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L11n431 L11n433

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DrawMorseLink

PD Presentation:

X8192 X14,4,15,3 X12,14,7,13 X2738 X22,10,13,9 X6,22,1,21 X15,20,16,21 X5,17,6,16 X11,19,12,18 X17,11,18,10 X19,5,20,4

Gauss Code:

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

Jones Polynomial:

1 - q + q2 + q4 + q5 + q7 - q8 + q9

A2 (sl(3)) Invariant:

1 + q2 + q8 + 2q10 + 2q12 + 4q14 + 3q16 + 4q18 + 3q20 + 3q22 + 2q24 + q28

HOMFLY-PT Polynomial:

a-8z-2 + a-8 + a-8z2 - 2a-6z-2 - 3a-6 + a-4z-2 - 5a-4z2 - 5a-4z4 - a-4z6 + 2a-2 + 4a-2z2 + a-2z4

Kauffman Polynomial:

5a-10z2 - 5a-10z4 + a-10z6 + 5a-9z3 - 5a-9z5 + a-9z7 - a-8z-2 + 7a-8 - 17a-8z2 + 16a-8z4 - 7a-8z6 + a-8z8 + 2a-7z-1 - 9a-7z + 13a-7z3 - 7a-7z5 + a-7z7 - 2a-6z-2 + 11a-6 - 25a-6z2 + 21a-6z4 - 8a-6z6 + a-6z8 + 2a-5z-1 - 9a-5z + 13a-5z3 - 7a-5z5 + a-5z7 - a-4z-2 + 3a-4 + 3a-4z2 - 5a-4z4 + a-4z6 + 5a-3z3 - 5a-3z5 + a-3z7 - 2a-2 + 6a-2z2 - 5a-2z4 + a-2z6

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 19                     1 j = 17                   1 1 j = 15               1 1     j = 13             1 1 1     j = 11           1 3 1       j = 9         2 1 1         j = 7       1 4 2           j = 5     1 1 1             j = 3   1 2 1               j = 1                       j = -1 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, 432]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 432]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 4, 15, 3], X[12, 14, 7, 13], X[2, 7, 3, 8], > X[22, 10, 13, 9], X[6, 22, 1, 21], X[15, 20, 16, 21], X[5, 17, 6, 16], > X[11, 19, 12, 18], X[17, 11, 18, 10], X[19, 5, 20, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, 11, -8, -6}, {4, -1, 5, 10, -9, -3}, > {3, -2, -7, 8, -10, 9, -11, 7, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	2 4 5 7 8 9 1 - q + q + q + q + q - q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 8 10 12 14 16 18 20 22 24 1 + q + q + 2 q + 2 q + 4 q + 3 q + 4 q + 3 q + 3 q + 2 q + 28 > q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 432]][a, z]
Out[8]=	2 2 2 4 4 6 -8 3 2 1 2 1 z 5 z 4 z 5 z z z a - -- + -- + ----- - ----- + ----- + -- - ---- + ---- - ---- + -- - -- 6 2 8 2 6 2 4 2 8 4 2 4 2 4 a a a z a z a z a a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 432]][a, z]
Out[9]=	2 7 11 3 2 1 2 1 2 2 9 z 9 z 5 z -- + -- + -- - -- - ----- - ----- - ----- + ---- + ---- - --- - --- + ---- - 8 6 4 2 8 2 6 2 4 2 7 5 7 5 10 a a a a a z a z a z a z a z a a a 2 2 2 2 3 3 3 3 4 4 17 z 25 z 3 z 6 z 5 z 13 z 13 z 5 z 5 z 16 z > ----- - ----- + ---- + ---- + ---- + ----- + ----- + ---- - ---- + ----- + 8 6 4 2 9 7 5 3 10 8 a a a a a a a a a a 4 4 4 5 5 5 5 6 6 6 6 21 z 5 z 5 z 5 z 7 z 7 z 5 z z 7 z 8 z z > ----- - ---- - ---- - ---- - ---- - ---- - ---- + --- - ---- - ---- + -- + 6 4 2 9 7 5 3 10 8 6 4 a a a a a a a a a a a 6 7 7 7 7 8 8 z z z z z z z > -- + -- + -- + -- + -- + -- + -- 2 9 7 5 3 8 6 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 5 1 q 3 5 7 5 2 7 2 9 2 2 q + q + ---- + -- + q t + q t + q t + q t + 4 q t + 2 q t + 2 t q t 7 3 9 3 11 3 9 4 11 4 13 4 11 5 13 5 > 2 q t + q t + q t + q t + 3 q t + q t + q t + q t + 15 5 13 6 15 6 17 7 17 8 19 8 > q t + q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n432

L11n431 L11n433