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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X9,14,10,15 X8493 X5,11,6,10 X11,20,12,21 X19,22,20,5 X13,19,14,18 X21,12,22,13 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

q-13/2 - q-11/2 + 2q-9/2 - 4q-7/2 + 3q-5/2 - 4q-3/2 + 3q-1/2 - 3q1/2 + 2q3/2 - q5/2

A2 (sl(3)) Invariant:

- q-22 - 2q-20 - q-18 + q-14 + 3q-12 + 2q-10 + 2q-8 + q-6 + q-2 + q2 + q4 + q8

HOMFLY-PT Polynomial:

- a-1z-1 - 2a-1z - a-1z3 + 2az-1 + 5az + 4az3 + az5 - 3a3z-1 - 5a3z - 2a3z3 + 3a5z-1 + 2a5z - a7z-1

Kauffman Polynomial:

- a-1z-1 + 4a-1z - 7a-1z3 + 5a-1z5 - a-1z7 + 5z2 - 13z4 + 10z6 - 2z8 - 2az-1 + 11az - 20az3 + 9az5 + 2az7 - az9 - 2a2 + 12a2z2 - 27a2z4 + 20a2z6 - 4a2z8 - 3a3z-1 + 12a3z - 18a3z3 + 8a3z5 + 2a3z7 - a3z9 + 8a4z2 - 15a4z4 + 10a4z6 - 2a4z8 - 3a5z-1 + 6a5z - 6a5z3 + 4a5z5 - a5z7 + 2a6 - a6z4 - a7z-1 + a7z - a7z3 + a8 - a8z2

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 6                   1 j = 4                 1   j = 2               2 1   j = 0           1 2 1     j = -2         1 4 2       j = -4       1 3 3         j = -6       3 2           j = -8   1 2 3             j = -10   1 2               j = -12 1 1                 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, 9]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 9]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[9, 14, 10, 15], > X[8, 4, 9, 3], X[5, 11, 6, 10], X[11, 20, 12, 21], X[19, 22, 20, 5], > X[13, 19, 14, 18], 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, -9, 4, 11, -2, 3, 9, > -8, 7, -10, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) -(11/2) 2 4 3 4 3 q - q + ---- - ---- + ---- - ---- + ------- - 3 Sqrt[q] + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 3/2 5/2 > 2 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 -18 -14 3 2 2 -6 -2 2 4 8 -q - --- - q + q + --- + --- + -- + q + q + q + q + q 20 12 10 8 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 9]][a, z]
Out[8]=	3 5 7 3 1 2 a 3 a 3 a a 2 z 3 5 z 3 -(---) + --- - ---- + ---- - -- - --- + 5 a z - 5 a z + 2 a z - -- + 4 a z - a z z z z z a a 3 3 5 > 2 a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 9]][a, z]
Out[9]=	3 5 7 2 6 8 1 2 a 3 a 3 a a 4 z 3 -2 a + 2 a + a - --- - --- - ---- - ---- - -- + --- + 11 a z + 12 a z + a z z z z z a 3 5 7 2 2 2 4 2 8 2 7 z 3 > 6 a z + a z + 5 z + 12 a z + 8 a z - a z - ---- - 20 a z - a 5 3 3 5 3 7 3 4 2 4 4 4 6 4 5 z > 18 a z - 6 a z - a z - 13 z - 27 a z - 15 a z - a z + ---- + a 7 5 3 5 5 5 6 2 6 4 6 z 7 > 9 a z + 8 a z + 4 a z + 10 z + 20 a z + 10 a z - -- + 2 a z + a 3 7 5 7 8 2 8 4 8 9 3 9 > 2 a z - a z - 2 z - 4 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 1 1 1 1 1 2 2 1 + -- + -- + ------ + ------ + ------ + ------ + ----- + ------ + ----- + 4 2 14 5 12 5 12 4 10 4 8 4 10 3 8 3 q q q t q t q t q t q t q t q t 3 3 1 2 3 1 2 t 2 2 2 > ----- + ----- + ----- + ---- + ---- + ---- + 2 t + --- + t + 2 q t + 8 2 6 2 4 2 6 4 2 2 q t q t q t q t q t q t q 2 3 4 3 6 4 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n9

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