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L11n287 L11n289

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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X11,18,12,19 X9,21,10,20 X19,5,20,10 X4,15,1,16 X17,22,18,11 X21,16,22,17

Gauss Code:

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

Jones Polynomial:

- q-10 + q-9 - 2q-8 + 3q-7 - 2q-6 + 4q-5 - 2q-4 + 3q-3 - q-2 + q-1

A2 (sl(3)) Invariant:

- q-34 - 2q-30 - 2q-28 - 2q-26 - q-24 + 2q-22 + 3q-20 + 7q-18 + 6q-16 + 6q-14 + 4q-12 + 3q-10 + 2q-8 + q-6 + q-4

HOMFLY-PT Polynomial:

2a4z-2 + 7a4 + 11a4z2 + 6a4z4 + a4z6 - 5a6z-2 - 14a6 - 20a6z2 - 17a6z4 - 7a6z6 - a6z8 + 4a8z-2 + 9a8 + 11a8z2 + 6a8z4 + a8z6 - a10z-2 - 2a10 - a10z2

Kauffman Polynomial:

- 2a4z-2 + 9a4 - 18a4z2 + 17a4z4 - 7a4z6 + a4z8 + 5a5z-1 - 13a5z + 9a5z3 + 4a5z5 - 5a5z7 + a5z9 - 5a6z-2 + 21a6 - 45a6z2 + 51a6z4 - 25a6z6 + 4a6z8 + 9a7z-1 - 29a7z + 34a7z3 - 12a7z5 - 2a7z7 + a7z9 - 4a8z-2 + 18a8 - 32a8z2 + 32a8z4 - 17a8z6 + 3a8z8 + 5a9z-1 - 21a9z + 27a9z3 - 16a9z5 + 3a9z7 - a10z-2 + 5a10 - 4a10z2 - 2a10z4 + a10z6 + a11z-1 - 4a11z + 2a11z3 + a12z2 + a13z

Khovanov Homology:

trqj r = -7 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               3 1   j = -7           1 1 1     j = -9           4 2       j = -11       2 2 2         j = -13       3 2           j = -15   1 2 2             j = -17   1 2               j = -19 1 1                 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, 288]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 288]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[3, 8, 4, 9], X[13, 2, 14, 3], > X[14, 7, 15, 8], X[11, 18, 12, 19], X[9, 21, 10, 20], X[19, 5, 20, 10], > X[4, 15, 1, 16], X[17, 22, 18, 11], X[21, 16, 22, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -9}, {-2, -1, 5, 3, -7, 8}, > {-6, 2, -4, -5, 9, 11, -10, 6, -8, 7, -11, 10}]
In[6]:=	Jones[L][q]
Out[6]=	-10 -9 2 3 2 4 2 3 -2 1 -q + q - -- + -- - -- + -- - -- + -- - q + - 8 7 6 5 4 3 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 2 2 2 -24 2 3 7 6 6 4 3 2 -q - --- - --- - --- - q + --- + --- + --- + --- + --- + --- + --- + -- + 30 28 26 22 20 18 16 14 12 10 8 q q q q q q q q q q q -6 -4 > q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 288]][a, z]
Out[8]=	4 6 8 10 4 6 8 10 2 a 5 a 4 a a 4 2 6 2 7 a - 14 a + 9 a - 2 a + ---- - ---- + ---- - --- + 11 a z - 20 a z + 2 2 2 2 z z z z 8 2 10 2 4 4 6 4 8 4 4 6 6 6 > 11 a z - a z + 6 a z - 17 a z + 6 a z + a z - 7 a z + 8 6 6 8 > a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 288]][a, z]
Out[9]=	4 6 8 10 5 7 9 4 6 8 10 2 a 5 a 4 a a 5 a 9 a 5 a 9 a + 21 a + 18 a + 5 a - ---- - ---- - ---- - --- + ---- + ---- + ---- + 2 2 2 2 z z z z z z z 11 a 5 7 9 11 13 4 2 6 2 > --- - 13 a z - 29 a z - 21 a z - 4 a z + a z - 18 a z - 45 a z - z 8 2 10 2 12 2 5 3 7 3 9 3 11 3 > 32 a z - 4 a z + a z + 9 a z + 34 a z + 27 a z + 2 a z + 4 4 6 4 8 4 10 4 5 5 7 5 9 5 > 17 a z + 51 a z + 32 a z - 2 a z + 4 a z - 12 a z - 16 a z - 4 6 6 6 8 6 10 6 5 7 7 7 9 7 > 7 a z - 25 a z - 17 a z + a z - 5 a z - 2 a z + 3 a z + 4 8 6 8 8 8 5 9 7 9 > a z + 4 a z + 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-7 3 1 1 1 1 1 2 2 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 21 7 19 7 19 6 17 6 15 6 17 5 15 5 q q t q t q t q t q t q t q t 2 3 2 2 2 2 4 1 > ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 15 4 13 4 11 4 13 3 11 3 11 2 9 2 7 2 q t q t q t q t q t q t q t q t 2 2 1 t t > ---- + ---- + -- + -- 9 7 5 q q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n288

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