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L11a413 L11a415

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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X22,16,11,15 X20,14,21,13 X14,22,15,21 X8,18,9,17 X16,8,17,7 X18,10,19,9 X10,20,5,19 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

q-2 - q-1 + 5 - 6q + 10q2 - 11q3 + 12q4 - 11q5 + 9q6 - 6q7 + 3q8 - q9

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

- a-6z-2 - 3a-6 - 5a-6z2 - 4a-6z4 - a-6z6 + 4a-4z-2 + 13a-4 + 18a-4z2 + 14a-4z4 + 6a-4z6 + a-4z8 - 5a-2z-2 - 17a-2 - 21a-2z2 - 11a-2z4 - 2a-2z6 + 2z-2 + 7 + 5z2 + z4

Kauffman Polynomial:

a-11z3 + 3a-10z4 + a-9z - 4a-9z3 + 6a-9z5 - a-8 + 5a-8z2 - 13a-8z4 + 9a-8z6 + a-7z-1 - 2a-7z + 9a-7z3 - 20a-7z5 + 10a-7z7 - a-6z-2 + 3a-6z2 + 3a-6z4 - 17a-6z6 + 8a-6z8 + 5a-5z-1 - 19a-5z + 33a-5z3 - 23a-5z5 - 3a-5z7 + 4a-5z9 - 4a-4z-2 + 13a-4 - 20a-4z2 + 37a-4z4 - 32a-4z6 + 6a-4z8 + a-4z10 + 9a-3z-1 - 35a-3z + 39a-3z3 - a-3z5 - 16a-3z7 + 5a-3z9 - 5a-2z-2 + 22a-2 - 39a-2z2 + 36a-2z4 - 13a-2z6 - a-2z8 + a-2z10 + 5a-1z-1 - 19a-1z + 20a-1z3 - 4a-1z5 - 3a-1z7 + a-1z9 - 2z-2 + 11 - 21z2 + 18z4 - 7z6 + z8

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 19                       1 j = 17                     2   j = 15                   4 1   j = 13                 5 2     j = 11               6 4       j = 9             6 5         j = 7           7 8           j = 5         3 4             j = 3       4 8               j = 1     1 2                 j = -1     4                   j = -3 1 1                     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, Alternating, 414]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 414]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[22, 16, 11, 15], X[20, 14, 21, 13], > X[14, 22, 15, 21], X[8, 18, 9, 17], X[16, 8, 17, 7], X[18, 10, 19, 9], > X[10, 20, 5, 19], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 7, -6, 8, -9}, > {11, -2, 4, -5, 3, -7, 6, -8, 9, -4, 5, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-2 1 2 3 4 5 6 7 8 9 5 + q - - - 6 q + 10 q - 11 q + 12 q - 11 q + 9 q - 6 q + 3 q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 2 3 2 4 6 8 10 12 14 16 6 + q + -- + -- + 4 q + 6 q + 4 q + 2 q + 4 q - 2 q + 2 q - 2 q - 4 2 q q 18 22 24 26 > q - 2 q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 414]][a, z]
Out[8]=	2 2 2 3 13 17 2 1 4 5 2 5 z 18 z 21 z 7 - -- + -- - -- + -- - ----- + ----- - ----- + 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 4 6 6 6 8 4 4 z 14 z 11 z z 6 z 2 z z > z - ---- + ----- - ----- - -- + ---- - ---- + -- 6 4 2 6 4 2 4 a a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 414]][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 3 z 2 z 19 z 35 z 19 z 2 5 z 3 z 20 z 39 z z > -- - --- - ---- - ---- - ---- - 21 z + ---- + ---- - ----- - ----- + --- - 9 7 5 3 a 8 6 4 2 11 a a a a a a a a a 3 3 3 3 3 4 4 4 4 4 z 9 z 33 z 39 z 20 z 4 3 z 13 z 3 z 37 z > ---- + ---- + ----- + ----- + ----- + 18 z + ---- - ----- + ---- + ----- + 9 7 5 3 a 10 8 6 4 a a a a a a a a 4 5 5 5 5 5 6 6 6 36 z 6 z 20 z 23 z z 4 z 6 9 z 17 z 32 z > ----- + ---- - ----- - ----- - -- - ---- - 7 z + ---- - ----- - ----- - 2 9 7 5 3 a 8 6 4 a a a a a a a a 6 7 7 7 7 8 8 8 9 9 13 z 10 z 3 z 16 z 3 z 8 8 z 6 z z 4 z 5 z > ----- + ----- - ---- - ----- - ---- + z + ---- + ---- - -- + ---- + ---- + 2 7 5 3 a 6 4 2 5 3 a a a a a a a a a 9 10 10 z z z > -- + --- + --- a 4 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 5 1 1 1 4 q 2 q 4 q 5 8 q + 3 q + ----- + ----- + ----- + ---- + -- + --- + ---- + 4 q t + 5 4 3 4 3 3 2 2 t t q t q t q t q t t 7 7 2 9 2 9 3 11 3 11 4 13 4 > 7 q t + 8 q t + 6 q t + 5 q t + 6 q t + 4 q t + 5 q t + 13 5 15 5 15 6 17 6 19 7 > 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a414

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