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L11n392 L11n394

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X12,6,13,5 X8493 X13,22,14,19 X9,20,10,21 X19,10,20,11 X21,14,22,15 X18,12,5,11 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-7 + 3q-6 - 4q-5 + 7q-4 - 5q-3 + 6q-2 - 4q-1 + 3 - q - q2 + q3

A2 (sl(3)) Invariant:

- q-22 + 2q-18 + 2q-16 + 6q-14 + 5q-12 + 6q-10 + 5q-8 + q-6 + 2q-4 - 2q-2 - q4 + q6 + q10

HOMFLY-PT Polynomial:

a-2 + a-2z2 - 1 - z2 + a2z-2 - 4a2z2 - 4a2z4 - a2z6 - 2a4z-2 + a4 + 5a4z2 + 2a4z4 + a6z-2 - a6 - a6z2

Kauffman Polynomial:

- a-2 + 4a-2z2 - 5a-2z4 + a-2z6 - 4a-1z + 10a-1z3 - 7a-1z5 + a-1z7 + 1 + 2z2 - z6 - 13az + 30az3 - 19az5 + 3az7 + a2z-2 + 5a2 - 23a2z2 + 41a2z4 - 27a2z6 + 5a2z8 - 2a3z-1 - 11a3z + 30a3z3 - 15a3z5 - 4a3z7 + 2a3z9 + 2a4z-2 + 4a4 - 31a4z2 + 55a4z4 - 39a4z6 + 8a4z8 - 2a5z-1 - 3a5z + 14a5z3 - 7a5z5 - 5a5z7 + 2a5z9 + a6z-2 - 10a6z2 + 19a6z4 - 14a6z6 + 3a6z8 - a7z + 4a7z3 - 4a7z5 + a7z7

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 = 7                       1 j = 5                         j = 3                 2 1 1   j = 1               3 1       j = -1             4 4 1       j = -3           3 3 2         j = -5         3 4             j = -7       4 3 1             j = -9     2 5                 j = -11   1 2                   j = -13   2                     j = -15 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, 393]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 393]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[12, 6, 13, 5], > 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[18, 12, 5, 11], 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]=	-7 3 4 7 5 6 4 2 3 3 - q + -- - -- + -- - -- + -- - - - q - q + q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 2 6 5 6 5 -6 2 2 4 6 10 -q + --- + --- + --- + --- + --- + -- + q + -- - -- - q + q + q 18 16 14 12 10 8 4 2 q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 393]][a, z]
Out[8]=	2 4 6 2 -2 4 6 a 2 a a 2 z 2 2 4 2 6 2 -1 + a + a - a + -- - ---- + -- - z + -- - 4 a z + 5 a z - a z - 2 2 2 2 z z z a 2 4 4 4 2 6 > 4 a z + 2 a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 393]][a, z]
Out[9]=	2 4 6 3 5 -2 2 4 a 2 a a 2 a 2 a 4 z 3 1 - a + 5 a + 4 a + -- + ---- + -- - ---- - ---- - --- - 13 a z - 11 a z - 2 2 2 z z a z z z 2 3 5 7 2 4 z 2 2 4 2 6 2 10 z > 3 a z - a z + 2 z + ---- - 23 a z - 31 a z - 10 a z + ----- + 2 a a 4 3 3 3 5 3 7 3 5 z 2 4 4 4 > 30 a z + 30 a z + 14 a z + 4 a z - ---- + 41 a z + 55 a z + 2 a 5 6 6 4 7 z 5 3 5 5 5 7 5 6 z > 19 a z - ---- - 19 a z - 15 a z - 7 a z - 4 a z - z + -- - a 2 a 7 2 6 4 6 6 6 z 7 3 7 5 7 7 7 > 27 a z - 39 a z - 14 a z + -- + 3 a z - 4 a z - 5 a z + a z + a 2 8 4 8 6 8 3 9 5 9 > 5 a z + 8 a z + 3 a z + 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 1 2 1 2 2 5 4 -- + - + 3 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 q q t q t q t q t q t q t q t 3 3 1 4 3 3 4 t 3 > ----- + ----- + ----- + ----- + ----- + ---- + --- + - + q t + 2 q t + 7 3 5 3 7 2 5 2 3 2 3 q t q q t q t q t q t q t q t 3 2 3 3 7 4 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n393

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