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L11n216 L11n218

Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X8,9,1,10 X3,12,4,13 X15,22,16,9 X17,3,18,2 X21,4,22,5 X5,15,6,14 X13,21,14,20 X11,16,12,17 X19,7,20,6 X7,19,8,18

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 3q-11/2 - 6q-9/2 + 10q-7/2 - 12q-5/2 + 12q-3/2 - 12q-1/2 + 8q1/2 - 6q3/2 + 2q5/2

A2 (sl(3)) Invariant:

q-20 - q-18 + 2q-14 - 3q-12 + q-10 - q-8 - q-6 + 3q-4 + 6 + q2 + q4 + 2q6 - 2q8

HOMFLY-PT Polynomial:

- a-1z-1 + 2a-1z + 2a-1z3 + az-1 - 3az - 5az3 - 2az5 - a3z3 - a3z5 + a5z + a5z3

Kauffman Polynomial:

3a-2z2 - 3a-2z4 - a-1z-1 - 4a-1z + 11a-1z3 - 5a-1z5 - a-1z7 + 1 + 8z2 - 9z4 + 5z6 - 3z8 - az-1 - 6az + 15az3 - 9az5 + 2az7 - 2az9 + 11a2z2 - 23a2z4 + 19a2z6 - 8a2z8 - 6a3z3 + 8a3z5 - 2a3z7 - 2a3z9 + 4a4z2 - 11a4z4 + 11a4z6 - 5a4z8 + 2a5z - 8a5z3 + 11a5z5 - 5a5z7 - 2a6z2 + 6a6z4 - 3a6z6 + 2a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 6                   2 j = 4                 4   j = 2               4 2   j = 0             8 4     j = -2           6 6       j = -4         6 6         j = -6       4 6           j = -8     2 6             j = -10   1 4               j = -12   2                 j = -14 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, 217]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 217]]
Out[4]=	PD[X[10, 1, 11, 2], X[8, 9, 1, 10], X[3, 12, 4, 13], X[15, 22, 16, 9], > X[17, 3, 18, 2], X[21, 4, 22, 5], X[5, 15, 6, 14], X[13, 21, 14, 20], > X[11, 16, 12, 17], X[19, 7, 20, 6], X[7, 19, 8, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 5, -3, 6, -7, 10, -11, -2}, > {2, -1, -9, 3, -8, 7, -4, 9, -5, 11, -10, 8, -6, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 3 6 10 12 12 12 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 8 Sqrt[q] - 6 q + 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 > 2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 2 3 -10 -8 -6 3 2 4 6 8 6 + q - q + --- - --- + q - q - q + -- + q + q + 2 q - 2 q 14 12 4 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 217]][a, z]
Out[8]=	3 1 a 2 z 5 2 z 3 3 3 5 3 5 3 5 -(---) + - + --- - 3 a z + a z + ---- - 5 a z - a z + a z - 2 a z - a z a z z a a
In[9]:=	Kauffman[Link[11, NonAlternating, 217]][a, z]
Out[9]=	2 1 a 4 z 5 2 3 z 2 2 4 2 1 - --- - - - --- - 6 a z + 2 a z + 8 z + ---- + 11 a z + 4 a z - a z z a 2 a 3 4 6 2 11 z 3 3 3 5 3 7 3 4 3 z > 2 a z + ----- + 15 a z - 6 a z - 8 a z + 2 a z - 9 z - ---- - a 2 a 5 2 4 4 4 6 4 5 z 5 3 5 5 5 > 23 a z - 11 a z + 6 a z - ---- - 9 a z + 8 a z + 11 a z - a 7 7 5 6 2 6 4 6 6 6 z 7 3 7 > a z + 5 z + 19 a z + 11 a z - 3 a z - -- + 2 a z - 2 a z - a 5 7 8 2 8 4 8 9 3 9 > 5 a z - 3 z - 8 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 1 2 1 4 2 6 4 6 8 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 6 6 6 2 2 2 4 2 6 3 > ----- + ---- + ---- + 4 t + 4 q t + 2 q t + 4 q t + 2 q t 4 2 4 2 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n217

L11n216 L11n218