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L11n247 L11n249

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DrawMorseLink

PD Presentation:

X12,1,13,2 X8493 X14,6,15,5 X18,8,19,7 X20,9,21,10 X10,11,1,12 X6,14,7,13 X4,18,5,17 X15,11,16,22 X2,19,3,20 X21,17,22,16

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 4q-5/2 - 9q-3/2 + 12q-1/2 - 15q1/2 + 15q3/2 - 14q5/2 + 10q7/2 - 6q9/2 + 2q11/2

A2 (sl(3)) Invariant:

q-10 - 2q-8 + 3q-6 + 5 - 2q2 + 4q4 - 2q6 + q8 + 2q10 - 2q12 + 3q14 - q16 - q18

HOMFLY-PT Polynomial:

a-5z + a-3z3 + a-3z5 - a-1z-1 - 3a-1z - 6a-1z3 - 4a-1z5 - a-1z7 + az-1 + 2az + 2az3 + az5

Kauffman Polynomial:

3a-6z2 - 3a-6z4 - 2a-5z + 7a-5z3 - 5a-5z5 - a-5z7 + a-4z2 - a-4z4 + a-4z6 - 3a-4z8 - 2a-3z3 + 7a-3z5 - 4a-3z7 - 2a-3z9 - 2a-2z2 - 3a-2z4 + 14a-2z6 - 10a-2z8 - a-1z-1 + 6a-1z - 19a-1z3 + 28a-1z5 - 11a-1z7 - 2a-1z9 + 1 - z2 + 9z6 - 7z8 - az-1 + 4az - 9az3 + 15az5 - 8az7 - a2z2 + 5a2z4 - 4a2z6 + a3z3 - a3z5

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12                   2 j = 10                 4   j = 8               6 2   j = 6             8 4     j = 4           7 6       j = 2         8 8         j = 0       6 9           j = -2     3 6             j = -4   1 6               j = -6   3                 j = -8 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 248]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 248]]
Out[4]=	PD[X[12, 1, 13, 2], X[8, 4, 9, 3], X[14, 6, 15, 5], X[18, 8, 19, 7], > X[20, 9, 21, 10], X[10, 11, 1, 12], X[6, 14, 7, 13], X[4, 18, 5, 17], > X[15, 11, 16, 22], X[2, 19, 3, 20], X[21, 17, 22, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -8, 3, -7, 4, -2, 5, -6}, > {6, -1, 7, -3, -9, 11, 8, -4, 10, -5, -11, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 4 9 12 3/2 5/2 7/2 -q + ---- - ---- + ------- - 15 Sqrt[q] + 15 q - 14 q + 10 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 > 6 q + 2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 2 3 2 4 6 8 10 12 14 16 18 5 + q - -- + -- - 2 q + 4 q - 2 q + q + 2 q - 2 q + 3 q - q - q 8 6 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 248]][a, z]
Out[8]=	3 3 5 5 7 1 a z 3 z z 6 z 3 z 4 z 5 z -(---) + - + -- - --- + 2 a z + -- - ---- + 2 a z + -- - ---- + a z - -- a z z 5 a 3 a 3 a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 248]][a, z]
Out[9]=	2 2 2 3 3 1 a 2 z 6 z 2 3 z z 2 z 2 2 7 z 2 z 1 - --- - - - --- + --- + 4 a z - z + ---- + -- - ---- - a z + ---- - ---- - a z z 5 a 6 4 2 5 3 a a a a a a 3 4 4 4 5 5 5 19 z 3 3 3 3 z z 3 z 2 4 5 z 7 z 28 z > ----- - 9 a z + a z - ---- - -- - ---- + 5 a z - ---- + ---- + ----- + a 6 4 2 5 3 a a a a a a 6 6 7 7 7 5 3 5 6 z 14 z 2 6 z 4 z 11 z > 15 a z - a z + 9 z + -- + ----- - 4 a z - -- - ---- - ----- - 4 2 5 3 a a a a a 8 8 9 9 7 8 3 z 10 z 2 z 2 z > 8 a z - 7 z - ---- - ----- - ---- - ---- 4 2 3 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 6 3 6 6 2 4 9 + 8 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 8 q t + 7 q t + 8 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 2 6 2 6 3 8 3 8 4 10 4 12 5 > 6 q t + 8 q t + 4 q t + 6 q t + 2 q t + 4 q t + 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n248

L11n247 L11n249