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L11n280 L11n282

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X9,18,10,19 X22,17,11,18 X20,11,21,12 X16,21,17,22 X4,15,1,16 X19,10,20,5

Gauss Code:

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

Jones Polynomial:

q-10 - 4q-9 + 6q-8 - 8q-7 + 9q-6 - 7q-5 + 8q-4 - 3q-3 + 2q-2

A2 (sl(3)) Invariant:

- 4q-28 - 2q-26 - 2q-24 - q-22 + 5q-20 + 4q-18 + 9q-16 + 6q-14 + 5q-12 + 5q-10 + 2q-6

HOMFLY-PT Polynomial:

2a4z-2 + 6a4 + 6a4z2 + 2a4z4 - 5a6z-2 - 10a6 - 8a6z2 - 4a6z4 - a6z6 + 4a8z-2 + 4a8 + 2a8z2 + a8z4 - a10z-2

Kauffman Polynomial:

- 2a4z-2 + 6a4 - 7a4z2 + 3a4z4 + 5a5z-1 - 10a5z + 6a5z3 - a5z5 + a5z7 - 5a6z-2 + 10a6 - 8a6z2 + a6z4 + a6z6 + a6z8 + 9a7z-1 - 15a7z + 7a7z3 - 6a7z5 + 5a7z7 - 4a8z-2 + 5a8 + a8z2 - 11a8z4 + 7a8z6 + a8z8 + 5a9z-1 - 5a9z - 3a9z3 - a9z5 + 4a9z7 - a10z-2 + 2a10z2 - 8a10z4 + 6a10z6 + a11z-1 - 4a11z3 + 4a11z5 + a12z4

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -3                 2 j = -5               3 2 j = -7             5     j = -9           2 3     j = -11         7 5       j = -13       3 4         j = -15     3 5           j = -17   1 3             j = -19   3               j = -21 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, 281]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 281]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[8, 4, 9, 3], X[2, 14, 3, 13], > X[14, 7, 15, 8], X[9, 18, 10, 19], X[22, 17, 11, 18], X[20, 11, 21, 12], > X[16, 21, 17, 22], X[4, 15, 1, 16], X[19, 10, 20, 5]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -10}, {-2, -1, 5, -3, -6, 11}, > {8, 2, 4, -5, 10, -9, 7, 6, -11, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-10 4 6 8 9 7 8 3 2 q - -- + -- - -- + -- - -- + -- - -- + -- 9 8 7 6 5 4 3 2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-4 2 2 -22 5 4 9 6 5 5 2 --- - --- - --- - q + --- + --- + --- + --- + --- + --- + -- 28 26 24 20 18 16 14 12 10 6 q q q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 281]][a, z]
Out[8]=	4 6 8 10 4 6 8 2 a 5 a 4 a a 4 2 6 2 8 2 6 a - 10 a + 4 a + ---- - ---- + ---- - --- + 6 a z - 8 a z + 2 a z + 2 2 2 2 z z z z 4 4 6 4 8 4 6 6 > 2 a z - 4 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 281]][a, z]
Out[9]=	4 6 8 10 5 7 9 11 4 6 8 2 a 5 a 4 a a 5 a 9 a 5 a a 6 a + 10 a + 5 a - ---- - ---- - ---- - --- + ---- + ---- + ---- + --- - 2 2 2 2 z z z z z z z z 5 7 9 4 2 6 2 8 2 10 2 > 10 a z - 15 a z - 5 a z - 7 a z - 8 a z + a z + 2 a z + 5 3 7 3 9 3 11 3 4 4 6 4 8 4 > 6 a z + 7 a z - 3 a z - 4 a z + 3 a z + a z - 11 a z - 10 4 12 4 5 5 7 5 9 5 11 5 6 6 8 6 > 8 a z + a z - a z - 6 a z - a z + 4 a z + a z + 7 a z + 10 6 5 7 7 7 9 7 6 8 8 8 > 6 a z + a z + 5 a z + 4 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 3 1 3 3 5 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 7 17 7 17 6 15 6 15 5 13 5 q q q t q t q t q t q t q t q t 4 7 5 2 3 5 3 > ------ + ------ + ------ + ----- + ----- + ----- + ---- 13 4 11 4 11 3 9 3 9 2 7 2 5 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 L11n281

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