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DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X22,10,7,9 X2738 X4,15,5,16 X5,13,6,12 X11,16,12,17 X17,6,18,1 X14,20,15,19 X20,14,21,13 X18,21,19,22

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 2q-9/2 - 3q-7/2 + 3q-5/2 - 4q-3/2 + 3q-1/2 - 3q1/2 + 2q3/2 - q5/2

A2 (sl(3)) Invariant:

q-18 + q-16 + q-12 + q-8 + q-6 + q-2 + q2 + q4 + q8

HOMFLY-PT Polynomial:

- a-1z-1 - 2a-1z - a-1z3 + 2az-1 + 5az + 4az3 + az5 - 2a3z-1 - 5a3z - 2a3z3 + a5z-1 + a5z

Kauffman Polynomial:

- a-1z-1 + 4a-1z - 7a-1z3 + 5a-1z5 - a-1z7 + 5z2 - 13z4 + 10z6 - 2z8 - 2az-1 + 12az - 20az3 + 9az5 + 2az7 - az9 - a2 + 10a2z2 - 26a2z4 + 20a2z6 - 4a2z8 - 2a3z-1 + 12a3z - 20a3z3 + 9a3z5 + 2a3z7 - a3z9 + 5a4z2 - 13a4z4 + 10a4z6 - 2a4z8 - a5z-1 + 4a5z - 7a5z3 + 5a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 6                 1 j = 4               1   j = 2             2 1   j = 0           2 2     j = -2         2 2 1     j = -4       2 3         j = -6     1 2 1         j = -8   1 2             j = -10   1               j = -12 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, 127]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 127]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[22, 10, 7, 9], X[2, 7, 3, 8], > X[4, 15, 5, 16], X[5, 13, 6, 12], X[11, 16, 12, 17], X[17, 6, 18, 1], > X[14, 20, 15, 19], X[20, 14, 21, 13], X[18, 21, 19, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -5, -6, 8}, > {4, -1, 3, -2, -7, 6, 10, -9, 5, 7, -8, -11, 9, -10, 11, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 2 3 3 4 3 3/2 5/2 -q + ---- - ---- + ---- - ---- + ------- - 3 Sqrt[q] + 2 q - q 9/2 7/2 5/2 3/2 Sqrt[q] q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 -12 -8 -6 -2 2 4 8 q + q + q + q + q + q + q + q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 127]][a, z]
Out[8]=	3 5 3 1 2 a 2 a a 2 z 3 5 z 3 -(---) + --- - ---- + -- - --- + 5 a z - 5 a z + a z - -- + 4 a z - a z z z z a a 3 3 5 > 2 a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 127]][a, z]
Out[9]=	3 5 2 1 2 a 2 a a 4 z 3 5 2 -a - --- - --- - ---- - -- + --- + 12 a z + 12 a z + 4 a z + 5 z + a z z z z a 3 2 2 4 2 7 z 3 3 3 5 3 4 > 10 a z + 5 a z - ---- - 20 a z - 20 a z - 7 a z - 13 z - a 5 2 4 4 4 5 z 5 3 5 5 5 6 > 26 a z - 13 a z + ---- + 9 a z + 9 a z + 5 a z + 10 z + a 7 2 6 4 6 z 7 3 7 5 7 8 2 8 > 20 a z + 10 a z - -- + 2 a z + 2 a z - a z - 2 z - 4 a z - a 4 8 9 3 9 > 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-2 2 1 1 1 2 1 2 2 2 + q + 2 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 q t q t q t q t q t q t q t 1 3 2 2 2 2 4 6 2 > ----- + ----- + ----- + - + ---- + q t + q t + q t 6 2 4 2 2 2 t 2 q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n127

L11n126 L11n128