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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X14,5,15,6 X11,21,12,20 X17,11,18,22 X21,17,22,16 X10,13,5,14 X8,20,9,19 X18,8,19,7 X2,9,3,10 X4,16,1,15

Gauss Code:

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

Jones Polynomial:

q-3 - 3q-2 + 6q-1 - 7 + 11q - 9q2 + 10q3 - 7q4 + 4q5 - 2q6

A2 (sl(3)) Invariant:

q-8 - q-6 + 2q-4 + q-2 + 4 + 7q2 + 5q4 + 10q6 + 3q8 + 4q10 - q12 - 3q14 - q16 - 3q18 - q22

HOMFLY-PT Polynomial:

- a-6z-2 - 2a-6 - a-6z2 + 4a-4z-2 + 8a-4 + 9a-4z2 + 5a-4z4 + a-4z6 - 5a-2z-2 - 10a-2 - 14a-2z2 - 13a-2z4 - 6a-2z6 - a-2z8 + 2z-2 + 4 + 5z2 + 4z4 + z6

Kauffman Polynomial:

a-7z-1 - 5a-7z + 3a-7z3 - a-6z-2 + 3a-6 - 4a-6z2 + 2a-6z4 + a-6z6 + 5a-5z-1 - 18a-5z + 25a-5z3 - 13a-5z5 + 4a-5z7 - 4a-4z-2 + 12a-4 - 26a-4z2 + 35a-4z4 - 19a-4z6 + 5a-4z8 + 9a-3z-1 - 24a-3z + 31a-3z3 - 15a-3z5 + 2a-3z9 - 5a-2z-2 + 15a-2 - 37a-2z2 + 50a-2z4 - 34a-2z6 + 9a-2z8 + 5a-1z-1 - 11a-1z + 14a-1z3 - 11a-1z5 - a-1z7 + 2a-1z9 - 2z-2 + 7 - 13z2 + 14z4 - 13z6 + 4z8 + 5az3 - 9az5 + 3az7 + 2a2z2 - 3a2z4 + a2z6

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 13                   2 j = 11                 2   j = 9               5 2   j = 7             5 2     j = 5           5 6       j = 3         6 4         j = 1       3 7           j = -1     3 4             j = -3   1 4               j = -5   2                 j = -7 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, 319]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 319]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 5, 15, 6], X[11, 21, 12, 20], > X[17, 11, 18, 22], X[21, 17, 22, 16], X[10, 13, 5, 14], X[8, 20, 9, 19], > X[18, 8, 19, 7], X[2, 9, 3, 10], X[4, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {3, -1, 9, -8, 10, -7}, > {-4, -2, 7, -3, 11, 6, -5, -9, 8, 4, -6, 5}]
In[6]:=	Jones[L][q]
Out[6]=	-3 3 6 2 3 4 5 6 -7 + q - -- + - + 11 q - 9 q + 10 q - 7 q + 4 q - 2 q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 2 -2 2 4 6 8 10 12 14 4 + q - q + -- + q + 7 q + 5 q + 10 q + 3 q + 4 q - q - 3 q - 4 q 16 18 22 > q - 3 q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 319]][a, z]
Out[8]=	2 2 2 2 8 10 2 1 4 5 2 z 9 z 14 z 4 - -- + -- - -- + -- - ----- + ----- - ----- + 5 z - -- + ---- - ----- + 6 4 2 2 6 2 4 2 2 2 6 4 2 a a a z a z a z a z a a a 4 4 6 6 8 4 5 z 13 z 6 z 6 z z > 4 z + ---- - ----- + z + -- - ---- - -- 4 2 4 2 2 a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 319]][a, z]
Out[9]=	3 12 15 2 1 4 5 1 5 9 5 7 + -- + -- + -- - -- - ----- - ----- - ----- + ---- + ---- + ---- + --- - 6 4 2 2 6 2 4 2 2 2 7 5 3 a z a a a z a z a z a z a z a z a z 2 2 2 3 5 z 18 z 24 z 11 z 2 4 z 26 z 37 z 2 2 3 z > --- - ---- - ---- - ---- - 13 z - ---- - ----- - ----- + 2 a z + ---- + 7 5 3 a 6 4 2 7 a a a a a a a 3 3 3 4 4 4 25 z 31 z 14 z 3 4 2 z 35 z 50 z 2 4 > ----- + ----- + ----- + 5 a z + 14 z + ---- + ----- + ----- - 3 a z - 5 3 a 6 4 2 a a a a a 5 5 5 6 6 6 13 z 15 z 11 z 5 6 z 19 z 34 z 2 6 > ----- - ----- - ----- - 9 a z - 13 z + -- - ----- - ----- + a z + 5 3 a 6 4 2 a a a a a 7 7 8 8 9 9 4 z z 7 8 5 z 9 z 2 z 2 z > ---- - -- + 3 a z + 4 z + ---- + ---- + ---- + ---- 5 a 4 2 3 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 2 1 4 3 4 3 q 3 7 q + 6 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 4 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 5 q t + 6 q t + 5 q t + 2 q t + 5 q t + 2 q t + 2 q t + 13 5 > 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n319

L11n318 L11n320