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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X9,22,10,19 X8493 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 + 2q-3 - 2q-2 + 3q-1 + 1 + q + q2 - 2q3 + 2q4 - 2q5 + q6

A2 (sl(3)) Invariant:

- q-12 - q-8 + 2q-6 + 4q-4 + 7q-2 + 9 + 5q2 + 4q4 - q6 - q8 - q10 - q12 + q14 + q18

HOMFLY-PT Polynomial:

2a-4 + 3a-4z2 + a-4z4 + a-2z-2 - 6a-2 - 11a-2z2 - 6a-2z4 - a-2z6 - 2z-2 + 6 + 11z2 + 6z4 + z6 + a2z-2 - 2a2 - 3a2z2 - a2z4

Kauffman Polynomial:

2a-6z2 - 4a-6z4 + a-6z6 - 4a-5z + 8a-5z3 - 9a-5z5 + 2a-5z7 + 4a-4 - 8a-4z2 + 11a-4z4 - 10a-4z6 + 2a-4z8 - 14a-3z + 26a-3z3 - 10a-3z5 - 3a-3z7 + a-3z9 + a-2z-2 + 12a-2 - 40a-2z2 + 59a-2z4 - 33a-2z6 + 5a-2z8 - 2a-1z-1 - 18a-1z + 32a-1z3 - 4a-1z5 - 9a-1z7 + 2a-1z9 + 2z-2 + 13 - 44z2 + 61z4 - 33z6 + 5z8 - 2az-1 - 10az + 20az3 - 8az5 - 3az7 + az9 + a2z-2 + 4a2 - 14a2z2 + 17a2z4 - 11a2z6 + 2a2z8 - 2a3z + 6a3z3 - 5a3z5 + a3z7

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                       1 j = 11                     1   j = 9                 1 2 1   j = 7               1 2 1     j = 5             2 3 2       j = 3           3 3 2         j = 1         2 6 2           j = -1       2 3 5             j = -3     1 2                 j = -5   1 2 1                 j = -7   1                     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, 396]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 396]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[9, 22, 10, 19], > X[8, 4, 9, 3], 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 2 2 3 2 3 4 5 6 1 - q + -- - -- + - + q + q - 2 q + 2 q - 2 q + q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -8 2 4 7 2 4 6 8 10 12 14 18 9 - q - q + -- + -- + -- + 5 q + 4 q - q - q - q - q + q + q 6 4 2 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 396]][a, z]
Out[8]=	2 2 2 2 6 2 2 1 a 2 3 z 11 z 2 2 4 6 + -- - -- - 2 a - -- + ----- + -- + 11 z + ---- - ----- - 3 a z + 6 z + 4 2 2 2 2 2 4 2 a a z a z z a a 4 4 6 z 6 z 2 4 6 z > -- - ---- - a z + z - -- 4 2 2 a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 396]][a, z]
Out[9]=	2 4 12 2 2 1 a 2 2 a 4 z 14 z 18 z 13 + -- + -- + 4 a + -- + ----- + -- - --- - --- - --- - ---- - ---- - 4 2 2 2 2 2 a z z 5 3 a a a z a z z a a 2 2 2 3 3 3 2 2 z 8 z 40 z 2 2 8 z 26 z > 10 a z - 2 a z - 44 z + ---- - ---- - ----- - 14 a z + ---- + ----- + 6 4 2 5 3 a a a a a 3 4 4 4 32 z 3 3 3 4 4 z 11 z 59 z 2 4 > ----- + 20 a z + 6 a z + 61 z - ---- + ----- + ----- + 17 a z - a 6 4 2 a a a 5 5 5 6 6 6 9 z 10 z 4 z 5 3 5 6 z 10 z 33 z > ---- - ----- - ---- - 8 a z - 5 a z - 33 z + -- - ----- - ----- - 5 3 a 6 4 2 a a a a a 7 7 7 8 8 2 6 2 z 3 z 9 z 7 3 7 8 2 z 5 z > 11 a z + ---- - ---- - ---- - 3 a z + a z + 5 z + ---- + ---- + 5 3 a 4 2 a a a a 9 9 2 8 z 2 z 9 > 2 a z + -- + ---- + a z 3 a a
In[10]:=	Kh[L][q, t]
Out[10]=	5 3 1 1 1 2 1 1 2 2 - + 6 q + 3 q + ----- + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 9 5 7 4 5 4 5 3 3 3 5 2 3 2 2 q t q t q t q t q t q t q t q t 3 2 q 3 5 3 2 5 2 7 2 5 3 > --- + --- + 2 q t + 3 q t + 2 q t + 2 q t + 3 q t + q t + 2 q t + q t t 7 3 9 3 7 4 9 4 9 5 11 5 13 6 > 2 q t + q t + q t + 2 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n396

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