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L11a510 L11a512

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DrawMorseLink

PD Presentation:

X8192 X14,3,15,4 X20,12,21,11 X18,10,19,9 X22,16,13,15 X12,20,7,19 X10,22,11,21 X16,6,17,5 X2738 X4,13,5,14 X6,18,1,17

Gauss Code:

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

Jones Polynomial:

q-3 - 3q-2 + 7q-1 - 10 + 15q - 16q2 + 17q3 - 14q4 + 11q5 - 6q6 + 3q7 - q8

A2 (sl(3)) Invariant:

q-10 - q-6 + 3q-4 + 2 + 5q2 + 6q6 + 2q8 + 4q10 + 4q12 - q14 + 4q16 - q18 - q20 + q22 - q24

HOMFLY-PT Polynomial:

- a-6 - 2a-6z2 - a-6z4 + a-4z-2 + 4a-4 + 5a-4z2 + 3a-4z4 + a-4z6 - 2a-2z-2 - 5a-2 - 3a-2z2 + a-2z4 + a-2z6 + z-2 + 1 - 3z2 - 2z4 + a2 + a2z2

Kauffman Polynomial:

- 2a-9z3 + a-9z5 + 2a-8z2 - 6a-8z4 + 3a-8z6 - a-7z + 6a-7z3 - 10a-7z5 + 5a-7z7 + 2a-6 - 9a-6z2 + 17a-6z4 - 14a-6z6 + 6a-6z8 - 3a-5z + 9a-5z3 - 2a-5z5 - 4a-5z7 + 4a-5z9 - a-4z-2 + 10a-4 - 35a-4z2 + 53a-4z4 - 35a-4z6 + 10a-4z8 + a-4z10 + 2a-3z-1 - 8a-3z + 5a-3z3 + 7a-3z5 - 13a-3z7 + 7a-3z9 - 2a-2z-2 + 12a-2 - 29a-2z2 + 35a-2z4 - 27a-2z6 + 8a-2z8 + a-2z10 + 2a-1z-1 - 6a-1z + 9a-1z3 - 10a-1z5 - a-1z7 + 3a-1z9 - z-2 + 4 - 2z2 + 2z4 - 8z6 + 4z8 + 5az3 - 8az5 + 3az7 - a2 + 3a2z2 - 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 r = 6 r = 7 j = 17                       1 j = 15                     2   j = 13                   4 1   j = 11                 7 2     j = 9               8 5       j = 7             9 6         j = 5           8 9           j = 3         7 8             j = 1       4 9               j = -1     3 6                 j = -3   1 5                   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, Alternating, 511]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 511]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 3, 15, 4], X[20, 12, 21, 11], X[18, 10, 19, 9], > X[22, 16, 13, 15], X[12, 20, 7, 19], X[10, 22, 11, 21], X[16, 6, 17, 5], > X[2, 7, 3, 8], X[4, 13, 5, 14], X[6, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 8, -11}, {9, -1, 4, -7, 3, -6}, > {10, -2, 5, -8, 11, -4, 6, -3, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-3 3 7 2 3 4 5 6 7 8 -10 + q - -- + - + 15 q - 16 q + 17 q - 14 q + 11 q - 6 q + 3 q - q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -6 3 2 6 8 10 12 14 16 18 2 + q - q + -- + 5 q + 6 q + 2 q + 4 q + 4 q - q + 4 q - q - 4 q 20 22 24 > q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 511]][a, z]
Out[8]=	2 2 2 -6 4 5 2 -2 1 2 2 2 z 5 z 3 z 1 - a + -- - -- + a + z + ----- - ----- - 3 z - ---- + ---- - ---- + 4 2 4 2 2 2 6 4 2 a a a z a z a a a 4 4 4 6 6 2 2 4 z 3 z z z z > a z - 2 z - -- + ---- + -- + -- + -- 6 4 2 4 2 a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 511]][a, z]
Out[9]=	2 10 12 2 -2 1 2 2 2 z 3 z 8 z 4 + -- + -- + -- - a - z - ----- - ----- + ---- + --- - -- - --- - --- - 6 4 2 4 2 2 2 3 a z 7 5 3 a a a a z a z a z a a a 2 2 2 2 3 3 3 6 z 2 2 z 9 z 35 z 29 z 2 2 2 z 6 z 9 z > --- - 2 z + ---- - ---- - ----- - ----- + 3 a z - ---- + ---- + ---- + a 8 6 4 2 9 7 5 a a a a a a a 3 3 4 4 4 4 5 5 z 9 z 3 4 6 z 17 z 53 z 35 z 2 4 z > ---- + ---- + 5 a z + 2 z - ---- + ----- + ----- + ----- - 3 a z + -- - 3 a 8 6 4 2 9 a a a a a a 5 5 5 5 6 6 6 10 z 2 z 7 z 10 z 5 6 3 z 14 z 35 z > ----- - ---- + ---- - ----- - 8 a z - 8 z + ---- - ----- - ----- - 7 5 3 a 8 6 4 a a a a a a 6 7 7 7 7 8 8 27 z 2 6 5 z 4 z 13 z z 7 8 6 z 10 z > ----- + a z + ---- - ---- - ----- - -- + 3 a z + 4 z + ---- + ----- + 2 7 5 3 a 6 4 a a a a a a 8 9 9 9 10 10 8 z 4 z 7 z 3 z z z > ---- + ---- + ---- + ---- + --- + --- 2 5 3 a 4 2 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 2 1 5 3 6 4 q 3 9 q + 7 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 8 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 > 8 q t + 9 q t + 9 q t + 6 q t + 8 q t + 5 q t + 7 q t + 11 5 13 5 13 6 15 6 17 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 L11a511

L11a510 L11a512