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L11n167 L11n169

Knotscape

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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X18,11,19,12 X19,7,20,22 X15,21,16,20 X21,17,22,16 X12,17,13,18 X6718 X4,13,5,14

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 2q-17/2 - 4q-15/2 + 6q-13/2 - 7q-11/2 + 7q-9/2 - 7q-7/2 + 4q-5/2 - 3q-3/2 + q-1/2

A2 (sl(3)) Invariant:

q-28 + 2q-24 - q-20 + q-18 - q-16 + 3q-14 + q-12 + 2q-10 + 2q-8 - q-6 + q-4 - q-2

HOMFLY-PT Polynomial:

2a3z + 3a3z3 + a3z5 - a5z-1 - 10a5z - 13a5z3 - 6a5z5 - a5z7 + a7z-1 + 5a7z + 4a7z3 + a7z5

Kauffman Polynomial:

2a2z2 - a2z4 - 2a3z + 6a3z3 - 3a3z5 + 5a4z2 - 7a4z4 + 3a4z6 - a4z8 + a5z-1 - 9a5z + 14a5z3 - 12a5z5 + 4a5z7 - a5z9 - a6 + 10a6z2 - 18a6z4 + 10a6z6 - 3a6z8 + a7z-1 - 5a7z + 6a7z3 - 5a7z5 + 2a7z7 - a7z9 + 5a8z2 - 7a8z4 + 5a8z6 - 2a8z8 + a9z3 + 3a9z5 - 2a9z7 - 2a10z2 + 5a10z4 - 2a10z6 - 2a11z + 3a11z3 - 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                 2   j = -4               3 2   j = -6             4 1     j = -8           3 3       j = -10         4 4         j = -12       2 3           j = -14     2 4             j = -16   1 3               j = -18   1                 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, 168]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 168]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[14, 5, 15, 6], > X[18, 11, 19, 12], X[19, 7, 20, 22], X[15, 21, 16, 20], X[21, 17, 22, 16], > X[12, 17, 13, 18], X[6, 7, 1, 8], X[4, 13, 5, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 4, -10}, > {10, -1, 2, -3, 5, -9, 11, -4, -7, 8, 9, -5, -6, 7, -8, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 2 4 6 7 7 7 4 3 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 -20 -18 -16 3 -12 2 2 -6 -4 -2 q + --- - q + q - q + --- + q + --- + -- - q + q - q 24 14 10 8 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 168]][a, z]
Out[8]=	5 7 a a 3 5 7 3 3 5 3 7 3 3 5 -(--) + -- + 2 a z - 10 a z + 5 a z + 3 a z - 13 a z + 4 a z + a z - z z 5 5 7 5 5 7 > 6 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 168]][a, z]
Out[9]=	5 7 6 a a 3 5 7 11 2 2 4 2 -a + -- + -- - 2 a z - 9 a z - 5 a z - 2 a z + 2 a z + 5 a z + z z 6 2 8 2 10 2 3 3 5 3 7 3 9 3 > 10 a z + 5 a z - 2 a z + 6 a z + 14 a z + 6 a z + a z + 11 3 2 4 4 4 6 4 8 4 10 4 3 5 > 3 a z - a z - 7 a z - 18 a z - 7 a z + 5 a z - 3 a z - 5 5 7 5 9 5 11 5 4 6 6 6 8 6 > 12 a z - 5 a z + 3 a z - a z + 3 a z + 10 a z + 5 a z - 10 6 5 7 7 7 9 7 4 8 6 8 8 8 > 2 a z + 4 a z + 2 a z - 2 a z - a z - 3 a z - 2 a z - 5 9 7 9 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 1 3 2 4 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 3 4 4 3 3 4 1 3 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 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 L11n168

L11n167 L11n169