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L11n268 L11n270

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X3,11,4,10 X7,17,8,16 X15,5,16,8 X18,11,19,12 X22,17,9,18 X20,13,21,14 X12,19,13,20 X14,21,15,22 X2536 X9,1,10,4

Gauss Code:

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

Jones Polynomial:

- q-8 + q-7 - 2q-6 + 2q-5 - 3q-4 + 3q-3 + 2q-1 + 1 + q2

A2 (sl(3)) Invariant:

- q-24 - q-22 - 3q-20 - 3q-18 - 3q-16 - 4q-14 - q-12 + q-10 + 6q-8 + 8q-6 + 8q-4 + 8q-2 + 5 + 4q2 + 2q4 + q6

HOMFLY-PT Polynomial:

3z-2 + 7 + 5z2 + z4 - 8a2z-2 - 20a2 - 20a2z2 - 8a2z4 - a2z6 + 7a4z-2 + 18a4 + 18a4z2 + 7a4z4 + a4z6 - 2a6z-2 - 5a6 - 4a6z2 - a6z4

Kauffman Polynomial:

- 3z-2 + 13 - 23z2 + 21z4 - 8z6 + z8 + 8az-1 - 24az + 24az3 - 9az5 + az7 - 8a2z-2 + 28a2 - 46a2z2 + 33a2z4 - 10a2z6 + a2z8 + 15a3z-1 - 45a3z + 46a3z3 - 17a3z5 + 2a3z7 - 7a4z-2 + 22a4 - 32a4z2 + 24a4z4 - 8a4z6 + a4z8 + 7a5z-1 - 21a5z + 26a5z3 - 12a5z5 + 2a5z7 - 2a6z-2 + 7a6 - 9a6z2 + 9a6z4 - 5a6z6 + a6z8 - a7z-1 + 3a7z - 3a7z5 + a7z7 + a8 - 3a8z4 + a8z6 - a9z-1 + 3a9z - 4a9z3 + a9z5

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 5                       1 j = 3                       1 j = 1                   1     j = -1               3         j = -3             1 4 1       j = -5           3             j = -7         1 2 1           j = -9       2 3               j = -11       1 1               j = -13   1 2                   j = -15                         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, 269]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 269]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 11, 4, 10], X[7, 17, 8, 16], X[15, 5, 16, 8], > X[18, 11, 19, 12], X[22, 17, 9, 18], X[20, 13, 21, 14], X[12, 19, 13, 20], > X[14, 21, 15, 22], X[2, 5, 3, 6], X[9, 1, 10, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -3, 4}, > {-11, 2, 5, -8, 7, -9, -4, 3, 6, -5, 8, -7, 9, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-8 -7 2 2 3 3 2 2 1 - q + q - -- + -- - -- + -- + - + q 6 5 4 3 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 3 3 3 4 -12 -10 6 8 8 8 5 - q - q - --- - --- - --- - --- - q + q + -- + -- + -- + -- + 20 18 16 14 8 6 4 2 q q q q q q q q 2 4 6 > 4 q + 2 q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 269]][a, z]
Out[8]=	2 4 6 2 4 6 3 8 a 7 a 2 a 2 2 2 7 - 20 a + 18 a - 5 a + -- - ---- + ---- - ---- + 5 z - 20 a z + 2 2 2 2 z z z z 4 2 6 2 4 2 4 4 4 6 4 2 6 4 6 > 18 a z - 4 a z + z - 8 a z + 7 a z - a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 269]][a, z]
Out[9]=	2 4 6 3 5 2 4 6 8 3 8 a 7 a 2 a 8 a 15 a 7 a 13 + 28 a + 22 a + 7 a + a - -- - ---- - ---- - ---- + --- + ----- + ---- - 2 2 2 2 z z z z z z z 7 9 a a 3 5 7 9 2 2 2 > -- - -- - 24 a z - 45 a z - 21 a z + 3 a z + 3 a z - 23 z - 46 a z - z z 4 2 6 2 3 3 3 5 3 9 3 4 > 32 a z - 9 a z + 24 a z + 46 a z + 26 a z - 4 a z + 21 z + 2 4 4 4 6 4 8 4 5 3 5 5 5 > 33 a z + 24 a z + 9 a z - 3 a z - 9 a z - 17 a z - 12 a z - 7 5 9 5 6 2 6 4 6 6 6 8 6 7 > 3 a z + a z - 8 z - 10 a z - 8 a z - 5 a z + a z + a z + 3 7 5 7 7 7 8 2 8 4 8 6 8 > 2 a z + 2 a z + a z + z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 3 1 1 2 1 2 1 3 1 -- + - + ------ + ------ + ------ + ------ + ----- + ------ + ----- + ----- + 3 q 17 7 13 6 13 5 11 4 9 4 11 3 9 3 7 3 q q t q t q t q t q t q t q t q t 2 3 1 1 t 2 3 4 5 4 > ----- + ----- + ---- + ---- + -- + q t + q t + q t 7 2 5 2 7 3 3 q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n269

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