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L11n238 L11n240

Knotscape

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DrawMorseLink

PD Presentation:

X12,1,13,2 X7,16,8,17 X5,1,6,10 X3746 X9,5,10,4 X18,14,19,13 X22,20,11,19 X20,15,21,16 X14,21,15,22 X2,11,3,12 X17,8,18,9

Gauss Code:

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

Jones Polynomial:

- q-11/2 + q-9/2 - 2q-7/2 + q-5/2 - q-1/2 + q1/2 - 2q3/2 + 2q5/2 - 2q7/2 + q9/2

A2 (sl(3)) Invariant:

q-18 + 2q-16 + 2q-14 + 2q-12 - 2q-6 - q-4 + q-2 + 1 + 3q2 + q4 + q6 - q10 - q14

HOMFLY-PT Polynomial:

a-3z-1 + 2a-3z + a-3z3 - 4a-1z-1 - 8a-1z - 5a-1z3 - a-1z5 + 6az-1 + 10az + 6az3 + az5 - 5a3z-1 - 7a3z - 2a3z3 + 2a5z-1 + a5z

Kauffman Polynomial:

a-4 - 3a-4z2 + 4a-4z4 - a-4z6 - a-3z-1 + 4a-3z - 9a-3z3 + 9a-3z5 - 2a-3z7 + 3a-2 - 9a-2z2 + 4a-2z4 + 3a-2z6 - a-2z8 - 4a-1z-1 + 18a-1z - 29a-1z3 + 18a-1z5 - 3a-1z7 + 3 - 7z2 - z4 + 5z6 - z8 - 6az-1 + 28az - 42az3 + 22az5 - 3az7 + a2 - a2z2 - 6a2z4 + 6a2z6 - a2z8 - 5a3z-1 + 22a3z - 33a3z3 + 19a3z5 - 3a3z7 + a4 - 5a4z4 + 5a4z6 - a4z8 - 2a5z-1 + 8a5z - 11a5z3 + 6a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                       1 j = 8                     1   j = 6                   1 1   j = 4               1 2 1     j = 2             1 1 1       j = 0           1 3 2         j = -2         1 2 2           j = -4       1 1 1             j = -6     1 1 1               j = -8   1 2                   j = -10                         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, 239]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 239]]
Out[4]=	PD[X[12, 1, 13, 2], X[7, 16, 8, 17], X[5, 1, 6, 10], X[3, 7, 4, 6], > X[9, 5, 10, 4], X[18, 14, 19, 13], X[22, 20, 11, 19], X[20, 15, 21, 16], > X[14, 21, 15, 22], X[2, 11, 3, 12], X[17, 8, 18, 9]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -4, 5, -3, 4, -2, 11, -5, 3}, > {10, -1, 6, -9, 8, 2, -11, -6, 7, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) -(9/2) 2 -(5/2) 1 3/2 5/2 -q + q - ---- + q - ------- + Sqrt[q] - 2 q + 2 q - 7/2 Sqrt[q] q 7/2 9/2 > 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 2 2 2 -4 -2 2 4 6 10 14 1 + q + --- + --- + --- - -- - q + q + 3 q + q + q - q - q 16 14 12 6 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 239]][a, z]
Out[8]=	3 5 3 1 4 6 a 5 a 2 a 2 z 8 z 3 5 z ---- - --- + --- - ---- + ---- + --- - --- + 10 a z - 7 a z + a z + -- - 3 a z z z z 3 a 3 a z a a 3 5 5 z 3 3 3 z 5 > ---- + 6 a z - 2 a z - -- + a z a a
In[9]:=	Kauffman[Link[11, NonAlternating, 239]][a, z]
Out[9]=	3 5 -4 3 2 4 1 4 6 a 5 a 2 a 4 z 18 z 3 + a + -- + a + a - ---- - --- - --- - ---- - ---- + --- + ---- + 28 a z + 2 3 a z z z z 3 a a a z a 2 2 3 3 3 5 2 3 z 9 z 2 2 9 z 29 z 3 > 22 a z + 8 a z - 7 z - ---- - ---- - a z - ---- - ----- - 42 a z - 4 2 3 a a a a 4 4 5 5 3 3 5 3 4 4 z 4 z 2 4 4 4 9 z 18 z > 33 a z - 11 a z - z + ---- + ---- - 6 a z - 5 a z + ---- + ----- + 4 2 3 a a a a 6 6 5 3 5 5 5 6 z 3 z 2 6 4 6 > 22 a z + 19 a z + 6 a z + 5 z - -- + ---- + 6 a z + 5 a z - 4 2 a a 7 7 8 2 z 3 z 7 3 7 5 7 8 z 2 8 4 8 > ---- - ---- - 3 a z - 3 a z - a z - z - -- - a z - a z 3 a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 2 1 1 1 1 1 3 + -- + q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ----- + 2 12 6 8 5 8 4 6 4 6 3 4 3 6 2 4 2 q q t q t q t q t q t q t q t q t 1 1 1 2 2 4 2 2 4 2 4 3 > ----- + - + ---- + ---- + 2 t + q t + q t + q t + 2 q t + q t + 2 2 t 4 2 q t q t q t 6 3 6 4 8 4 10 5 > q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n239

L11n238 L11n240