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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X17,1,18,4 X9,14,10,15 X3849 X5,11,6,10 X11,19,12,18 X13,20,14,21 X19,5,20,22 X21,12,22,13 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-18 + q-16 + q-12 - q-10 - q-8 - q-4 + q-2 + 1 + 2q2 + 3q4 + q6 + 2q8 - q12

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-4z2 + 3a-3z - 3a-3z3 - a-2 + a-2z4 - a-2z6 - 2a-1z-1 + 13a-1z - 21a-1z3 + 13a-1z5 - 3a-1z7 - 2 + 11z2 - 18z4 + 13z6 - 3z8 - 4az-1 + 22az - 37az3 + 21az5 - az7 - az9 - 3a2 + 17a2z2 - 33a2z4 + 24a2z6 - 5a2z8 - 3a3z-1 + 16a3z - 26a3z3 + 13a3z5 + a3z7 - a3z9 - a4 + 7a4z2 - 14a4z4 + 10a4z6 - 2a4z8 - a5z-1 + 4a5z - 7a5z3 + 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                 2   j = 4               2 2   j = 2             4 2     j = 0           3 4 1     j = -2         3 4 1       j = -4       2 3           j = -6     1 3 1           j = -8   1 2               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, 5]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 5]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[17, 1, 18, 4], X[9, 14, 10, 15], > X[3, 8, 4, 9], X[5, 11, 6, 10], X[11, 19, 12, 18], X[13, 20, 14, 21], > X[19, 5, 20, 22], X[21, 12, 22, 13], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, -5, 3}, {-6, -1, 2, 5, -4, 6, -7, 10, -8, 4, 11, -2, -3, 7, > -9, 8, -10, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 2 3 4 5 5 3/2 5/2 -q + ---- - ---- + ---- - ---- + ------- - 5 Sqrt[q] + 3 q - 3 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]=	-18 -16 -12 -10 -8 -4 -2 2 4 6 8 12 1 + q + q + q - q - q - q + q + 2 q + 3 q + q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 5]][a, z]
Out[8]=	3 5 3 -2 4 a 3 a a z 4 z 3 5 2 z 3 --- + --- - ---- + -- + -- - --- + 7 a z - 5 a z + a z - ---- + 4 a z - a z z z z 3 a a a 3 3 5 > 2 a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 5]][a, z]
Out[9]=	3 5 -2 2 4 2 4 a 3 a a 3 z 13 z 3 -2 - a - 3 a - a - --- - --- - ---- - -- + --- + ---- + 22 a z + 16 a z + a z z z z 3 a a 2 3 3 5 2 z 2 2 4 2 3 z 21 z 3 > 4 a z + 11 z - -- + 17 a z + 7 a z - ---- - ----- - 37 a z - 4 3 a a a 4 5 3 3 5 3 4 z 2 4 4 4 13 z 5 > 26 a z - 7 a z - 18 z + -- - 33 a z - 14 a z + ----- + 21 a z + 2 a a 6 7 3 5 5 5 6 z 2 6 4 6 3 z 7 > 13 a z + 5 a z + 13 z - -- + 24 a z + 10 a z - ---- - a z + 2 a a 3 7 5 7 8 2 8 4 8 9 3 9 > a z - a z - 3 z - 5 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-2 2 1 1 1 2 1 3 2 4 + q + 4 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 q t q t q t q t q t q t q t 1 3 3 3 4 2 4 4 2 > ----- + ----- + ----- + - + ---- + t + 2 q t + 2 q t + 2 q t + 6 2 4 2 2 2 t 2 q t q t q t q t 6 2 8 3 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n5

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