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L11n201 L11n203

Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X16,8,17,7 X11,18,12,19 X19,3,20,2 X3,12,4,13 X13,21,14,20 X5,15,6,14 X6,9,7,10 X15,22,16,9 X8,18,1,17 X21,4,22,5

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 2q-9/2 - 2q-7/2 + 3q-5/2 - 4q-3/2 + 3q-1/2 - 4q1/2 + 2q3/2 - 2q5/2 + q7/2

A2 (sl(3)) Invariant:

q-16 - q-10 - q-8 + 3q-2 + 3 + 3q2 + 2q4 - q10

HOMFLY-PT Polynomial:

- a-1z-1 + 3a-1z + 4a-1z3 + a-1z5 + az-1 - 6az - 11az3 - 6az5 - az7 + 3a3z + 4a3z3 + a3z5

Kauffman Polynomial:

- a-4z2 + a-3z - 2a-3z3 + a-2z2 - 2a-2z4 - a-1z-1 - 2a-1z + 6a-1z3 - 3a-1z5 + 1 + 6z2 - 7z4 + 4z6 - z8 - az-1 - 6az + 15az3 - 11az5 + 5az7 - az9 + 11a2z2 - 22a2z4 + 15a2z6 - 3a2z8 - 2a3z + a3z3 - 3a3z5 + 4a3z7 - a3z9 + 7a4z2 - 17a4z4 + 11a4z6 - 2a4z8 + a5z - 6a5z3 + 5a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 8                   1 j = 6                 1   j = 4               1 1   j = 2             3 1     j = 0           2 3       j = -2         2 1         j = -4       1 2           j = -6     1 2             j = -8   1 1               j = -10   1                 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, 202]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 202]]
Out[4]=	PD[X[10, 1, 11, 2], X[16, 8, 17, 7], X[11, 18, 12, 19], X[19, 3, 20, 2], > X[3, 12, 4, 13], X[13, 21, 14, 20], X[5, 15, 6, 14], X[6, 9, 7, 10], > X[15, 22, 16, 9], X[8, 18, 1, 17], X[21, 4, 22, 5]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -5, 11, -7, -8, 2, -10}, > {8, -1, -3, 5, -6, 7, -9, -2, 10, 3, -4, 6, -11, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 2 2 3 4 3 3/2 5/2 -q + ---- - ---- + ---- - ---- + ------- - 4 Sqrt[q] + 2 q - 2 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 7/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -10 -8 3 2 4 10 3 + q - q - q + -- + 3 q + 2 q - q 2 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 202]][a, z]
Out[8]=	3 5 1 a 3 z 3 4 z 3 3 3 z 5 -(---) + - + --- - 6 a z + 3 a z + ---- - 11 a z + 4 a z + -- - 6 a z + a z z a a a 3 5 7 > a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 202]][a, z]
Out[9]=	2 2 1 a z 2 z 3 5 2 z z 2 2 1 - --- - - + -- - --- - 6 a z - 2 a z + a z + 6 z - -- + -- + 11 a z + a z z 3 a 4 2 a a a 3 3 4 4 2 2 z 6 z 3 3 3 5 3 4 2 z > 7 a z - ---- + ---- + 15 a z + a z - 6 a z - 7 z - ---- - 3 a 2 a a 5 2 4 4 4 3 z 5 3 5 5 5 6 > 22 a z - 17 a z - ---- - 11 a z - 3 a z + 5 a z + 4 z + a 2 6 4 6 7 3 7 5 7 8 2 8 4 8 > 15 a z + 11 a z + 5 a z + 4 a z - a z - z - 3 a z - 2 a z - 9 3 9 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 1 1 2 1 2 3 + 3 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 2 2 1 2 4 4 2 6 2 8 3 > ----- + - + ---- + q t + q t + q t + q t + q t 2 2 t 2 q t q t

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L11n201 L11n203