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L11n194 L11n196

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DrawMorseLink

PD Presentation:

X8192 X2,11,3,12 X12,3,13,4 X16,5,17,6 X6718 X4,15,5,16 X20,14,21,13 X9,18,10,19 X19,10,20,11 X22,18,7,17 X14,22,15,21

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 3q-17/2 - 6q-15/2 + 8q-13/2 - 10q-11/2 + 10q-9/2 - 9q-7/2 + 6q-5/2 - 4q-3/2 + q-1/2

A2 (sl(3)) Invariant:

q-28 - q-26 + 2q-24 + 3q-18 - q-16 + 3q-14 - q-12 + q-10 + 2q-8 - q-6 + 2q-4 - q-2

HOMFLY-PT Polynomial:

2a3z3 + a3z5 - a5z-1 - 6a5z - 9a5z3 - 5a5z5 - a5z7 + a7z-1 + 3a7z + 3a7z3 + a7z5

Kauffman Polynomial:

a2z2 - a2z4 + 5a3z3 - 4a3z5 + 2a4z2 - 2a4z4 - a4z8 + a5z-1 - 6a5z + 10a5z3 - 7a5z5 + a5z7 - a5z9 - a6 + 3a6z2 - 6a6z4 + 6a6z6 - 4a6z8 + a7z-1 - 4a7z + a7z3 + 5a7z5 - 3a7z7 - a7z9 + a8z4 + 3a8z6 - 3a8z8 + a9z - 2a9z3 + 7a9z5 - 4a9z7 - 2a10z2 + 6a10z4 - 3a10z6 - a11z + 2a11z3 - 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                 3   j = -4               4 2   j = -6             5 2     j = -8           5 4       j = -10         5 5         j = -12       3 5           j = -14     3 5             j = -16   1 4               j = -18   2                 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, 195]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 195]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[16, 5, 17, 6], > X[6, 7, 1, 8], X[4, 15, 5, 16], X[20, 14, 21, 13], X[9, 18, 10, 19], > X[19, 10, 20, 11], X[22, 18, 7, 17], X[14, 22, 15, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -6, 4, -5}, > {5, -1, -8, 9, 2, -3, 7, -11, 6, -4, 10, 8, -9, -7, 11, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 3 6 8 10 10 9 6 4 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 -26 2 3 -16 3 -12 -10 2 -6 2 -2 q - q + --- + --- - q + --- - q + q + -- - q + -- - q 24 18 14 8 4 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 195]][a, z]
Out[8]=	5 7 a a 5 7 3 3 5 3 7 3 3 5 5 5 -(--) + -- - 6 a z + 3 a z + 2 a z - 9 a z + 3 a z + a z - 5 a z + z z 7 5 5 7 > a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 195]][a, z]
Out[9]=	5 7 6 a a 5 7 9 11 2 2 4 2 6 2 -a + -- + -- - 6 a z - 4 a z + a z - a z + a z + 2 a z + 3 a z - z z 10 2 3 3 5 3 7 3 9 3 11 3 2 4 > 2 a z + 5 a z + 10 a z + a z - 2 a z + 2 a z - a z - 4 4 6 4 8 4 10 4 3 5 5 5 7 5 > 2 a z - 6 a z + a z + 6 a z - 4 a z - 7 a z + 5 a z + 9 5 11 5 6 6 8 6 10 6 5 7 7 7 > 7 a z - a z + 6 a z + 3 a z - 3 a z + a z - 3 a z - 9 7 4 8 6 8 8 8 5 9 7 9 > 4 a z - a z - 4 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 1 2 1 4 3 5 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 5 5 5 5 4 5 2 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 L11n195

L11n194 L11n196