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L11n213 L11n215

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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X9,18,10,19 X17,22,18,9 X21,1,22,8 X20,13,21,14 X5,14,6,15 X7,16,8,17 X15,6,16,7 X4,20,5,19

Gauss Code:

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

Jones Polynomial:

q-23/2 - 3q-21/2 + 6q-19/2 - 9q-17/2 + 10q-15/2 - 10q-13/2 + 9q-11/2 - 7q-9/2 + 3q-7/2 - 2q-5/2

A2 (sl(3)) Invariant:

- q-34 + q-32 - 2q-30 + q-26 - q-24 + 3q-22 - 2q-20 + 2q-18 + q-16 + q-14 + 4q-12 + 2q-8

HOMFLY-PT Polynomial:

- 2a5z-1 - 9a5z - 8a5z3 - 2a5z5 + 3a7z-1 + 10a7z + 10a7z3 + 5a7z5 + a7z7 - a9z-1 - 3a9z - 3a9z3 - a9z5

Kauffman Polynomial:

2a5z-1 - 10a5z + 10a5z3 - 3a5z5 - 3a6 + 7a6z2 - 4a6z4 + 2a6z6 - a6z8 + 3a7z-1 - 14a7z + 20a7z3 - 9a7z5 + 2a7z7 - a7z9 - 3a8 + 11a8z2 - 8a8z4 + 7a8z6 - 4a8z8 + a9z-1 - 4a9z + 6a9z3 + 2a9z5 - 3a9z7 - a9z9 - a10 + 3a10z4 - 3a10z8 - a11z - a11z3 + 5a11z5 - 5a11z7 - 3a12z2 + 6a12z4 - 5a12z6 - a13z + 3a13z3 - 3a13z5 + a14z2 - 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                 2 1 j = -8               5 1   j = -10             4 2     j = -12           6 5       j = -14         5 5         j = -16       4 5           j = -18     2 5             j = -20   1 4               j = -22   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, 214]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 214]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[9, 18, 10, 19], > X[17, 22, 18, 9], X[21, 1, 22, 8], X[20, 13, 21, 14], X[5, 14, 6, 15], > X[7, 16, 8, 17], X[15, 6, 16, 7], X[4, 20, 5, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, -8, 10, -9, 6}, > {-4, -1, 2, -3, 7, 8, -10, 9, -5, 4, 11, -7, -6, 5}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) 3 6 9 10 10 9 7 3 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 -32 2 -26 -24 3 2 2 -16 -14 4 2 -q + q - --- + q - q + --- - --- + --- + q + q + --- + -- 30 22 20 18 12 8 q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 214]][a, z]
Out[8]=	5 7 9 -2 a 3 a a 5 7 9 5 3 7 3 9 3 ----- + ---- - -- - 9 a z + 10 a z - 3 a z - 8 a z + 10 a z - 3 a z - z z z 5 5 7 5 9 5 7 7 > 2 a z + 5 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 214]][a, z]
Out[9]=	5 7 9 6 8 10 2 a 3 a a 5 7 9 11 -3 a - 3 a - a + ---- + ---- + -- - 10 a z - 14 a z - 4 a z - a z - z z z 13 6 2 8 2 12 2 14 2 5 3 7 3 > a z + 7 a z + 11 a z - 3 a z + a z + 10 a z + 20 a z + 9 3 11 3 13 3 6 4 8 4 10 4 12 4 > 6 a z - a z + 3 a z - 4 a z - 8 a z + 3 a z + 6 a z - 14 4 5 5 7 5 9 5 11 5 13 5 6 6 > a z - 3 a z - 9 a z + 2 a z + 5 a z - 3 a z + 2 a z + 8 6 12 6 7 7 9 7 11 7 6 8 8 8 > 7 a z - 5 a z + 2 a z - 3 a z - 5 a z - a z - 4 a z - 10 8 7 9 9 9 > 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 2 1 2 1 4 2 5 4 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 5 5 5 6 5 4 2 5 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 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 2 > ---- + ---- 8 6 q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n214

L11n213 L11n215