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DrawMorseLink

PD Presentation:

X6172 X20,7,21,8 X4,21,1,22 X5,12,6,13 X8493 X9,16,10,17 X13,19,14,18 X17,15,18,14 X15,10,16,11 X11,22,12,5 X2,20,3,19

Gauss Code:

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

Jones Polynomial:

q-15/2 - 2q-13/2 + 3q-11/2 - 3q-9/2 + 2q-7/2 - 2q-5/2 - 2q1/2 + 2q3/2 - q5/2

A2 (sl(3)) Invariant:

- q-24 - q-22 - q-18 + q-16 + q-12 + 2q-10 + q-8 + 3q-6 + q-2 + 1 + q4 + q8

HOMFLY-PT Polynomial:

- a-1z-1 - 2a-1z - a-1z3 + 2az-1 + 6az + 5az3 + az5 - 3a3z-1 - 8a3z - 5a3z3 - a3z5 + 3a5z-1 + 5a5z + 2a5z3 - a7z-1 - a7z

Kauffman Polynomial:

- a-1z-1 + 4a-1z - 6a-1z3 + 5a-1z5 - a-1z7 + 5z2 - 14z4 + 11z6 - 2z8 - 2az-1 + 13az - 21az3 + 5az5 + 4az7 - az9 - 2a2 + 13a2z2 - 32a2z4 + 20a2z6 - 3a2z8 - 3a3z-1 + 18a3z - 27a3z3 + 7a3z5 + 4a3z7 - a3z9 + 7a4z2 - 17a4z4 + 12a4z6 - 2a4z8 - 3a5z-1 + 12a5z - 19a5z3 + 15a5z5 - 3a5z7 + 2a6 - 5a6z2 + 5a6z4 + 2a6z6 - a6z8 - a7z-1 + 3a7z - 7a7z3 + 8a7z5 - 2a7z7 + a8 - 4a8z2 + 4a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 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                   1 1   j = 0               3 2 1     j = -2             2 3 1       j = -4           2 2 2         j = -6         2 3 1           j = -8       2 2 1             j = -10     1 2 1               j = -12   1 2                   j = -14   1                     j = -16 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, 39]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 39]]
Out[4]=	PD[X[6, 1, 7, 2], X[20, 7, 21, 8], X[4, 21, 1, 22], X[5, 12, 6, 13], > X[8, 4, 9, 3], X[9, 16, 10, 17], X[13, 19, 14, 18], X[17, 15, 18, 14], > X[15, 10, 16, 11], X[11, 22, 12, 5], X[2, 20, 3, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {-4, -1, 2, -5, -6, 9, -10, 4, -7, 8, -9, 6, -8, 7, > 11, -2, 3, 10}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 2 3 3 2 2 3/2 5/2 q - ----- + ----- - ---- + ---- - ---- - 2 Sqrt[q] + 2 q - q 13/2 11/2 9/2 7/2 5/2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 -18 -16 -12 2 -8 3 -2 4 8 1 - q - q - q + q + q + --- + q + -- + q + q + q 10 6 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 39]][a, z]
Out[8]=	3 5 7 3 1 2 a 3 a 3 a a 2 z 3 5 7 z -(---) + --- - ---- + ---- - -- - --- + 6 a z - 8 a z + 5 a z - a z - -- + a z z z z z a a 3 3 3 5 3 5 3 5 > 5 a z - 5 a z + 2 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 39]][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 - --- - --- - ---- - ---- - -- + --- + 13 a z + 18 a z + a z z z z z a 3 5 7 2 2 2 4 2 6 2 8 2 6 z > 12 a z + 3 a z + 5 z + 13 a z + 7 a z - 5 a z - 4 a z - ---- - a 3 3 3 5 3 7 3 4 2 4 4 4 > 21 a z - 27 a z - 19 a z - 7 a z - 14 z - 32 a z - 17 a z + 5 6 4 8 4 5 z 5 3 5 5 5 7 5 6 > 5 a z + 4 a z + ---- + 5 a z + 7 a z + 15 a z + 8 a z + 11 z + a 7 2 6 4 6 6 6 8 6 z 7 3 7 5 7 > 20 a z + 12 a z + 2 a z - a z - -- + 4 a z + 4 a z - 3 a z - a 7 7 8 2 8 4 8 6 8 9 3 9 > 2 a z - 2 z - 3 a z - 2 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 1 1 1 2 1 2 2 3 + -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 4 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 q q q t q t q t q t q t q t q t 1 2 2 1 3 2 1 2 2 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + ---- + 2 t + 10 3 8 3 6 3 8 2 6 2 4 2 6 4 2 q t q t q t q t q t q t q t q t q t t 2 2 2 2 3 4 3 6 4 > -- + t + q t + q t + q t + q t 2 q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n39

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