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L11n209 L11n211

Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X5,14,6,15 X22,18,9,17 X4,19,5,20 X21,6,22,7 X16,7,17,8 X8,9,1,10 X18,14,19,13 X15,21,16,20

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-28 + q-26 + 2q-24 + q-22 + q-18 + 2q-14 + q-12 + q-10 + q-8 - q-6 - q-2

HOMFLY-PT Polynomial:

a3z-1 + 4a3z + 4a3z3 + a3z5 - 3a5z-1 - 15a5z - 17a5z3 - 7a5z5 - a5z7 + 2a7z-1 + 7a7z + 5a7z3 + a7z5

Kauffman Polynomial:

- a2 + 3a2z2 - a2z4 + a3z-1 - 4a3z + 6a3z3 - 2a3z5 - 3a4 + 14a4z2 - 15a4z4 + 6a4z6 - a4z8 + 3a5z-1 - 18a5z + 28a5z3 - 21a5z5 + 7a5z7 - a5z9 - 3a6 + 15a6z2 - 23a6z4 + 11a6z6 - 2a6z8 + 2a7z-1 - 10a7z + 16a7z3 - 15a7z5 + 6a7z7 - a7z9 + 3a8z2 - 6a8z4 + 4a8z6 - a8z8 + a9z - 2a9z3 + 3a9z5 - a9z7 - a10z2 + 3a10z4 - a10z6 - 3a11z + 4a11z3 - 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                 1   j = -4               2 2   j = -6             2       j = -8           2 2       j = -10         2 2         j = -12       1 2           j = -14     1 2             j = -16     1               j = -18 1 1                 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, 210]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 210]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[5, 14, 6, 15], > X[22, 18, 9, 17], X[4, 19, 5, 20], X[21, 6, 22, 7], X[16, 7, 17, 8], > X[8, 9, 1, 10], X[18, 14, 19, 13], X[15, 21, 16, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -6, -4, 7, 8, -9}, > {9, -1, 2, -3, 10, 4, -11, -8, 5, -10, 6, 11, -7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) -(17/2) 2 3 4 4 4 2 2 -q + q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q 1 > ------- Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -26 2 -22 -18 2 -12 -10 -8 -6 -2 q + q + --- + q + q + --- + q + q + q - q - q 24 14 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 210]][a, z]
Out[8]=	3 5 7 a 3 a 2 a 3 5 7 3 3 5 3 7 3 -- - ---- + ---- + 4 a z - 15 a z + 7 a z + 4 a z - 17 a z + 5 a z + z z z 3 5 5 5 7 5 5 7 > a z - 7 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 210]][a, z]
Out[9]=	3 5 7 2 4 6 a 3 a 2 a 3 5 7 9 -a - 3 a - 3 a + -- + ---- + ---- - 4 a z - 18 a z - 10 a z + a z - z z z 11 2 2 4 2 6 2 8 2 10 2 3 3 > 3 a z + 3 a z + 14 a z + 15 a z + 3 a z - a z + 6 a z + 5 3 7 3 9 3 11 3 2 4 4 4 6 4 > 28 a z + 16 a z - 2 a z + 4 a z - a z - 15 a z - 23 a z - 8 4 10 4 3 5 5 5 7 5 9 5 11 5 > 6 a z + 3 a z - 2 a z - 21 a z - 15 a z + 3 a z - a z + 4 6 6 6 8 6 10 6 5 7 7 7 9 7 4 8 > 6 a z + 11 a z + 4 a z - a z + 7 a z + 6 a z - a z - a z - 6 8 8 8 5 9 7 9 > 2 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 -2 1 1 1 1 1 2 1 -- + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 20 8 18 8 18 7 16 6 14 6 14 5 12 5 q q t q t q t q t q t q t q t 2 2 2 2 2 2 2 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + t 12 4 10 4 10 3 8 3 8 2 6 2 4 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 L11n210

L11n209 L11n211