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L11n390 L11n392

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X8493 X13,22,14,19 X9,20,10,21 X19,10,20,11 X21,14,22,15 X11,18,12,5 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-32 - 2q-30 - 2q-28 - 2q-26 + q-24 + 4q-20 + 5q-18 + 5q-16 + 8q-14 + 3q-12 + 5q-10 + q-8 + q-6 + q-4 - q-2 + 1

HOMFLY-PT Polynomial:

a2 + 2a2z2 + a2z4 + 2a4z-2 + 4a4 - 3a4z4 - a4z6 - 5a6z-2 - 10a6 - 4a6z2 + 4a8z-2 + 6a8 + 2a8z2 - a10z-2 - a10

Kauffman Polynomial:

- a2 + 3a2z2 - 3a2z4 + a2z6 + 4a3z3 - 9a3z5 + 3a3z7 - 2a4z-2 + 6a4 - 3a4z2 + a4z4 - 8a4z6 + 3a4z8 + 5a5z-1 - 15a5z + 19a5z3 - 14a5z5 + a5z7 + a5z9 - 5a6z-2 + 16a6 - 23a6z2 + 25a6z4 - 17a6z6 + 4a6z8 + 9a7z-1 - 30a7z + 35a7z3 - 11a7z5 - 2a7z7 + a7z9 - 4a8z-2 + 13a8 - 25a8z2 + 33a8z4 - 15a8z6 + 2a8z8 + 5a9z-1 - 20a9z + 29a9z3 - 12a9z5 + a9z7 - a10z-2 + 3a10 - 8a10z2 + 12a10z4 - 7a10z6 + a10z8 + a11z-1 - 5a11z + 9a11z3 - 6a11z5 + a11z7

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 1                       1 j = -1                     2   j = -3                   4 1   j = -5                 3 4     j = -7             1 5 2       j = -9           1 2 3         j = -11         1 6 5           j = -13       1 1 3             j = -15       1 3               j = -17   1 1 1                 j = -19                         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, 391]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 391]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[5, 12, 6, 13], > X[8, 4, 9, 3], X[13, 22, 14, 19], X[9, 20, 10, 21], X[19, 10, 20, 11], > X[21, 14, 22, 15], X[11, 18, 12, 5], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {-8, 7, -9, 6}, > {-4, -1, 2, -5, -7, 8, -10, 4, -6, 9, 11, -2, 3, 10}]
In[6]:=	Jones[L][q]
Out[6]=	-10 -9 2 5 5 7 5 6 3 1 - q + q - -- + -- - -- + -- - -- + -- - - 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 2 2 2 -24 4 5 5 8 3 5 -8 1 - q - --- - --- - --- + q + --- + --- + --- + --- + --- + --- + q + 30 28 26 20 18 16 14 12 10 q q q q q q q q q -6 -4 -2 > q + q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 391]][a, z]
Out[8]=	4 6 8 10 2 4 6 8 10 2 a 5 a 4 a a 2 2 6 2 a + 4 a - 10 a + 6 a - a + ---- - ---- + ---- - --- + 2 a z - 4 a z + 2 2 2 2 z z z z 8 2 2 4 4 4 4 6 > 2 a z + a z - 3 a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 391]][a, z]
Out[9]=	4 6 8 10 5 7 2 4 6 8 10 2 a 5 a 4 a a 5 a 9 a -a + 6 a + 16 a + 13 a + 3 a - ---- - ---- - ---- - --- + ---- + ---- + 2 2 2 2 z z z z z z 9 11 5 a a 5 7 9 11 2 2 4 2 > ---- + --- - 15 a z - 30 a z - 20 a z - 5 a z + 3 a z - 3 a z - z z 6 2 8 2 10 2 3 3 5 3 7 3 9 3 > 23 a z - 25 a z - 8 a z + 4 a z + 19 a z + 35 a z + 29 a z + 11 3 2 4 4 4 6 4 8 4 10 4 3 5 > 9 a z - 3 a z + a z + 25 a z + 33 a z + 12 a z - 9 a z - 5 5 7 5 9 5 11 5 2 6 4 6 6 6 > 14 a z - 11 a z - 12 a z - 6 a z + a z - 8 a z - 17 a z - 8 6 10 6 3 7 5 7 7 7 9 7 11 7 > 15 a z - 7 a z + 3 a z + a z - 2 a z + a z + a z + 4 8 6 8 8 8 10 8 5 9 7 9 > 3 a z + 4 a z + 2 a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 4 1 1 1 1 1 1 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 9 17 8 17 7 17 6 15 6 13 6 15 5 q q q t q t q t q t q t q t q t 1 1 3 6 1 5 2 1 > ------ + ------ + ------ + ------ + ----- + ------ + ----- + ----- + 13 5 11 5 13 4 11 4 9 4 11 3 9 3 7 3 q t q t q t q t q t q t q t q t 3 5 2 3 t 2 t 2 > ----- + ----- + ---- + ---- + -- + --- + q t 9 2 7 2 7 5 3 q q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n391

L11n390 L11n392