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L11n208 L11n210

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 X19,5,20,4 X6,22,7,21 X16,7,17,8 X8,9,1,10 X13,18,14,19 X20,15,21,16

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

A2 (sl(3)) Invariant:

q-28 + q-24 - q-22 - 2q-20 - q-16 + 3q-14 + 2q-12 + 3q-10 + 3q-8 + q-4 - q-2

HOMFLY-PT Polynomial:

- a3z-1 + a3z + 3a3z3 + a3z5 + a5z-1 - 7a5z - 12a5z3 - 6a5z5 - a5z7 + 4a7z + 4a7z3 + a7z5

Kauffman Polynomial:

2a2z2 - a2z4 - a3z-1 - 2a3z + 7a3z3 - 3a3z5 + a4 + 6a4z2 - 9a4z4 + 4a4z6 - a4z8 - a5z-1 - 6a5z + 15a5z3 - 14a5z5 + 5a5z7 - a5z9 + 12a6z2 - 24a6z4 + 13a6z6 - 3a6z8 - 3a7z + 3a7z3 - 5a7z5 + 3a7z7 - a7z9 + 5a8z2 - 10a8z4 + 7a8z6 - 2a8z8 - 2a9z3 + 5a9z5 - 2a9z7 - 3a10z2 + 6a10z4 - 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               2 2   j = -6             4 1     j = -8           2 2       j = -10         4 4         j = -12       2 3           j = -14     1 3             j = -16   1 2               j = -18   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, 209]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 209]]
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[19, 5, 20, 4], X[6, 22, 7, 21], X[16, 7, 17, 8], > X[8, 9, 1, 10], X[13, 18, 14, 19], X[20, 15, 21, 16]]
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) 2 3 5 6 6 6 3 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 -24 -22 2 -16 3 2 3 3 -4 -2 q + q - q - --- - q + --- + --- + --- + -- + q - q 20 14 12 10 8 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 209]][a, z]
Out[8]=	3 5 a a 3 5 7 3 3 5 3 7 3 3 5 -(--) + -- + a z - 7 a z + 4 a z + 3 a z - 12 a z + 4 a z + a z - z z 5 5 7 5 5 7 > 6 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 209]][a, z]
Out[9]=	3 5 4 a a 3 5 7 11 2 2 4 2 a - -- - -- - 2 a z - 6 a z - 3 a z - a z + 2 a z + 6 a z + z z 6 2 8 2 10 2 3 3 5 3 7 3 9 3 > 12 a z + 5 a z - 3 a z + 7 a z + 15 a z + 3 a z - 2 a z + 11 3 2 4 4 4 6 4 8 4 10 4 3 5 > 3 a z - a z - 9 a z - 24 a z - 10 a z + 6 a z - 3 a z - 5 5 7 5 9 5 11 5 4 6 6 6 8 6 > 14 a z - 5 a z + 5 a z - a z + 4 a z + 13 a z + 7 a z - 10 6 5 7 7 7 9 7 4 8 6 8 8 8 > 2 a z + 5 a z + 3 a z - 2 a z - a z - 3 a z - 2 a z - 5 9 7 9 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 1 2 1 3 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 7 16 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 3 4 4 2 2 4 1 2 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 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 L11n209

L11n208 L11n210