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L11n307 L11n309

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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X7,17,8,16 X9,21,10,20 X15,9,16,8 X19,5,20,10 X13,19,14,18 X17,11,18,22 X21,15,22,14 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

q-3 - 2q-2 + 4q-1 - 3 + 5q - 3q2 + 3q3 - q4 + q6 - q7

A2 (sl(3)) Invariant:

q-10 + 2q-4 + 2q-2 + 5 + 5q2 + 5q4 + 6q6 + 3q8 + 4q10 - q14 - q16 - 2q18 - q20 - q22

HOMFLY-PT Polynomial:

- a-6z-2 - 2a-6 - a-6z2 + 4a-4z-2 + 9a-4 + 6a-4z2 + a-4z4 - 5a-2z-2 - 11a-2 - 8a-2z2 - 2a-2z4 + 2z-2 + 3 - z2 - z4 + a2 + a2z2

Kauffman Polynomial:

a-7z-1 - 4a-7z + 9a-7z3 - 6a-7z5 + a-7z7 - a-6z-2 + 3a-6 - 9a-6z2 + 13a-6z4 - 7a-6z6 + a-6z8 + 5a-5z-1 - 19a-5z + 23a-5z3 - 10a-5z5 + a-5z7 - 4a-4z-2 + 15a-4 - 29a-4z2 + 33a-4z4 - 15a-4z6 + 2a-4z8 + 9a-3z-1 - 33a-3z + 37a-3z3 - 10a-3z5 - 3a-3z7 + a-3z9 - 5a-2z-2 + 20a-2 - 38a-2z2 + 43a-2z4 - 23a-2z6 + 4a-2z8 + 5a-1z-1 - 18a-1z + 26a-1z3 - 13a-1z5 - a-1z7 + a-1z9 - 2z-2 + 8 - 15z2 + 19z4 - 14z6 + 3z8 + 3az3 - 7az5 + 2az7 - a2 + 3a2z2 - 4a2z4 + a2z6

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 15                       1 j = 13                         j = 11                 1 1 1   j = 9               2 1       j = 7             2 1 1       j = 5           4 5 1         j = 3         2 1 1           j = 1       2 5 1             j = -1     2 2 1               j = -3     2                   j = -5 1 2                     j = -7 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, 308]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 308]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[7, 17, 8, 16], X[9, 21, 10, 20], > X[15, 9, 16, 8], X[19, 5, 20, 10], X[13, 19, 14, 18], X[17, 11, 18, 22], > X[21, 15, 22, 14], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -3, 5, -4, 6}, > {11, -2, -7, 9, -5, 3, -8, 7, -6, 4, -9, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-3 2 4 2 3 4 6 7 -3 + q - -- + - + 5 q - 3 q + 3 q - q + q - q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 2 2 2 4 6 8 10 14 16 18 5 + q + -- + -- + 5 q + 5 q + 6 q + 3 q + 4 q - q - q - 2 q - 4 2 q q 20 22 > q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 308]][a, z]
Out[8]=	2 2 2 2 9 11 2 2 1 4 5 2 z 6 z 8 z 3 - -- + -- - -- + a + -- - ----- + ----- - ----- - z - -- + ---- - ---- + 6 4 2 2 6 2 4 2 2 2 6 4 2 a a a z a z a z a z a a a 4 4 2 2 4 z 2 z > a z - z + -- - ---- 4 2 a a
In[9]:=	Kauffman[Link[11, NonAlternating, 308]][a, z]
Out[9]=	3 15 20 2 2 1 4 5 1 5 9 5 8 + -- + -- + -- - a - -- - ----- - ----- - ----- + ---- + ---- + ---- + --- - 6 4 2 2 6 2 4 2 2 2 7 5 3 a z a a a z a z a z a z a z a z a z 2 2 2 3 4 z 19 z 33 z 18 z 2 9 z 29 z 38 z 2 2 9 z > --- - ---- - ---- - ---- - 15 z - ---- - ----- - ----- + 3 a z + ---- + 7 5 3 a 6 4 2 7 a a a a a a a 3 3 3 4 4 4 23 z 37 z 26 z 3 4 13 z 33 z 43 z 2 4 > ----- + ----- + ----- + 3 a z + 19 z + ----- + ----- + ----- - 4 a z - 5 3 a 6 4 2 a a a a a 5 5 5 5 6 6 6 6 z 10 z 10 z 13 z 5 6 7 z 15 z 23 z > ---- - ----- - ----- - ----- - 7 a z - 14 z - ---- - ----- - ----- + 7 5 3 a 6 4 2 a a a a a a 7 7 7 7 8 8 8 9 9 2 6 z z 3 z z 7 8 z 2 z 4 z z z > a z + -- + -- - ---- - -- + 2 a z + 3 z + -- + ---- + ---- + -- + -- 7 5 3 a 6 4 2 3 a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	1 3 1 1 2 2 2 2 2 q - + 5 q + 2 q + ----- + ----- + ----- + ----- + ---- + --- + --- + q t + q 7 4 5 4 5 3 3 2 2 q t t q t q t q t q t q t 3 5 3 2 5 2 7 2 5 3 7 3 9 3 > q t + 4 q t + q t + 5 q t + 2 q t + q t + q t + 2 q t + 7 4 9 4 11 4 11 5 11 6 15 7 > q t + q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n308

L11n307 L11n309