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L11n215 L11n217

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DrawMorseLink

PD Presentation:

X10,1,11,2 X12,3,13,4 X18,8,19,7 X14,6,15,5 X17,9,18,22 X21,17,22,16 X20,13,21,14 X6,16,7,15 X4,20,5,19 X2,9,3,10 X8,11,1,12

Gauss Code:

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

Jones Polynomial:

q-5/2 - 3q-3/2 + 4q-1/2 - 8q1/2 + 8q3/2 - 8q5/2 + 7q7/2 - 5q9/2 + 3q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-8 + q-6 + q-4 + q-2 + 5 + q2 + 2q4 - q8 + q10 - 2q12 + q14 - q18 + q20

HOMFLY-PT Polynomial:

- a-5z - a-5z3 + a-3z + 2a-3z3 + a-3z5 - a-1z-1 + a-1z + 2a-1z3 + a-1z5 + az-1 - az - az3

Kauffman Polynomial:

2a-7z3 - a-7z5 - 2a-6z2 + 7a-6z4 - 3a-6z6 + 2a-5z - 6a-5z3 + 10a-5z5 - 4a-5z7 + 2a-4z2 - 6a-4z4 + 7a-4z6 - 3a-4z8 + 2a-3z - 8a-3z3 + 6a-3z5 - a-3z7 - a-3z9 + 8a-2z2 - 20a-2z4 + 13a-2z6 - 4a-2z8 - a-1z-1 - 2a-1z + 6a-1z3 - 8a-1z5 + 3a-1z7 - a-1z9 + 1 + 6z2 - 8z4 + 3z6 - z8 - az-1 - 2az + 6az3 - 3az5 + 2a2z2 - a2z4

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 14                   1 j = 12                 2   j = 10               3 1   j = 8             4 2     j = 6           4 3       j = 4         4 4         j = 2       4 4           j = 0     2 6             j = -2   1 2               j = -4   2                 j = -6 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, 216]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 216]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 3, 13, 4], X[18, 8, 19, 7], X[14, 6, 15, 5], > X[17, 9, 18, 22], X[21, 17, 22, 16], X[20, 13, 21, 14], X[6, 16, 7, 15], > X[4, 20, 5, 19], X[2, 9, 3, 10], X[8, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -9, 4, -8, 3, -11}, > {10, -1, 11, -2, 7, -4, 8, 6, -5, -3, 9, -7, -6, 5}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 3 4 3/2 5/2 7/2 9/2 q - ---- + ------- - 8 Sqrt[q] + 8 q - 8 q + 7 q - 5 q + 3/2 Sqrt[q] q 11/2 13/2 > 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 -4 -2 2 4 8 10 12 14 18 20 5 - q + q + q + q + q + 2 q - q + q - 2 q + q - q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 216]][a, z]
Out[8]=	3 3 3 5 5 1 a z z z z 2 z 2 z 3 z z -(---) + - - -- + -- + - - a z - -- + ---- + ---- - a z + -- + -- a z z 5 3 a 5 3 a 3 a a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 216]][a, z]
Out[9]=	2 2 2 1 a 2 z 2 z 2 z 2 2 z 2 z 8 z 2 2 1 - --- - - + --- + --- - --- - 2 a z + 6 z - ---- + ---- + ---- + 2 a z + a z z 5 3 a 6 4 2 a a a a a 3 3 3 3 4 4 4 2 z 6 z 8 z 6 z 3 4 7 z 6 z 20 z 2 4 > ---- - ---- - ---- + ---- + 6 a z - 8 z + ---- - ---- - ----- - a z - 7 5 3 a 6 4 2 a a a a a a 5 5 5 5 6 6 6 7 z 10 z 6 z 8 z 5 6 3 z 7 z 13 z 4 z > -- + ----- + ---- - ---- - 3 a z + 3 z - ---- + ---- + ----- - ---- - 7 5 3 a 6 4 2 5 a a a a a a a 7 7 8 8 9 9 z 3 z 8 3 z 4 z z z > -- + ---- - z - ---- - ---- - -- - -- 3 a 4 2 3 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 2 2 2 4 4 2 6 + 4 q + ----- + ----- + ----- + - + ---- + 4 q t + 4 q t + 4 q t + 6 3 4 2 2 2 t 2 q t q t q t q t 6 2 6 3 8 3 8 4 10 4 10 5 12 5 > 4 q t + 3 q t + 4 q t + 2 q t + 3 q t + q t + 2 q t + 14 6 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n216

L11n215 L11n217