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L11n394 L11n396

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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X15,1,16,4 X9,22,10,19 X3849 X21,17,22,16 X11,5,12,18 X5,21,6,20 X17,11,18,10 X19,12,20,13 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

- q-4 + 3q-3 - 4q-2 + 7q-1 - 7 + 9q - 6q2 + 6q3 - 4q4 + q5

A2 (sl(3)) Invariant:

- q-12 + q-10 + 2q-6 + 5q-4 + 3q-2 + 6 + 4q2 + 4q4 + 4q6 + q10 - 2q12 - q14 + q16

HOMFLY-PT Polynomial:

a-4z2 + a-2z-2 - 4a-2z2 - 2a-2z4 - 2z-2 + 5z2 + 4z4 + z6 + a2z-2 - 2a2z2 - a2z4

Kauffman Polynomial:

a-6z2 - a-5z + 4a-5z3 - a-4z2 - a-4z4 + 2a-4z6 - 4a-3z + 17a-3z3 - 16a-3z5 + 5a-3z7 + a-2z-2 - 13a-2z2 + 22a-2z4 - 18a-2z6 + 5a-2z8 - 2a-1z-1 - 6a-1z + 22a-1z3 - 17a-1z5 - a-1z7 + 2a-1z9 + 2z-2 + 1 - 19z2 + 42z4 - 34z6 + 8z8 - 2az-1 - 4az + 13az3 - 5az5 - 5az7 + 2az9 + a2z-2 - 8a2z2 + 19a2z4 - 14a2z6 + 3a2z8 - a3z + 4a3z3 - 4a3z5 + a3z7

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 11                   1 j = 9                 3   j = 7               4 2   j = 5             3 3     j = 3           6 4 1     j = 1         5 7         j = -1       2 3 1         j = -3     2 5             j = -5   1 2               j = -7   2                 j = -9 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, 395]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 395]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[15, 1, 16, 4], X[9, 22, 10, 19], > X[3, 8, 4, 9], X[21, 17, 22, 16], X[11, 5, 12, 18], X[5, 21, 6, 20], > X[17, 11, 18, 10], X[19, 12, 20, 13], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, -5, 3}, {-10, 8, -6, 4}, > {-8, -1, 2, 5, -4, 9, -7, 10, 11, -2, -3, 6, -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-4 3 4 7 2 3 4 5 -7 - q + -- - -- + - + 9 q - 6 q + 6 q - 4 q + q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 2 5 3 2 4 6 10 12 14 16 6 - q + q + -- + -- + -- + 4 q + 4 q + 4 q + q - 2 q - q + q 6 4 2 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 395]][a, z]
Out[8]=	2 2 2 4 -2 1 a 2 z 4 z 2 2 4 2 z 2 4 6 -- + ----- + -- + 5 z + -- - ---- - 2 a z + 4 z - ---- - a z + z 2 2 2 2 4 2 2 z a z z a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 395]][a, z]
Out[9]=	2 2 2 1 a 2 2 a z 4 z 6 z 3 2 z 1 + -- + ----- + -- - --- - --- - -- - --- - --- - 4 a z - a z - 19 z + -- - 2 2 2 2 a z z 5 3 a 6 z a z z a a a 2 2 3 3 3 z 13 z 2 2 4 z 17 z 22 z 3 3 3 4 > -- - ----- - 8 a z + ---- + ----- + ----- + 13 a z + 4 a z + 42 z - 4 2 5 3 a a a a a 4 4 5 5 6 z 22 z 2 4 16 z 17 z 5 3 5 6 2 z > -- + ----- + 19 a z - ----- - ----- - 5 a z - 4 a z - 34 z + ---- - 4 2 3 a 4 a a a a 6 7 7 8 18 z 2 6 5 z z 7 3 7 8 5 z 2 8 > ----- - 14 a z + ---- - -- - 5 a z + a z + 8 z + ---- + 3 a z + 2 3 a 2 a a a 9 2 z 9 > ---- + 2 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	1 3 1 2 1 2 2 5 2 3 - + 7 q + 6 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t q t 5 q 3 5 3 2 5 2 7 2 7 3 9 3 > --- + 4 q t + 3 q t + q t + 3 q t + 4 q t + 2 q t + 3 q t + t 11 4 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n395

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