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L11n26 L11n28

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-30 - q-28 - q-26 - 3q-24 - q-22 + q-18 + 3q-16 + 2q-14 + 3q-12 + 2q-10 + 2q-8 + 2q-6 + 1

HOMFLY-PT Polynomial:

- az - 2a3z-1 - 3a3z + 2a5z-1 + 4a5z + 2a5z3 + a7z-1 + a7z + a7z3 - a9z-1 - a9z

Kauffman Polynomial:

- az - a2z2 - 2a3z-1 + 7a3z - 3a3z3 + 2a4 - 5a4z2 + 7a4z4 - 2a4z6 - 2a5z-1 + 13a5z - 25a5z3 + 19a5z5 - 4a5z7 - 4a6 + 11a6z2 - 18a6z4 + 14a6z6 - 3a6z8 + a7z-1 + 5a7z - 22a7z3 + 13a7z5 + a7z7 - a7z9 - 9a8 + 31a8z2 - 42a8z4 + 23a8z6 - 4a8z8 + a9z-1 - 6a9z5 + 5a9z7 - a9z9 - 4a10 + 16a10z2 - 17a10z4 + 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                   1 j = -2                 2 2 j = -4               1   1 j = -6             3 3     j = -8           2 1       j = -10         1 3 1       j = -12       3 2           j = -14       1             j = -16   1 3               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, 27]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 27]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[17, 1, 18, 4], X[5, 12, 6, 13], > X[3, 8, 4, 9], X[13, 22, 14, 5], X[21, 14, 22, 15], X[11, 18, 12, 19], > X[9, 20, 10, 21], X[19, 10, 20, 11], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, -5, 3}, {-4, -1, 2, 5, -9, 10, -8, 4, -6, 7, 11, -2, -3, 8, > -10, 9, -7, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) -(17/2) 3 4 3 4 3 3 -(3/2) q - q + ----- - ----- + ----- - ---- + ---- - ---- + q - 15/2 13/2 11/2 9/2 7/2 5/2 q q q q q q 1 > ------- Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 -26 3 -22 -18 3 2 3 2 2 2 1 - q - q - q - --- - q + q + --- + --- + --- + --- + -- + -- 24 16 14 12 10 8 6 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 27]][a, z]
Out[8]=	3 5 7 9 -2 a 2 a a a 3 5 7 9 5 3 7 3 ----- + ---- + -- - -- - a z - 3 a z + 4 a z + a z - a z + 2 a z + a z z z z z
In[9]:=	Kauffman[Link[11, NonAlternating, 27]][a, z]
Out[9]=	3 5 7 9 4 6 8 10 2 a 2 a a a 3 5 2 a - 4 a - 9 a - 4 a - ---- - ---- + -- + -- - a z + 7 a z + 13 a z + z z z z 7 2 2 4 2 6 2 8 2 10 2 3 3 > 5 a z - a z - 5 a z + 11 a z + 31 a z + 16 a z - 3 a z - 5 3 7 3 4 4 6 4 8 4 10 4 > 25 a z - 22 a z + 7 a z - 18 a z - 42 a z - 17 a z + 5 5 7 5 9 5 4 6 6 6 8 6 10 6 > 19 a z + 13 a z - 6 a z - 2 a z + 14 a z + 23 a z + 7 a z - 5 7 7 7 9 7 6 8 8 8 10 8 7 9 9 9 > 4 a z + a z + 5 a z - 3 a z - 4 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 2 1 1 3 1 3 2 1 1 + q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 3 2 1 1 3 3 1 2 > ------ + ----- + ------ + ----- + ----- + ----- + ----- + ---- 10 4 8 4 10 3 8 3 6 3 6 2 4 2 2 q t 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 L11n27

L11n26 L11n28