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L11n387 L11n389

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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X5,12,6,13 X8493 X22,11,19,12 X18,22,5,21 X20,10,21,9 X10,17,11,18 X16,19,17,20 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

- 3q-6 + 6q-5 - 9q-4 + 12q-3 - 10q-2 + 11q-1 - 7 + 5q - q2

A2 (sl(3)) Invariant:

- 2q-20 - 4q-18 - 3q-14 + q-12 + 4q-10 + 3q-8 + 9q-6 + 4q-4 + 8q-2 + 4 + q2 + 3q4 - q6

HOMFLY-PT Polynomial:

2z-2 + 3 - z4 - 5a2z-2 - 8a2 - 2a2z2 + 2a2z4 + a2z6 + 4a4z-2 + 7a4 + 2a4z2 - a4z4 - a6z-2 - 2a6

Kauffman Polynomial:

a-1z5 - 2z-2 + 5 - 8z4 + 5z6 + 5az-1 - 8az + 6az3 - 13az5 + 7az7 - 5a2z-2 + 11a2 - 3a2z2 - 16a2z4 + 5a2z6 + 3a2z8 + 9a3z-1 - 21a3z + 24a3z3 - 27a3z5 + 13a3z7 - 4a4z-2 + 10a4 - 3a4z2 - 8a4z4 + 3a4z6 + 3a4z8 + 5a5z-1 - 19a5z + 24a5z3 - 13a5z5 + 6a5z7 - a6z-2 + 3a6 + 3a6z6 + a7z-1 - 6a7z + 6a7z3

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 5                 1 j = 3               4   j = 1             3 1   j = -1           8 4     j = -3         6 7       j = -5       6 4         j = -7     3 6           j = -9   3 6             j = -11   3               j = -13 3

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, 388]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 388]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[5, 12, 6, 13], > X[8, 4, 9, 3], X[22, 11, 19, 12], X[18, 22, 5, 21], X[20, 10, 21, 9], > X[10, 17, 11, 18], X[16, 19, 17, 20], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {10, -8, 7, -6}, > {-4, -1, 2, -5, 8, -9, 6, 4, 11, -2, 3, -10, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	3 6 9 12 10 11 2 -7 - -- + -- - -- + -- - -- + -- + 5 q - q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 4 3 -12 4 3 9 4 8 2 4 6 4 - --- - --- - --- + q + --- + -- + -- + -- + -- + q + 3 q - q 20 18 14 10 8 6 4 2 q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 388]][a, z]
Out[8]=	2 4 6 2 4 6 2 5 a 4 a a 2 2 4 2 4 3 - 8 a + 7 a - 2 a + -- - ---- + ---- - -- - 2 a z + 2 a z - z + 2 2 2 2 z z z z 2 4 4 4 2 6 > 2 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 388]][a, z]
Out[9]=	2 4 6 3 5 7 2 4 6 2 5 a 4 a a 5 a 9 a 5 a a 5 + 11 a + 10 a + 3 a - -- - ---- - ---- - -- + --- + ---- + ---- + -- - 2 2 2 2 z z z z z z z z 3 5 7 2 2 4 2 3 > 8 a z - 21 a z - 19 a z - 6 a z - 3 a z - 3 a z + 6 a z + 5 3 3 5 3 7 3 4 2 4 4 4 z 5 > 24 a z + 24 a z + 6 a z - 8 z - 16 a z - 8 a z + -- - 13 a z - a 3 5 5 5 6 2 6 4 6 6 6 7 > 27 a z - 13 a z + 5 z + 5 a z + 3 a z + 3 a z + 7 a z + 3 7 5 7 2 8 4 8 > 13 a z + 6 a z + 3 a z + 3 a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 8 3 3 3 6 3 6 6 4 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 3 q 13 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q q t q t q t q t q t q t q t q t 6 4 t 2 3 2 5 3 > ---- + --- + 3 q t + q t + 4 q t + q t 3 q q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n388

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