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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X17,1,18,4 X9,21,10,20 X3849 X21,18,22,19 X11,14,12,15 X5,13,6,12 X13,5,14,22 X19,11,20,10 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

q-9/2 - 3q-7/2 + 4q-5/2 - 7q-3/2 + 8q-1/2 - 8q1/2 + 7q3/2 - 6q5/2 + 3q7/2 - q9/2

A2 (sl(3)) Invariant:

- q-14 + q-12 + 4q-6 + q-4 + 2q-2 + 1 - q2 + q4 - 2q6 + q8 - q12 + 2q14 + q16

HOMFLY-PT Polynomial:

- a-5z-1 + 3a-3z-1 + 4a-3z - 4a-1z-1 - 8a-1z - 4a-1z3 + 2az-1 + 5az + 3az3 + az5 - a3z - a3z3

Kauffman Polynomial:

- a-5z-1 + 2a-5z - a-5z3 - a-4 + 3a-4z2 - 3a-4z4 - 3a-3z-1 + 14a-3z - 17a-3z3 + 7a-3z5 - 2a-3z7 - 3a-2 + 15a-2z2 - 23a-2z4 + 12a-2z6 - 3a-2z8 - 4a-1z-1 + 24a-1z - 40a-1z3 + 22a-1z5 - 3a-1z7 - a-1z9 - 2 + 15z2 - 28z4 + 22z6 - 6z8 - 2az-1 + 15az - 34az3 + 26az5 - 4az7 - az9 - a2 + 2a2z2 - 5a2z4 + 9a2z6 - 3a2z8 + 3a3z - 10a3z3 + 11a3z5 - 3a3z7 - a4z2 + 3a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 10                   1 j = 8                 2   j = 6               4 1   j = 4             3 2     j = 2           5 4       j = 0         5 5         j = -2       2 3           j = -4     2 5             j = -6   1 2               j = -8   2                 j = -10 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, 23]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 23]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[17, 1, 18, 4], X[9, 21, 10, 20], > X[3, 8, 4, 9], X[21, 18, 22, 19], X[11, 14, 12, 15], X[5, 13, 6, 12], > X[13, 5, 14, 22], X[19, 11, 20, 10], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, -5, 3}, {-8, -1, 2, 5, -4, 10, -7, 8, -9, 7, 11, -2, -3, 6, > -10, 4, -6, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 3 4 7 8 3/2 5/2 7/2 q - ---- + ---- - ---- + ------- - 8 Sqrt[q] + 7 q - 6 q + 3 q - 7/2 5/2 3/2 Sqrt[q] q q q 9/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 4 -4 2 2 4 6 8 12 14 16 1 - q + q + -- + q + -- - q + q - 2 q + q - q + 2 q + q 6 2 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 23]][a, z]
Out[8]=	3 1 3 4 2 a 4 z 8 z 3 4 z 3 3 3 -(----) + ---- - --- + --- + --- - --- + 5 a z - a z - ---- + 3 a z - a z + 5 3 a z z 3 a a a z a z a 5 > a z
In[9]:=	Kauffman[Link[11, NonAlternating, 23]][a, z]
Out[9]=	-4 3 2 1 3 4 2 a 2 z 14 z 24 z -2 - a - -- - a - ---- - ---- - --- - --- + --- + ---- + ---- + 15 a z + 2 5 3 a z z 5 3 a a a z a z a a 2 2 3 3 3 3 2 3 z 15 z 2 2 4 2 z 17 z 40 z > 3 a z + 15 z + ---- + ----- + 2 a z - a z - -- - ----- - ----- - 4 2 5 3 a a a a a 4 4 5 3 3 3 4 3 z 23 z 2 4 4 4 7 z > 34 a z - 10 a z - 28 z - ---- - ----- - 5 a z + 3 a z + ---- + 4 2 3 a a a 5 6 7 22 z 5 3 5 6 12 z 2 6 4 6 2 z > ----- + 26 a z + 11 a z + 22 z + ----- + 9 a z - a z - ---- - a 2 3 a a 7 8 9 3 z 7 3 7 8 3 z 2 8 z 9 > ---- - 4 a z - 3 a z - 6 z - ---- - 3 a z - -- - a z a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 2 2 5 2 5 3 5 + 5 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + ---- + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t q t q t 2 4 4 2 6 2 6 3 8 3 10 4 > 4 q t + 3 q t + 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n23

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