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L11n319 L11n321

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,14,6,15 X3849 X15,2,16,3 X16,7,17,8 X9,18,10,19 X4,17,1,18 X19,13,20,22 X13,10,14,11 X21,5,22,12 X11,21,12,20

Gauss Code:

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

Jones Polynomial:

q-8 - q-7 + 2q-6 - 2q-5 + 4q-4 - 2q-3 + 3q-2 - 2q-1 + 2 - q

A2 (sl(3)) Invariant:

q-28 + 2q-24 + 2q-22 + 2q-20 + 3q-18 + 3q-16 + 5q-14 + 3q-12 + 4q-10 + 2q-8 + q-6 - q2

HOMFLY-PT Polynomial:

- 2a2 - 6a2z2 - 5a2z4 - a2z6 + a4z-2 + 9a4 + 19a4z2 + 17a4z4 + 7a4z6 + a4z8 - 2a6z-2 - 9a6 - 11a6z2 - 6a6z4 - a6z6 + a8z-2 + 2a8 + a8z2

Kauffman Polynomial:

- 2az + 6az3 - 5az5 + az7 + 3a2 - 10a2z2 + 17a2z4 - 11a2z6 + 2a2z8 - 7a3z + 15a3z3 - 5a3z5 - 3a3z7 + a3z9 - a4z-2 + 11a4 - 32a4z2 + 41a4z4 - 23a4z6 + 4a4z8 + 2a5z-1 - 12a5z + 17a5z3 - 5a5z5 - 3a5z7 + a5z9 - 2a6z-2 + 11a6 - 25a6z2 + 25a6z4 - 12a6z6 + 2a6z8 + 2a7z-1 - 8a7z + 9a7z3 - 5a7z5 + a7z7 - a8z-2 + 3a8 - 2a8z2 + a8z4 - a9z + a9z3 - a10 + a10z2

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 3                   1 j = 1                 1   j = -1               1 1   j = -3           1 3 1     j = -5         1 2 2       j = -7         4 2         j = -9     2 2 2           j = -11   1 4 3             j = -13   1 2               j = -15 1 1                 j = -17 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, 320]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 320]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 14, 6, 15], X[3, 8, 4, 9], X[15, 2, 16, 3], > X[16, 7, 17, 8], X[9, 18, 10, 19], X[4, 17, 1, 18], X[19, 13, 20, 22], > X[13, 10, 14, 11], X[21, 5, 22, 12], X[11, 21, 12, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -7}, {-2, -1, 5, 3, -6, 9, -11, 10}, > {-9, 2, -4, -5, 7, 6, -8, 11, -10, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-8 -7 2 2 4 2 3 2 2 + q - q + -- - -- + -- - -- + -- - - - q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 2 2 2 3 3 5 3 4 2 -6 2 q + --- + --- + --- + --- + --- + --- + --- + --- + -- + q - q 24 22 20 18 16 14 12 10 8 q q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 320]][a, z]
Out[8]=	4 6 8 2 4 6 8 a 2 a a 2 2 4 2 6 2 -2 a + 9 a - 9 a + 2 a + -- - ---- + -- - 6 a z + 19 a z - 11 a z + 2 2 2 z z z 8 2 2 4 4 4 6 4 2 6 4 6 6 6 4 8 > a z - 5 a z + 17 a z - 6 a z - a z + 7 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 320]][a, z]
Out[9]=	4 6 8 5 7 2 4 6 8 10 a 2 a a 2 a 2 a 3 a + 11 a + 11 a + 3 a - a - -- - ---- - -- + ---- + ---- - 2 a z - 2 2 2 z z z z z 3 5 7 9 2 2 4 2 6 2 > 7 a z - 12 a z - 8 a z - a z - 10 a z - 32 a z - 25 a z - 8 2 10 2 3 3 3 5 3 7 3 9 3 > 2 a z + a z + 6 a z + 15 a z + 17 a z + 9 a z + a z + 2 4 4 4 6 4 8 4 5 3 5 5 5 > 17 a z + 41 a z + 25 a z + a z - 5 a z - 5 a z - 5 a z - 7 5 2 6 4 6 6 6 7 3 7 5 7 > 5 a z - 11 a z - 23 a z - 12 a z + a z - 3 a z - 3 a z + 7 7 2 8 4 8 6 8 3 9 5 9 > a z + 2 a z + 4 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 1 1 1 1 1 2 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 17 6 15 6 15 5 13 5 11 5 13 4 11 4 q q q t q t q t q t q t q t q t 2 3 2 2 4 1 2 2 1 t > ----- + ------ + ----- + ----- + ----- + ----- + ---- + ---- + ---- + -- + 9 4 11 3 9 3 9 2 7 2 5 2 7 5 3 3 q t q t q t q t q t q t q t q t q t q 2 t t 2 3 3 > - + -- + q t + q t q q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n320

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