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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X5,12,6,13 X3849 X9,16,10,17 X11,20,12,21 X17,22,18,5 X21,18,22,19 X19,10,20,11 X13,2,14,3

Gauss Code:

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

Jones Polynomial:

q-25/2 - q-23/2 + 2q-21/2 - 3q-19/2 + 3q-17/2 - 3q-15/2 + 2q-13/2 - 3q-11/2 + q-9/2 - q-7/2

A2 (sl(3)) Invariant:

- q-42 - q-38 - 2q-36 - q-34 - q-32 + q-30 + 2q-26 + 2q-24 + 2q-22 + 3q-20 + 2q-18 + 2q-16 + q-12

HOMFLY-PT Polynomial:

- 2a7z-1 - 8a7z - 11a7z3 - 6a7z5 - a7z7 + 2a9z-1 + 2a9z - 5a9z3 - 5a9z5 - a9z7 + a11z-1 + 5a11z + 5a11z3 + a11z5 - a13z-1 - a13z

Kauffman Polynomial:

- 2a7z-1 + 8a7z - 11a7z3 + 6a7z5 - a7z7 + 3a8 - 5a8z2 - a8z4 + 4a8z6 - a8z8 - 2a9z-1 + 6a9z - 6a9z3 - a9z5 + 4a9z7 - a9z9 + 6a10z2 - 19a10z4 + 15a10z6 - 3a10z8 + a11z-1 - 3a11z + 2a11z3 - 3a11z5 + 4a11z7 - a11z9 - 3a12 + 13a12z2 - 19a12z4 + 11a12z6 - 2a12z8 + a13z-1 - 4a13z3 + 4a13z5 - a13z7 + a14z2 - a14z4 + a15z - a15z3 + a16 - a16z2

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n16

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