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L11n13 L11n15

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X17,1,18,4 X5,10,6,11 X3849 X11,22,12,5 X13,20,14,21 X19,14,20,15 X21,12,22,13 X9,18,10,19 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

q-19/2 - q-17/2 + 2q-15/2 - 3q-13/2 + 3q-11/2 - 4q-9/2 + 3q-7/2 - 3q-5/2 + 2q-3/2 - 2q-1/2

A2 (sl(3)) Invariant:

- q-30 - q-28 - q-26 - 2q-24 + q-20 + q-18 + 2q-16 + q-14 + q-12 + q-10 + q-8 + 2q-6 + q-4 + q-2 + 2

HOMFLY-PT Polynomial:

- az-1 - 2az + a3z-1 + a3z + a3z3 - a5z-1 + a5z3 + 2a7z-1 + 2a7z + a7z3 - a9z-1 - a9z

Kauffman Polynomial:

az-1 - 3az - a2 - a2z4 + a3z-1 - 3a3z + 4a3z3 - 2a3z5 + 4a4z4 - 2a4z6 + a5z-1 - 3a5z + a5z3 + 5a5z5 - 2a5z7 - 4a6 + 13a6z2 - 14a6z4 + 9a6z6 - 2a6z8 + 2a7z-1 - 6a7z + 4a7z3 - 4a7z5 + 4a7z7 - a7z9 - 7a8 + 26a8z2 - 35a8z4 + 18a8z6 - 3a8z8 + a9z-1 - 3a9z + 7a9z3 - 11a9z5 + 6a9z7 - a9z9 - 3a10 + 13a10z2 - 16a10z4 + 7a10z6 - a10z8

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = 0                   2 j = -2                 2 2 j = -4               1     j = -6             2 2     j = -8           2 1       j = -10         1 2         j = -12       2 2           j = -14       1             j = -16   1 2               j = -18                     j = -20 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 14]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 14]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[17, 1, 18, 4], X[5, 10, 6, 11], > X[3, 8, 4, 9], X[11, 22, 12, 5], X[13, 20, 14, 21], X[19, 14, 20, 15], > X[21, 12, 22, 13], X[9, 18, 10, 19], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, -5, 3}, {-4, -1, 2, 5, -10, 4, -6, 9, -7, 8, 11, -2, -3, 10, > -8, 7, -9, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) -(17/2) 2 3 3 4 3 3 2 q - q + ----- - ----- + ----- - ---- + ---- - ---- + ---- - 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q 2 > ------- Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 -26 2 -20 -18 2 -14 -12 -10 -8 2 - q - q - q - --- + q + q + --- + q + q + q + q + 24 16 q q 2 -4 -2 > -- + q + q 6 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 14]][a, z]
Out[8]=	3 5 7 9 a a a 2 a a 3 7 9 3 3 5 3 -(-) + -- - -- + ---- - -- - 2 a z + a z + 2 a z - a z + a z + a z + z z z z z 7 3 > a z
In[9]:=	Kauffman[Link[11, NonAlternating, 14]][a, z]
Out[9]=	3 5 7 9 2 6 8 10 a a a 2 a a 3 5 -a - 4 a - 7 a - 3 a + - + -- + -- + ---- + -- - 3 a z - 3 a z - 3 a z - z z z z z 7 9 6 2 8 2 10 2 3 3 5 3 > 6 a z - 3 a z + 13 a z + 26 a z + 13 a z + 4 a z + a z + 7 3 9 3 2 4 4 4 6 4 8 4 10 4 > 4 a z + 7 a z - a z + 4 a z - 14 a z - 35 a z - 16 a z - 3 5 5 5 7 5 9 5 4 6 6 6 8 6 > 2 a z + 5 a z - 4 a z - 11 a z - 2 a z + 9 a z + 18 a z + 10 6 5 7 7 7 9 7 6 8 8 8 10 8 > 7 a z - 2 a z + 4 a z + 6 a z - 2 a z - 3 a z - a z - 7 9 9 9 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 2 1 2 2 1 2 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 20 9 16 8 16 7 14 6 12 6 12 5 10 5 q q t q t q t q t q t q t q t 2 2 1 2 2 1 2 > ------ + ----- + ----- + ----- + ----- + ----- + ---- 10 4 8 4 8 3 6 3 6 2 4 2 2 q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n14

L11n13 L11n15