           j = -20   1 1             j = -22   1               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, NonAlternating, 176]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 176]]
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[12, 7, 13, 8], X[16, 13, 17, 14], X[22, 17, 7, 18], > 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]=	-(23/2) 2 2 3 3 2 3 -(9/2) -(7/2) -q + ----- - ----- + ----- - ----- + ----- - ----- + q - q 21/2 19/2 17/2 15/2 13/2 11/2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 -40 -38 2 -30 2 -26 2 2 3 2 2 -q + q + q - --- - q - --- + q + --- + --- + --- + --- + --- + 32 28 24 22 20 18 16 q q q q q q q -12 > q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 176]][a, z]
Out[8]=	7 9 11 -2 a 3 a a 7 9 11 13 7 3 9 3 ----- + ---- - --- - 8 a z + 2 a z + 4 a z - a z - 11 a z - 5 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, 176]][a, z]
Out[9]=	7 9 11 8 10 12 2 a 3 a a 7 9 13 8 2 3 a + 3 a + a - ---- - ---- - --- + 8 a z + 7 a z + a z - 5 a z + z z z 10 2 12 2 7 3 9 3 11 3 13 3 8 4 > 3 a z + 8 a z - 11 a z - 6 a z - a z - 6 a z - a z - 10 4 12 4 7 5 9 5 11 5 13 5 8 6 > 18 a z - 17 a z + 6 a z - a z - 2 a z + 5 a z + 4 a z + 10 6 12 6 7 7 9 7 11 7 13 7 8 8 > 15 a z + 11 a z - a z + 4 a z + 4 a z - a z - a z - 10 8 12 8 9 9 11 9 > 3 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 1 1 1 2 1 3 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 24 8 22 7 20 7 20 6 18 6 16 6 18 5 q t q t q t q t q t q t q t 2 1 3 1 2 1 1 2 1 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + ---- 16 5 16 4 14 4 12 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 L11n176

L11n175 L11n177