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L11n219 L11n221

Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X8,9,1,10 X3,12,4,13 X15,22,16,9 X2,17,3,18 X21,4,22,5 X14,5,15,6 X20,13,21,14 X11,16,12,17 X6,19,7,20 X18,7,19,8

Gauss Code:

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

Jones Polynomial:

- q-23/2 + 2q-21/2 - 2q-19/2 + 2q-17/2 - 3q-15/2 + 2q-13/2 - 2q-11/2 + q-9/2 - q-7/2

A2 (sl(3)) Invariant:

- q-42 + q-34 + q-30 + 2q-26 + 2q-24 + q-22 + q-20 + q-16 + q-12

HOMFLY-PT Polynomial:

- 4a7z - 10a7z3 - 6a7z5 - a7z7 - 2a9z-1 - 8a9z - 11a9z3 - 6a9z5 - a9z7 + 3a11z-1 + 9a11z + 6a11z3 + a11z5 - a13z-1 - a13z

Kauffman Polynomial:

4a7z - 10a7z3 + 6a7z5 - a7z7 + a8z2 - 6a8z4 + 5a8z6 - a8z8 + 2a9z-1 - 10a9z + 16a9z3 - 13a9z5 + 6a9z7 - a9z9 - 3a10 + 14a10z2 - 19a10z4 + 11a10z6 - 2a10z8 + 3a11z-1 - 16a11z + 29a11z3 - 20a11z5 + 7a11z7 - a11z9 - 3a12 + 12a12z2 - 13a12z4 + 6a12z6 - a12z8 + a13z-1 - 3a13z + 3a13z3 - a13z5 - a14 - a14z2 - a15z

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -6                 1 j = -8               1 1 j = -10             1     j = -12           1 1     j = -14         2 1       j = -16         1         j = -18     2 2           j = -20                   j = -22 1 2               j = -24 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, 220]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 220]]
Out[4]=	PD[X[10, 1, 11, 2], X[8, 9, 1, 10], X[3, 12, 4, 13], X[15, 22, 16, 9], > X[2, 17, 3, 18], X[21, 4, 22, 5], X[14, 5, 15, 6], X[20, 13, 21, 14], > X[11, 16, 12, 17], X[6, 19, 7, 20], X[18, 7, 19, 8]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -5, -3, 6, 7, -10, 11, -2}, > {2, -1, -9, 3, 8, -7, -4, 9, 5, -11, 10, -8, -6, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) 2 2 2 3 2 2 -(9/2) -(7/2) -q + ----- - ----- + ----- - ----- + ----- - ----- + q - q 21/2 19/2 17/2 15/2 13/2 11/2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 -34 -30 2 2 -22 -20 -16 -12 -q + q + q + --- + --- + q + q + q + q 26 24 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 220]][a, z]
Out[8]=	9 11 13 -2 a 3 a a 7 9 11 13 7 3 9 3 ----- + ----- - --- - 4 a z - 8 a z + 9 a z - a z - 10 a z - 11 a z + z z z 11 3 7 5 9 5 11 5 7 7 9 7 > 6 a z - 6 a z - 6 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 220]][a, z]
Out[9]=	9 11 13 10 12 14 2 a 3 a a 7 9 11 -3 a - 3 a - a + ---- + ----- + --- + 4 a z - 10 a z - 16 a z - z z z 13 15 8 2 10 2 12 2 14 2 7 3 > 3 a z - a z + a z + 14 a z + 12 a z - a z - 10 a z + 9 3 11 3 13 3 8 4 10 4 12 4 > 16 a z + 29 a z + 3 a z - 6 a z - 19 a z - 13 a z + 7 5 9 5 11 5 13 5 8 6 10 6 12 6 > 6 a z - 13 a z - 20 a z - a z + 5 a z + 11 a z + 6 a z - 7 7 9 7 11 7 8 8 10 8 12 8 9 9 11 9 > a z + 6 a z + 7 a z - a z - 2 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 1 2 2 2 1 2 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 24 8 22 8 22 7 18 6 18 5 16 4 14 4 q t q t q t q t q t q t q t 1 1 1 1 1 > ------ + ------ + ------ + ------ + ---- 14 3 12 3 12 2 10 2 8 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n220

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