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L11n419 L11n421

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DrawMorseLink

PD Presentation:

X8192 X7,16,8,17 X3,10,4,11 X17,2,18,3 X18,9,19,10 X11,20,12,21 X14,6,15,5 X22,15,13,16 X6,14,1,13 X4,19,5,20 X21,12,22,7

Gauss Code:

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

Jones Polynomial:

q-7 + 2q-5 - q-4 + 2q-3 - q-2 + q-1

A2 (sl(3)) Invariant:

- q-34 + q-32 + q-26 + q-24 + 3q-22 + 3q-20 + 5q-18 + 4q-16 + 3q-14 + 2q-12 + 2q-10 + q-8 + q-6 + q-4

HOMFLY-PT Polynomial:

a4z-2 + 5a4 + 10a4z2 + 6a4z4 + a4z6 - 2a6z-2 - 8a6 - 16a6z2 - 16a6z4 - 7a6z6 - a6z8 + a8z-2 + 4a8 + 9a8z2 + 6a8z4 + a8z6 - a10 - a10z2

Kauffman Polynomial:

- a4z-2 + 6a4 - 15a4z2 + 16a4z4 - 7a4z6 + a4z8 + 2a5z-1 - 6a5z + 9a5z5 - 6a5z7 + a5z9 - 2a6z-2 + 12a6 - 33a6z2 + 41a6z4 - 20a6z6 + 3a6z8 + 2a7z-1 - 8a7z + 8a7z3 + 3a7z5 - 5a7z7 + a7z9 - a8z-2 + 8a8 - 20a8z2 + 25a8z4 - 13a8z6 + 2a8z8 - 3a9z + 8a9z3 - 6a9z5 + a9z7 + a10 - 2a10z2 - a11z

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = -1                 1 j = -3                   j = -5             2 1   j = -7         1 1 1     j = -9         2 1       j = -11     2 1 1         j = -13     2 1           j = -15 1 1 1             j = -17 1 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 420]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 420]]
Out[4]=	PD[X[8, 1, 9, 2], X[7, 16, 8, 17], X[3, 10, 4, 11], X[17, 2, 18, 3], > X[18, 9, 19, 10], X[11, 20, 12, 21], X[14, 6, 15, 5], X[22, 15, 13, 16], > X[6, 14, 1, 13], X[4, 19, 5, 20], X[21, 12, 22, 7]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -10, 7, -9}, {-2, -1, 5, 3, -6, 11}, > {9, -7, 8, 2, -4, -5, 10, 6, -11, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-7 2 -4 2 -2 1 q + -- - q + -- - q + - 5 3 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 -32 -26 -24 3 3 5 4 3 2 2 -8 -q + q + q + q + --- + --- + --- + --- + --- + --- + --- + q + 22 20 18 16 14 12 10 q q q q q q q -6 -4 > q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 420]][a, z]
Out[8]=	4 6 8 4 6 8 10 a 2 a a 4 2 6 2 8 2 5 a - 8 a + 4 a - a + -- - ---- + -- + 10 a z - 16 a z + 9 a z - 2 2 2 z z z 10 2 4 4 6 4 8 4 4 6 6 6 8 6 6 8 > a z + 6 a z - 16 a z + 6 a z + a z - 7 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 420]][a, z]
Out[9]=	4 6 8 5 7 4 6 8 10 a 2 a a 2 a 2 a 5 7 6 a + 12 a + 8 a + a - -- - ---- - -- + ---- + ---- - 6 a z - 8 a z - 2 2 2 z z z z z 9 11 4 2 6 2 8 2 10 2 7 3 > 3 a z - a z - 15 a z - 33 a z - 20 a z - 2 a z + 8 a z + 9 3 4 4 6 4 8 4 5 5 7 5 9 5 > 8 a z + 16 a z + 41 a z + 25 a z + 9 a z + 3 a z - 6 a z - 4 6 6 6 8 6 5 7 7 7 9 7 4 8 > 7 a z - 20 a z - 13 a z - 6 a z - 5 a z + a z + a z + 6 8 8 8 5 9 7 9 > 3 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-7 2 1 1 1 1 1 2 2 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 17 6 15 6 17 5 15 5 15 4 13 4 11 4 q q t q t q t q t q t q t q t 2 1 1 1 2 1 1 1 t t > ------ + ------ + ------ + ----- + ----- + ---- + ---- + -- + -- 13 3 11 3 11 2 9 2 7 2 9 7 5 q q t q t q t q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n420

L11n419 L11n421