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L11n414 L11n416

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DrawMorseLink

PD Presentation:

X8192 X5,15,6,14 X10,3,11,4 X13,5,14,4 X2738 X6,9,1,10 X18,12,19,11 X12,18,7,17 X20,16,21,15 X22,20,13,19 X16,22,17,21

Gauss Code:

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

Jones Polynomial:

3q-1 - 4 + 9q - 10q2 + 12q3 - 11q4 + 9q5 - 6q6 + 3q7 - q8

A2 (sl(3)) Invariant:

3q-4 + 2q-2 + 4 + 9q2 + 4q4 + 7q6 + 3q8 - 4q14 + q16 - q18 - q20 + 2q22 - q24 - q26

HOMFLY-PT Polynomial:

- a-8 - a-6z-2 + 2a-6 + 3a-6z2 + 4a-4z-2 + 4a-4 - a-4z2 - 2a-4z4 - 5a-2z-2 - 10a-2 - 8a-2z2 - 3a-2z4 + 2z-2 + 5 + 3z2

Kauffman Polynomial:

a-9z - 2a-9z3 + a-9z5 - a-8 + 3a-8z2 - 6a-8z4 + 3a-8z6 + a-7z-1 - 2a-7z + a-7z3 - 6a-7z5 + 4a-7z7 - a-6z-2 + 6a-6z2 - 9a-6z4 + 3a-6z8 + 5a-5z-1 - 19a-5z + 33a-5z3 - 26a-5z5 + 8a-5z7 + a-5z9 - 4a-4z-2 + 13a-4 - 19a-4z2 + 23a-4z4 - 17a-4z6 + 7a-4z8 + 9a-3z-1 - 35a-3z + 48a-3z3 - 28a-3z5 + 7a-3z7 + a-3z9 - 5a-2z-2 + 22a-2 - 38a-2z2 + 32a-2z4 - 14a-2z6 + 4a-2z8 + 5a-1z-1 - 19a-1z + 18a-1z3 - 9a-1z5 + 3a-1z7 - 2z-2 + 11 - 16z2 + 6z4

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                   1 j = 15                 2   j = 13               4 1   j = 11             5 2     j = 9           7 5       j = 7         5 4         j = 5       5 7           j = 3     4 5             j = 1   1 6               j = -1 2 3                 j = -3 3

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, 415]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 415]]
Out[4]=	PD[X[8, 1, 9, 2], X[5, 15, 6, 14], X[10, 3, 11, 4], X[13, 5, 14, 4], > X[2, 7, 3, 8], X[6, 9, 1, 10], X[18, 12, 19, 11], X[12, 18, 7, 17], > X[20, 16, 21, 15], X[22, 20, 13, 19], X[16, 22, 17, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -5, 3, 4, -2, -6}, {5, -1, 6, -3, 7, -8}, > {-4, 2, 9, -11, 8, -7, 10, -9, 11, -10}]
In[6]:=	Jones[L][q]
Out[6]=	3 2 3 4 5 6 7 8 -4 + - + 9 q - 10 q + 12 q - 11 q + 9 q - 6 q + 3 q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	3 2 2 4 6 8 14 16 18 20 22 4 + -- + -- + 9 q + 4 q + 7 q + 3 q - 4 q + q - q - q + 2 q - 4 2 q q 24 26 > q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 415]][a, z]
Out[8]=	2 2 2 -8 2 4 10 2 1 4 5 2 3 z z 8 z 5 - a + -- + -- - -- + -- - ----- + ----- - ----- + 3 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 2 z 3 z > ---- - ---- 4 2 a a
In[9]:=	Kauffman[Link[11, NonAlternating, 415]][a, z]
Out[9]=	-8 13 22 2 1 4 5 1 5 9 5 11 - a + -- + -- - -- - ----- - ----- - ----- + ---- + ---- + ---- + --- + 4 2 2 6 2 4 2 2 2 7 5 3 a z a a z a z a z a z a z a z a z 2 2 2 2 z 2 z 19 z 35 z 19 z 2 3 z 6 z 19 z 38 z > -- - --- - ---- - ---- - ---- - 16 z + ---- + ---- - ----- - ----- - 9 7 5 3 a 8 6 4 2 a a a a a a a a 3 3 3 3 3 4 4 4 4 2 z z 33 z 48 z 18 z 4 6 z 9 z 23 z 32 z > ---- + -- + ----- + ----- + ----- + 6 z - ---- - ---- + ----- + ----- + 9 7 5 3 a 8 6 4 2 a a a a a a a a 5 5 5 5 5 6 6 6 7 7 z 6 z 26 z 28 z 9 z 3 z 17 z 14 z 4 z 8 z > -- - ---- - ----- - ----- - ---- + ---- - ----- - ----- + ---- + ---- + 9 7 5 3 a 8 4 2 7 5 a a a a a a a a a 7 7 8 8 8 9 9 7 z 3 z 3 z 7 z 4 z z z > ---- + ---- + ---- + ---- + ---- + -- + -- 3 a 6 4 2 5 3 a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 2 3 q 3 5 5 2 7 2 6 q + 4 q + ----- + ---- + --- + - + 5 q t + 5 q t + 7 q t + 5 q t + 3 2 2 q t t q t q t 7 3 9 3 9 4 11 4 11 5 13 5 13 6 > 4 q t + 7 q t + 5 q t + 5 q t + 2 q t + 4 q t + q t + 15 6 17 7 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n415

L11n414 L11n416