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DrawMorseLink

PD Presentation:

X8192 X10,3,11,4 X15,20,16,21 X14,5,15,6 X4,13,5,14 X17,22,18,7 X21,16,22,17 X19,12,20,13 X11,18,12,19 X2738 X6,9,1,10

Gauss Code:

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

Jones Polynomial:

q-27/2 - q-25/2 + q-23/2 - q-21/2 - q-15/2 + q-13/2 - 2q-11/2 + q-9/2 - q-7/2

A2 (sl(3)) Invariant:

- 2q-42 - q-40 - q-38 - 2q-36 + q-34 + q-32 + 3q-30 + 2q-28 + 2q-26 + 2q-24 + q-20 + q-18 + q-16 + q-12

HOMFLY-PT Polynomial:

- 5a7z - 10a7z3 - 6a7z5 - a7z7 - 3a9z-1 - 7a9z - 10a9z3 - 6a9z5 - a9z7 + 5a11z-1 + 11a11z + 7a11z3 + a11z5 - 2a13z-1 - 2a13z

Kauffman Polynomial:

5a7z - 10a7z3 + 6a7z5 - a7z7 - 5a8z4 + 5a8z6 - a8z8 + 3a9z-1 - 10a9z + 12a9z3 - 11a9z5 + 6a9z7 - a9z9 - 5a10 + 17a10z2 - 24a10z4 + 13a10z6 - 2a10z8 + 5a11z-1 - 21a11z + 31a11z3 - 22a11z5 + 8a11z7 - a11z9 - 5a12 + 19a12z2 - 19a12z4 + 8a12z6 - a12z8 + 2a13z-1 - 4a13z + 3a13z3 - 4a14z2 + 5a14z4 - a14z6 + 2a15z - 6a15z3 + 5a15z5 - a15z7 + a16 - 6a16z2 + 5a16z4 - a16z6

Khovanov Homology:

trqj r = -11 r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -6                       1 j = -8                     1 1 j = -10                   1     j = -12               1 1 1     j = -14             1 2 1       j = -16           1 1 1         j = -18         1 3 2           j = -20       1                 j = -22       1 1               j = -24   1 1                   j = -26                         j = -28 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, 165]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 165]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[15, 20, 16, 21], X[14, 5, 15, 6], > X[4, 13, 5, 14], X[17, 22, 18, 7], X[21, 16, 22, 17], X[19, 12, 20, 13], > X[11, 18, 12, 19], X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -5, 4, -11}, > {10, -1, 11, -2, -9, 8, 5, -4, -3, 7, -6, 9, -8, 3, -7, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(27/2) -(25/2) -(23/2) -(21/2) -(15/2) -(13/2) 2 q - q + q - q - q + q - ----- + 11/2 q -(9/2) -(7/2) > q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 -40 -38 2 -34 -32 3 2 2 2 -20 -18 --- - q - q - --- + q + q + --- + --- + --- + --- + q + q + 42 36 30 28 26 24 q q q q q q -16 -12 > q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 165]][a, z]
Out[8]=	9 11 13 -3 a 5 a 2 a 7 9 11 13 7 3 ----- + ----- - ----- - 5 a z - 7 a z + 11 a z - 2 a z - 10 a z - z z z 9 3 11 3 7 5 9 5 11 5 7 7 9 7 > 10 a z + 7 a z - 6 a z - 6 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 165]][a, z]
Out[9]=	9 11 13 10 12 16 3 a 5 a 2 a 7 9 11 -5 a - 5 a + a + ---- + ----- + ----- + 5 a z - 10 a z - 21 a z - z z z 13 15 10 2 12 2 14 2 16 2 > 4 a z + 2 a z + 17 a z + 19 a z - 4 a z - 6 a z - 7 3 9 3 11 3 13 3 15 3 8 4 > 10 a z + 12 a z + 31 a z + 3 a z - 6 a z - 5 a z - 10 4 12 4 14 4 16 4 7 5 9 5 > 24 a z - 19 a z + 5 a z + 5 a z + 6 a z - 11 a z - 11 5 15 5 8 6 10 6 12 6 14 6 16 6 > 22 a z + 5 a z + 5 a z + 13 a z + 8 a z - a z - a z - 7 7 9 7 11 7 15 7 8 8 10 8 12 8 9 9 > a z + 6 a z + 8 a z - a z - a z - 2 a z - a z - a z - 11 9 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 1 1 1 1 1 1 q + q + ------- + ------- + ------ + ------ + ------ + ------ + ------ + 28 11 24 10 24 9 22 8 20 8 22 7 18 7 q t q t q t q t q t q t q t 3 1 2 1 1 1 2 1 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 18 6 16 6 18 5 16 5 14 5 16 4 14 4 12 4 q t q t q t q t q t q t q t q t 1 1 1 1 1 > ------ + ------ + ------ + ------ + ---- 14 3 12 3 12 2 10 2 8 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n165

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