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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X16,7,17,8 X8,15,5,16 X11,19,12,18 X17,9,18,22 X13,21,14,20 X19,13,20,12 X21,15,22,14 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-5 - q-4 + 4q-3 - 2q-2 + 5q-1 - 4 + 4q - 3q2 + 2q3 - 2q4

A2 (sl(3)) Invariant:

q-16 + 3q-14 + 4q-12 + 7q-10 + 8q-8 + 9q-6 + 7q-4 + 3q-2 + 1 - 3q2 - 3q4 - 3q6 - 3q8 - 2q10 - 2q12

HOMFLY-PT Polynomial:

- 2a-2z-2 - 7a-2 - 8a-2z2 - 2a-2z4 + 7z-2 + 21 + 24z2 + 12z4 + 2z6 - 8a2z-2 - 18a2 - 14a2z2 - 3a2z4 + 3a4z-2 + 4a4 + a4z2

Kauffman Polynomial:

- a-5z-1 + 3a-5z + a-4 + a-4z4 - a-3z-1 + 3a-3z - 4a-3z3 + 2a-3z5 - 2a-2z-2 + 7a-2 - 8a-2z2 - 2a-2z4 + 2a-2z6 + 7a-1z-1 - 21a-1z + 24a-1z3 - 17a-1z5 + 4a-1z7 - 7z-2 + 22 - 37z2 + 38z4 - 21z6 + 4z8 + 15az-1 - 45az + 50az3 - 21az5 + az9 - 8a2z-2 + 28a2 - 51a2z2 + 59a2z4 - 30a2z6 + 5a2z8 + 8a3z-1 - 24a3z + 22a3z3 - 2a3z5 - 4a3z7 + a3z9 - 3a4z-2 + 13a4 - 22a4z2 + 18a4z4 - 7a4z6 + a4z8

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 9                   2 j = 7                 1 1 j = 5               3 2   j = 3             1 1 1   j = 1           4 4       j = -1         1 1 1       j = -3       1 4           j = -5     3 1             j = -7   1 4               j = -9                     j = -11 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 270]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 270]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 7, 17, 8], X[8, 15, 5, 16], > X[11, 19, 12, 18], X[17, 9, 18, 22], X[13, 21, 14, 20], X[19, 13, 20, 12], > X[21, 15, 22, 14], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -4}, > {11, -2, -5, 8, -7, 9, 4, -3, -6, 5, -8, 7, -9, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-5 -4 4 2 5 2 3 4 -4 + q - q + -- - -- + - + 4 q - 3 q + 2 q - 2 q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 3 4 7 8 9 7 3 2 4 6 8 1 + q + --- + --- + --- + -- + -- + -- + -- - 3 q - 3 q - 3 q - 3 q - 14 12 10 8 6 4 2 q q q q q q q 10 12 > 2 q - 2 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 270]][a, z]
Out[8]=	2 4 2 7 2 4 7 2 8 a 3 a 2 8 z 2 2 21 - -- - 18 a + 4 a + -- - ----- - ---- + ---- + 24 z - ---- - 14 a z + 2 2 2 2 2 2 2 a z a z z z a 4 4 2 4 2 z 2 4 6 > a z + 12 z - ---- - 3 a z + 2 z 2 a
In[9]:=	Kauffman[Link[11, NonAlternating, 270]][a, z]
Out[9]=	2 4 -4 7 2 4 7 2 8 a 3 a 1 1 7 22 + a + -- + 28 a + 13 a - -- - ----- - ---- - ---- - ---- - ---- + --- + 2 2 2 2 2 2 5 3 a z a z a z z z a z a z 3 2 15 a 8 a 3 z 3 z 21 z 3 2 8 z > ---- + ---- + --- + --- - ---- - 45 a z - 24 a z - 37 z - ---- - z z 5 3 a 2 a a a 3 3 4 2 2 4 2 4 z 24 z 3 3 3 4 z > 51 a z - 22 a z - ---- + ----- + 50 a z + 22 a z + 38 z + -- - 3 a 4 a a 4 5 5 2 z 2 4 4 4 2 z 17 z 5 3 5 6 > ---- + 59 a z + 18 a z + ---- - ----- - 21 a z - 2 a z - 21 z + 2 3 a a a 6 7 2 z 2 6 4 6 4 z 3 7 8 2 8 4 8 > ---- - 30 a z - 7 a z + ---- - 4 a z + 4 z + 5 a z + a z + 2 a a 9 3 9 > a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	1 3 1 1 4 3 1 1 4 1 - + 4 q + q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 6 7 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t q t 1 4 q 3 5 3 2 5 2 7 2 7 3 9 3 > --- + --- + q t + 3 q t + q t + 2 q t + q t + q t + 2 q t q t t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n270

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