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L11n211 L11n213

Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X13,21,14,20 X3,12,4,13 X2,19,3,20 X14,5,15,6 X16,7,17,8 X8,9,1,10 X18,12,19,11 X6,15,7,16 X22,18,9,17 X21,4,22,5

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 2q-17/2 - 4q-15/2 + 5q-13/2 - 7q-11/2 + 7q-9/2 - 6q-7/2 + 4q-5/2 - 3q-3/2 + q-1/2

A2 (sl(3)) Invariant:

q-28 + 2q-24 + q-22 + q-20 + 3q-18 + 2q-14 - q-12 + q-8 - q-6 + q-4 - q-2

HOMFLY-PT Polynomial:

a3z-1 + 2a3z + 3a3z3 + a3z5 - 3a5z-1 - 11a5z - 13a5z3 - 6a5z5 - a5z7 + 2a7z-1 + 5a7z + 4a7z3 + a7z5

Kauffman Polynomial:

- a2 + 2a2z2 - a2z4 + a3z-1 - 2a3z + 6a3z3 - 3a3z5 - 3a4 + 13a4z2 - 12a4z4 + 4a4z6 - a4z8 + 3a5z-1 - 14a5z + 23a5z3 - 17a5z5 + 5a5z7 - a5z9 - 3a6 + 12a6z2 - 20a6z4 + 11a6z6 - 3a6z8 + 2a7z-1 - 6a7z + 3a7z3 - 3a7z5 + 2a7z7 - a7z9 - 4a8z4 + 5a8z6 - 2a8z8 + 5a9z - 11a9z3 + 10a9z5 - 3a9z7 - a10z2 + 5a10z4 - 2a10z6 - a11z + 3a11z3 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 j = 0                   1 j = -2                 2   j = -4               3 2   j = -6             3 1     j = -8           4 3       j = -10         3 3         j = -12       2 4           j = -14     2 3             j = -16     2               j = -18 1 2                 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, 212]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 212]]
Out[4]=	PD[X[10, 1, 11, 2], X[13, 21, 14, 20], X[3, 12, 4, 13], X[2, 19, 3, 20], > X[14, 5, 15, 6], X[16, 7, 17, 8], X[8, 9, 1, 10], X[18, 12, 19, 11], > X[6, 15, 7, 16], X[22, 18, 9, 17], X[21, 4, 22, 5]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, -3, 11, 5, -9, 6, -7}, > {7, -1, 8, 3, -2, -5, 9, -6, 10, -8, 4, 2, -11, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 2 4 5 7 7 6 4 3 1 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 2 -22 -20 3 2 -12 -8 -6 -4 -2 q + --- + q + q + --- + --- - q + q - q + q - q 24 18 14 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 212]][a, z]
Out[8]=	3 5 7 a 3 a 2 a 3 5 7 3 3 5 3 7 3 -- - ---- + ---- + 2 a z - 11 a z + 5 a z + 3 a z - 13 a z + 4 a z + z z z 3 5 5 5 7 5 5 7 > a z - 6 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 212]][a, z]
Out[9]=	3 5 7 2 4 6 a 3 a 2 a 3 5 7 9 -a - 3 a - 3 a + -- + ---- + ---- - 2 a z - 14 a z - 6 a z + 5 a z - z z z 11 2 2 4 2 6 2 10 2 3 3 5 3 > a z + 2 a z + 13 a z + 12 a z - a z + 6 a z + 23 a z + 7 3 9 3 11 3 2 4 4 4 6 4 8 4 > 3 a z - 11 a z + 3 a z - a z - 12 a z - 20 a z - 4 a z + 10 4 3 5 5 5 7 5 9 5 11 5 4 6 > 5 a z - 3 a z - 17 a z - 3 a z + 10 a z - a z + 4 a z + 6 6 8 6 10 6 5 7 7 7 9 7 4 8 > 11 a z + 5 a z - 2 a z + 5 a z + 2 a z - 3 a z - a z - 6 8 8 8 5 9 7 9 > 3 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 2 2 2 3 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 8 18 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 4 3 3 4 3 3 1 3 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + t 12 4 10 4 10 3 8 3 8 2 6 2 6 4 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 L11n212

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