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L11n250 L11n252

Knotscape

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DrawMorseLink

PD Presentation:

X12,1,13,2 X2,13,3,14 X8394 X11,17,12,16 X14,8,15,7 X6,16,7,15 X17,11,18,22 X4,20,5,19 X18,6,19,5 X20,9,21,10 X10,21,1,22

Gauss Code:

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

Jones Polynomial:

q-11/2 - 3q-9/2 + 7q-7/2 - 11q-5/2 + 11q-3/2 - 13q-1/2 + 11q1/2 - 8q3/2 + 5q5/2 - 2q7/2

A2 (sl(3)) Invariant:

- q-16 + q-14 - 3q-12 + q-10 + 3q-8 + 5q-4 - q-2 + 3 - q4 + 2q6 - 2q8 + q10 + q12

HOMFLY-PT Polynomial:

- a-3z - a-1z-1 - 2a-1z - 2a-1z3 - a-1z5 + az-1 + 7az + 10az3 + 5az5 + az7 - 4a3z - 3a3z3 - a3z5

Kauffman Polynomial:

- 2a-3z + 6a-3z3 - 3a-3z5 - 6a-2z2 + 9a-2z4 - 2a-2z6 - a-2z8 - a-1z-1 + 4a-1z - 5a-1z3 + 9a-1z5 - 3a-1z7 - a-1z9 + 1 - 8z2 + 7z4 + 7z6 - 6z8 - az-1 + 14az - 29az3 + 30az5 - 11az7 - az9 - 7a2z2 + 6a2z4 + 3a2z6 - 5a2z8 + 8a3z - 16a3z3 + 15a3z5 - 8a3z7 - 4a4z2 + 7a4z4 - 6a4z6 + 2a5z3 - 3a5z5 + a6z2 - a6z4

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                   2 j = 6                 3   j = 4               5 2   j = 2             6 3     j = 0           7 5       j = -2         6 8         j = -4       5 5           j = -6     2 6             j = -8   1 5               j = -10   2                 j = -12 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, 251]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 251]]
Out[4]=	PD[X[12, 1, 13, 2], X[2, 13, 3, 14], X[8, 3, 9, 4], X[11, 17, 12, 16], > X[14, 8, 15, 7], X[6, 16, 7, 15], X[17, 11, 18, 22], X[4, 20, 5, 19], > X[18, 6, 19, 5], X[20, 9, 21, 10], X[10, 21, 1, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -8, 9, -6, 5, -3, 10, -11}, > {-4, -1, 2, -5, 6, 4, -7, -9, 8, -10, 11, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 3 7 11 11 13 3/2 5/2 q - ---- + ---- - ---- + ---- - ------- + 11 Sqrt[q] - 8 q + 5 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 7/2 > 2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 3 -10 3 5 -2 4 6 8 10 12 3 - q + q - --- + q + -- + -- - q - q + 2 q - 2 q + q + q 12 8 4 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 251]][a, z]
Out[8]=	3 5 1 a z 2 z 3 2 z 3 3 3 z -(---) + - - -- - --- + 7 a z - 4 a z - ---- + 10 a z - 3 a z - -- + a z z 3 a a a a 5 3 5 7 > 5 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 251]][a, z]
Out[9]=	2 1 a 2 z 4 z 3 2 6 z 2 2 4 2 1 - --- - - - --- + --- + 14 a z + 8 a z - 8 z - ---- - 7 a z - 4 a z + a z z 3 a 2 a a 3 3 4 6 2 6 z 5 z 3 3 3 5 3 4 9 z > a z + ---- - ---- - 29 a z - 16 a z + 2 a z + 7 z + ---- + 3 a 2 a a 5 5 2 4 4 4 6 4 3 z 9 z 5 3 5 5 5 > 6 a z + 7 a z - a z - ---- + ---- + 30 a z + 15 a z - 3 a z + 3 a a 6 7 8 6 2 z 2 6 4 6 3 z 7 3 7 8 z > 7 z - ---- + 3 a z - 6 a z - ---- - 11 a z - 8 a z - 6 z - -- - 2 a 2 a a 9 2 8 z 9 > 5 a z - -- - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	8 1 2 1 5 2 6 5 5 7 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 4 q q t q t q t q t q t q t q t q t 6 2 2 2 4 2 4 3 6 3 8 4 > ---- + 5 t + 6 q t + 3 q t + 5 q t + 2 q t + 3 q t + 2 q t 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n251

L11n250 L11n252