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DrawMorseLink

PD Presentation:

X6172 X12,7,13,8 X4,13,1,14 X18,6,19,5 X8493 X9,21,10,20 X19,11,20,10 X14,18,15,17 X22,16,17,15 X16,22,5,21 X2,12,3,11

Gauss Code:

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

Jones Polynomial:

- q-2 + 4q-1 - 7 + 11q - 13q2 + 14q3 - 12q4 + 10q5 - 5q6 + 3q7

A2 (sl(3)) Invariant:

- q-6 + 2q-4 + 3q2 - 3q4 + 2q6 - q8 + q10 + 4q12 + q14 + 7q16 + 3q18 + 3q20 + 4q22 + q24 + q26

HOMFLY-PT Polynomial:

a-8z-2 + a-8 - 2a-6z-2 - 4a-6 - 3a-6z2 - a-6z4 + a-4z-2 + 3a-4 + 4a-4z2 + 3a-4z4 + a-4z6 - a-2 + 2a-2z4 + a-2z6 + 1 - z2 - z4

Kauffman Polynomial:

- a-8z-2 + 8a-8 - 14a-8z2 + 6a-8z4 + 2a-7z-1 - 10a-7z + 10a-7z3 - 6a-7z5 + 3a-7z7 - 2a-6z-2 + 15a-6 - 33a-6z2 + 33a-6z4 - 16a-6z6 + 5a-6z8 + 2a-5z-1 - 12a-5z + 20a-5z3 - 13a-5z5 + 3a-5z7 + 2a-5z9 - a-4z-2 + 9a-4 - 24a-4z2 + 33a-4z4 - 25a-4z6 + 10a-4z8 - 3a-3z + 14a-3z3 - 18a-3z5 + 6a-3z7 + 2a-3z9 + 2a-2 - 3a-2z2 - a-2z4 - 5a-2z6 + 5a-2z8 - a-1z + 3a-1z3 - 10a-1z5 + 6a-1z7 + 1 + 2z2 - 7z4 + 4z6 - az3 + az5

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 15                   3 j = 13                 4 2 j = 11               6 1   j = 9             6 4     j = 7           8 6       j = 5         5 6         j = 3       6 8           j = 1     3 7             j = -1   1 4               j = -3   3                 j = -5 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, 362]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 362]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 7, 13, 8], X[4, 13, 1, 14], X[18, 6, 19, 5], > X[8, 4, 9, 3], X[9, 21, 10, 20], X[19, 11, 20, 10], X[14, 18, 15, 17], > X[22, 16, 17, 15], X[16, 22, 5, 21], X[2, 12, 3, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {8, -4, -7, 6, 10, -9}, > {4, -1, 2, -5, -6, 7, 11, -2, 3, -8, 9, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-2 4 2 3 4 5 6 7 -7 - q + - + 11 q - 13 q + 14 q - 12 q + 10 q - 5 q + 3 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 2 2 4 6 8 10 12 14 16 18 -q + -- + 3 q - 3 q + 2 q - q + q + 4 q + q + 7 q + 3 q + 4 q 20 22 24 26 > 3 q + 4 q + q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 362]][a, z]
Out[8]=	2 2 4 -8 4 3 -2 1 2 1 2 3 z 4 z 4 z 1 + a - -- + -- - a + ----- - ----- + ----- - z - ---- + ---- - z - -- + 6 4 8 2 6 2 4 2 6 4 6 a a a z a z a z a a a 4 4 6 6 3 z 2 z z z > ---- + ---- + -- + -- 4 2 4 2 a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 362]][a, z]
Out[9]=	8 15 9 2 1 2 1 2 2 10 z 12 z 1 + -- + -- + -- + -- - ----- - ----- - ----- + ---- + ---- - ---- - ---- - 8 6 4 2 8 2 6 2 4 2 7 5 7 5 a a a a a z a z a z a z a z a a 2 2 2 2 3 3 3 3 z z 2 14 z 33 z 24 z 3 z 10 z 20 z 14 z > --- - - + 2 z - ----- - ----- - ----- - ---- + ----- + ----- + ----- + 3 a 8 6 4 2 7 5 3 a a a a a a a a 3 4 4 4 4 5 5 5 3 z 3 4 6 z 33 z 33 z z 6 z 13 z 18 z > ---- - a z - 7 z + ---- + ----- + ----- - -- - ---- - ----- - ----- - a 8 6 4 2 7 5 3 a a a a a a a 5 6 6 6 7 7 7 7 10 z 5 6 16 z 25 z 5 z 3 z 3 z 6 z 6 z > ----- + a z + 4 z - ----- - ----- - ---- + ---- + ---- + ---- + ---- + a 6 4 2 7 5 3 a a a a a a a 8 8 8 9 9 5 z 10 z 5 z 2 z 2 z > ---- + ----- + ---- + ---- + ---- 6 4 2 5 3 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 3 1 4 3 q 3 5 5 2 7 q + 6 q + ----- + ----- + ---- + --- + --- + 8 q t + 5 q t + 6 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 13 5 > 8 q t + 6 q t + 6 q t + 4 q t + 6 q t + q t + 4 q t + 13 6 15 6 > 2 q t + 3 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n362

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