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DrawMorseLink

PD Presentation:

X8192 X16,8,17,7 X14,6,15,5 X3,10,4,11 X4,14,5,13 X17,2,18,3 X9,19,10,18 X12,21,7,22 X22,11,13,12 X20,16,21,15 X6,19,1,20

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-16 + q-14 + 2q-12 + 2q-10 + 2q-8 + 3q-6 + 3q-4 + 6q-2 + 5 + 4q2 + q4 - q6 - q8 - q10

HOMFLY-PT Polynomial:

- 2a-2 - a-2z2 + z-2 + 7 + 5z2 + z4 - 2a2z-2 - 7a2 - 5a2z2 - a2z4 + a4z-2 + 2a4 + a4z2

Kauffman Polynomial:

- 2a-3z + a-3z3 + 4a-2 - 4a-2z2 + a-2z4 - 6a-1z + 5a-1z3 - a-1z5 - z-2 + 13 - 24z2 + 21z4 - 8z6 + z8 + 2az-1 - 7az - 3az3 + 13az5 - 7az7 + az9 - 2a2z-2 + 13a2 - 32a2z2 + 35a2z4 - 15a2z6 + 2a2z8 + 2a3z-1 - 3a3z - 7a3z3 + 14a3z5 - 7a3z7 + a3z9 - a4z-2 + 5a4 - 12a4z2 + 15a4z4 - 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 = 7                   1 j = 5                     j = 3             1 1 1   j = 1             2       j = -1         1 1 3       j = -3       1 2           j = -5       1 2           j = -7   1 1               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, 426]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 426]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 8, 17, 7], X[14, 6, 15, 5], X[3, 10, 4, 11], > X[4, 14, 5, 13], X[17, 2, 18, 3], X[9, 19, 10, 18], X[12, 21, 7, 22], > X[22, 11, 13, 12], X[20, 16, 21, 15], X[6, 19, 1, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 6, -4, -5, 3, -11}, {2, -1, -7, 4, 9, -8}, > {5, -3, 10, -2, -6, 7, 11, -10, 8, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-5 -4 -3 1 2 3 2 + q - q + q + - + q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 2 2 2 3 3 6 2 4 6 8 10 5 + q + q + --- + --- + -- + -- + -- + -- + 4 q + q - q - q - q 12 10 8 6 4 2 q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 426]][a, z]
Out[8]=	2 4 2 2 2 4 -2 2 a a 2 z 2 2 4 2 4 7 - -- - 7 a + 2 a + z - ---- + -- + 5 z - -- - 5 a z + a z + z - 2 2 2 2 a z z a 2 4 > a z
In[9]:=	Kauffman[Link[11, NonAlternating, 426]][a, z]
Out[9]=	2 4 3 4 2 4 -2 2 a a 2 a 2 a 2 z 6 z 13 + -- + 13 a + 5 a - z - ---- - -- + --- + ---- - --- - --- - 7 a z - 2 2 2 z z 3 a a z z a 2 3 3 3 2 4 z 2 2 4 2 z 5 z 3 > 3 a z - 24 z - ---- - 32 a z - 12 a z + -- + ---- - 3 a z - 2 3 a a a 4 5 3 3 4 z 2 4 4 4 z 5 3 5 > 7 a z + 21 z + -- + 35 a z + 15 a z - -- + 13 a z + 14 a z - 2 a a 6 2 6 4 6 7 3 7 8 2 8 4 8 > 8 z - 15 a z - 7 a z - 7 a z - 7 a z + z + 2 a z + a z + 9 3 9 > a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 1 1 1 1 1 2 2 1 - + 2 q + q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 6 7 5 7 4 5 3 3 3 5 2 3 2 2 q t q t q t q t q t q t q t q t 1 3 3 2 7 3 > --- + q t + q t + q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n426

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