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L11n212 L11n214

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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X9,18,10,19 X17,22,18,9 X21,1,22,8 X20,15,21,16 X7,16,8,17 X4,13,5,14 X6,20,7,19

Gauss Code:

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

Jones Polynomial:

q-23/2 - 2q-21/2 + 3q-19/2 - 5q-17/2 + 6q-15/2 - 6q-13/2 + 5q-11/2 - 4q-9/2 + 2q-7/2 - 2q-5/2

A2 (sl(3)) Invariant:

- q-34 - q-30 + q-26 + 2q-22 - q-20 + q-18 + q-16 + q-14 + 3q-12 + q-10 + 2q-8

HOMFLY-PT Polynomial:

- 2a5z-1 - 10a5z - 9a5z3 - 2a5z5 + 3a7z-1 + 12a7z + 13a7z3 + 6a7z5 + a7z7 - a9z-1 - 4a9z - 4a9z3 - a9z5

Kauffman Polynomial:

2a5z-1 - 11a5z + 12a5z3 - 3a5z5 - 3a6 + 9a6z2 - 7a6z4 + 4a6z6 - a6z8 + 3a7z-1 - 15a7z + 23a7z3 - 14a7z5 + 5a7z7 - a7z9 - 3a8 + 17a8z2 - 23a8z4 + 13a8z6 - 3a8z8 + a9z-1 - 3a9z + 3a9z3 - 5a9z5 + 3a9z7 - a9z9 - a10 + 5a10z2 - 12a10z4 + 7a10z6 - 2a10z8 - 4a11z3 + 4a11z5 - 2a11z7 - a12z2 + 3a12z4 - 2a12z6 - a13z + 4a13z3 - 2a13z5 + 2a14z2 - a14z4

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -4                   2 j = -6                 1 1 j = -8               3 1   j = -10             2 1     j = -12           4 3       j = -14         3 3         j = -16       2 3           j = -18     1 3             j = -20   1 2               j = -22   1                 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, 213]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 213]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[14, 5, 15, 6], > X[9, 18, 10, 19], X[17, 22, 18, 9], X[21, 1, 22, 8], X[20, 15, 21, 16], > X[7, 16, 8, 17], X[4, 13, 5, 14], X[6, 20, 7, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -10, 4, -11, -9, 7}, > {-5, -1, 2, -3, 10, -4, 8, 9, -6, 5, 11, -8, -7, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) 2 3 5 6 6 5 4 2 2 q - ----- + ----- - ----- + ----- - ----- + ----- - ---- + ---- - ---- 21/2 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 -30 -26 2 -20 -18 -16 -14 3 -10 2 -q - q + q + --- - q + q + q + q + --- + q + -- 22 12 8 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 213]][a, z]
Out[8]=	5 7 9 -2 a 3 a a 5 7 9 5 3 7 3 9 3 ----- + ---- - -- - 10 a z + 12 a z - 4 a z - 9 a z + 13 a z - 4 a z - z z z 5 5 7 5 9 5 7 7 > 2 a z + 6 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 213]][a, z]
Out[9]=	5 7 9 6 8 10 2 a 3 a a 5 7 9 13 -3 a - 3 a - a + ---- + ---- + -- - 11 a z - 15 a z - 3 a z - a z + z z z 6 2 8 2 10 2 12 2 14 2 5 3 7 3 > 9 a z + 17 a z + 5 a z - a z + 2 a z + 12 a z + 23 a z + 9 3 11 3 13 3 6 4 8 4 10 4 12 4 > 3 a z - 4 a z + 4 a z - 7 a z - 23 a z - 12 a z + 3 a z - 14 4 5 5 7 5 9 5 11 5 13 5 6 6 > a z - 3 a z - 14 a z - 5 a z + 4 a z - 2 a z + 4 a z + 8 6 10 6 12 6 7 7 9 7 11 7 6 8 > 13 a z + 7 a z - 2 a z + 5 a z + 3 a z - 2 a z - a z - 8 8 10 8 7 9 9 9 > 3 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 2 1 1 1 2 1 3 2 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 24 9 22 8 20 8 20 7 18 7 18 6 16 6 q q t q t q t q t q t q t q t 3 3 3 4 3 2 1 3 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 16 5 14 5 14 4 12 4 12 3 10 3 10 2 8 2 q t q t q t q t q t q t q t q t 1 1 > ---- + ---- 8 6 q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n213

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