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DrawMorseLink

PD Presentation:

X10,1,11,2 X16,11,17,12 X5,21,6,20 X12,4,13,3 X14,8,15,7 X6,14,7,13 X17,9,18,22 X21,19,22,18 X8,9,1,10 X19,5,20,4 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

q-3/2 - 3q-1/2 + 5q1/2 - 8q3/2 + 8q5/2 - 9q7/2 + 8q9/2 - 6q11/2 + 3q13/2 - q15/2

A2 (sl(3)) Invariant:

- q-4 + q-2 - 1 + q2 + 2q4 + 4q8 - q10 + 2q12 + 2q18 - q20 + q22

HOMFLY-PT Polynomial:

- a-5z-1 - 3a-5z - 3a-5z3 - a-5z5 + a-3z-1 + 7a-3z + 9a-3z3 + 5a-3z5 + a-3z7 - 2a-1z - 3a-1z3 - a-1z5

Kauffman Polynomial:

a-9z - a-9z3 + 2a-8z2 - 3a-8z4 + a-7z - 3a-7z3 - a-7z7 + a-6z2 - 6a-6z4 + 4a-6z6 - 2a-6z8 - a-5z-1 + 6a-5z - 13a-5z3 + 10a-5z5 - 2a-5z7 - a-5z9 + a-4 - a-4z2 - 6a-4z4 + 12a-4z6 - 5a-4z8 - a-3z-1 + 9a-3z - 20a-3z3 + 20a-3z5 - 4a-3z7 - a-3z9 - 2a-2z2 + 7a-2z6 - 3a-2z8 + 3a-1z - 9a-1z3 + 10a-1z5 - 3a-1z7 - 2z2 + 3z4 - z6

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 16                   1 j = 14                 2   j = 12               4 1   j = 10             5 3     j = 8           4 3       j = 6         4 5         j = 4       4 4           j = 2     2 5             j = 0   1 3               j = -2   2                 j = -4 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, 231]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 231]]
Out[4]=	PD[X[10, 1, 11, 2], X[16, 11, 17, 12], X[5, 21, 6, 20], X[12, 4, 13, 3], > X[14, 8, 15, 7], X[6, 14, 7, 13], X[17, 9, 18, 22], X[21, 19, 22, 18], > X[8, 9, 1, 10], X[19, 5, 20, 4], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 4, 10, -3, -6, 5, -9}, > {9, -1, 2, -4, 6, -5, 11, -2, -7, 8, -10, 3, -8, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(3/2) 3 3/2 5/2 7/2 9/2 11/2 q - ------- + 5 Sqrt[q] - 8 q + 8 q - 9 q + 8 q - 6 q + Sqrt[q] 13/2 15/2 > 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-4 -2 2 4 8 10 12 18 20 22 -1 - q + q + q + 2 q + 4 q - q + 2 q + 2 q - q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 231]][a, z]
Out[8]=	3 3 3 5 5 5 7 1 1 3 z 7 z 2 z 3 z 9 z 3 z z 5 z z z -(----) + ---- - --- + --- - --- - ---- + ---- - ---- - -- + ---- - -- + -- 5 3 5 3 a 5 3 a 5 3 a 3 a z a z a a a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 231]][a, z]
Out[9]=	2 2 2 2 -4 1 1 z z 6 z 9 z 3 z 2 2 z z z 2 z a - ---- - ---- + -- + -- + --- + --- + --- - 2 z + ---- + -- - -- - ---- - 5 3 9 7 5 3 a 8 6 4 2 a z a z a a a a a a a a 3 3 3 3 3 4 4 4 5 z 3 z 13 z 20 z 9 z 4 3 z 6 z 6 z 10 z > -- - ---- - ----- - ----- - ---- + 3 z - ---- - ---- - ---- + ----- + 9 7 5 3 a 8 6 4 5 a a a a a a a a 5 5 6 6 6 7 7 7 7 8 20 z 10 z 6 4 z 12 z 7 z z 2 z 4 z 3 z 2 z > ----- + ----- - z + ---- + ----- + ---- - -- - ---- - ---- - ---- - ---- - 3 a 6 4 2 7 5 3 a 6 a a a a a a a a 8 8 9 9 5 z 3 z z z > ---- - ---- - -- - -- 4 2 5 3 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 -2 2 3 2 q 4 6 6 2 5 q + 4 q + ----- + t + ----- + - + ---- + 4 q t + 4 q t + 5 q t + 4 3 2 2 t t q t q t 8 2 8 3 10 3 10 4 12 4 12 5 14 5 > 4 q t + 3 q t + 5 q t + 3 q t + 4 q t + q t + 2 q t + 16 6 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n231

L11n230 L11n232