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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X18,13,19,14 X17,11,18,22 X7,17,8,16 X21,8,22,9 X14,10,15,9 X20,16,21,15 X10,19,5,20 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

2q-5 - 3q-4 + 6q-3 - 5q-2 + 7q-1 - 5 + 4q - 3q2 + q3

A2 (sl(3)) Invariant:

q-18 + 4q-16 + 3q-14 + 4q-12 + 6q-10 + 3q-8 + 4q-6 + q-4 + q-2 + 1 - q2 + q4 - q6 - q8 + q10

HOMFLY-PT Polynomial:

a-2z2 - z2 - z4 + a2z-2 + 2a2 - a2z4 - 2a4z-2 - 2a4 + a4z2 + a6z-2

Kauffman Polynomial:

a-2z2 - 3a-2z4 + a-2z6 + 8a-1z3 - 11a-1z5 + 3a-1z7 - 4z2 + 10z4 - 11z6 + 3z8 + 8az3 - 9az5 + az9 + a2z-2 - 2a2 - 9a2z2 + 21a2z4 - 16a2z6 + 4a2z8 - 2a3z-1 + 2a3z + 3a3z5 - 3a3z7 + a3z9 + 2a4z-2 - 3a4 - 2a4z2 + 8a4z4 - 4a4z6 + a4z8 - 2a5z-1 + 2a5z + a5z5 + a6z-2 - 2a6 + 2a6z2

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                 1 j = 5               2   j = 3             2 1   j = 1           3 2     j = -1         4 2       j = -3       2 4         j = -5     4 3           j = -7   1 4             j = -9 1 2               j = -11 2

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, 300]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 300]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[18, 13, 19, 14], X[17, 11, 18, 22], > X[7, 17, 8, 16], X[21, 8, 22, 9], X[14, 10, 15, 9], X[20, 16, 21, 15], > X[10, 19, 5, 20], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -5, 6, 7, -9}, > {11, -2, 3, -7, 8, 5, -4, -3, 9, -8, -6, 4}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 6 5 7 2 3 -5 + -- - -- + -- - -- + - + 4 q - 3 q + q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 4 3 4 6 3 4 -4 -2 2 4 6 8 10 1 + q + --- + --- + --- + --- + -- + -- + q + q - q + q - q - q + q 16 14 12 10 8 6 q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 300]][a, z]
Out[8]=	2 4 6 2 2 4 a 2 a a 2 z 4 2 4 2 4 2 a - 2 a + -- - ---- + -- - z + -- + a z - z - a z 2 2 2 2 z z z a
In[9]:=	Kauffman[Link[11, NonAlternating, 300]][a, z]
Out[9]=	2 4 6 3 5 2 4 6 a 2 a a 2 a 2 a 3 5 2 -2 a - 3 a - 2 a + -- + ---- + -- - ---- - ---- + 2 a z + 2 a z - 4 z + 2 2 2 z z z z z 2 3 4 z 2 2 4 2 6 2 8 z 3 4 3 z > -- - 9 a z - 2 a z + 2 a z + ---- + 8 a z + 10 z - ---- + 2 a 2 a a 5 6 2 4 4 4 11 z 5 3 5 5 5 6 z > 21 a z + 8 a z - ----- - 9 a z + 3 a z + a z - 11 z + -- - a 2 a 7 2 6 4 6 3 z 3 7 8 2 8 4 8 9 3 9 > 16 a z - 4 a z + ---- - 3 a z + 3 z + 4 a z + a z + a z + a z a
In[10]:=	Kh[L][q, t]
Out[10]=	4 4 2 1 2 1 4 4 3 2 2 t -- + - + ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + 3 q 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q q t q t q t q t q t q t q t q t 2 3 2 3 3 5 3 7 4 > 3 q t + 2 q t + 2 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n300

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