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DrawMorseLink

PD Presentation:

X10,1,11,2 X3,14,4,15 X22,5,9,6 X6,9,7,10 X20,12,21,11 X18,14,19,13 X12,20,13,19 X7,16,8,17 X15,4,16,5 X17,8,18,1 X2,21,3,22

Gauss Code:

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

Jones Polynomial:

- q-19/2 + q-17/2 - q-15/2 - q-11/2 + q-7/2 - q-5/2 + q-3/2 - q-1/2

A2 (sl(3)) Invariant:

2q-30 + q-24 + q-22 + 3q-20 + 2q-18 + 2q-16 - q-14 - 2q-12 - q-10 + q-4 + q-2

HOMFLY-PT Polynomial:

- 5a3z - 5a3z3 - a3z5 + 6a5z + 10a5z3 + 6a5z5 + a5z7 - a7z-1 - 7a7z - 6a7z3 - a7z5 + a9z-1 + 2a9z

Kauffman Polynomial:

5a3z - 10a3z3 + 6a3z5 - a3z7 + 6a4z2 - 10a4z4 + 6a4z6 - a4z8 + 5a5z - 17a5z3 + 12a5z5 - 2a5z7 + 2a6z2 - 8a6z4 + 6a6z6 - a6z8 - a7z-1 + 9a7z - 21a7z3 + 13a7z5 - 2a7z7 + a8 - 6a8z2 + 6a8z4 - a8z6 - a9z-1 + 7a9z - 10a9z3 + 6a9z5 - a9z7 - 2a10z2 + 4a10z4 - a10z6 - 2a11z + 4a11z3 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 0                     1 j = -2                       j = -4               1 2 1   j = -6               1 1     j = -8           2 2 1       j = -10         1 1 1         j = -12       1 3 1           j = -14     1 1 1             j = -16     1 1               j = -18 1 1                   j = -20 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, 237]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 237]]
Out[4]=	PD[X[10, 1, 11, 2], X[3, 14, 4, 15], X[22, 5, 9, 6], X[6, 9, 7, 10], > X[20, 12, 21, 11], X[18, 14, 19, 13], X[12, 20, 13, 19], X[7, 16, 8, 17], > X[15, 4, 16, 5], X[17, 8, 18, 1], X[2, 21, 3, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, -2, 9, 3, -4, -8, 10}, > {4, -1, 5, -7, 6, 2, -9, 8, -10, -6, 7, -5, 11, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) -(17/2) -(15/2) -(11/2) -(7/2) -(5/2) -(3/2) -q + q - q - q + q - q + q - 1 > ------- Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	2 -24 -22 3 2 2 -14 2 -10 -4 -2 --- + q + q + --- + --- + --- - q - --- - q + q + q 30 20 18 16 12 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 237]][a, z]
Out[8]=	7 9 a a 3 5 7 9 3 3 5 3 7 3 -(--) + -- - 5 a z + 6 a z - 7 a z + 2 a z - 5 a z + 10 a z - 6 a z - z z 3 5 5 5 7 5 5 7 > a z + 6 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 237]][a, z]
Out[9]=	7 9 8 a a 3 5 7 9 11 4 2 a - -- - -- + 5 a z + 5 a z + 9 a z + 7 a z - 2 a z + 6 a z + z z 6 2 8 2 10 2 3 3 5 3 7 3 9 3 > 2 a z - 6 a z - 2 a z - 10 a z - 17 a z - 21 a z - 10 a z + 11 3 4 4 6 4 8 4 10 4 3 5 5 5 > 4 a z - 10 a z - 8 a z + 6 a z + 4 a z + 6 a z + 12 a z + 7 5 9 5 11 5 4 6 6 6 8 6 10 6 3 7 > 13 a z + 6 a z - a z + 6 a z + 6 a z - a z - a z - a z - 5 7 7 7 9 7 4 8 6 8 > 2 a z - 2 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 2 1 1 1 1 1 1 1 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 20 8 18 8 18 7 16 6 14 6 16 5 14 5 q q t q t q t q t q t q t q t 1 1 3 1 1 1 2 1 > ------ + ------ + ------ + ------ + ------ + ------ + ----- + ------ + 12 5 14 4 12 4 10 4 12 3 10 3 8 3 10 2 q t q t q t q t q t q t q t q t 2 1 1 1 t 2 > ----- + ---- + ---- + ---- + -- + t 8 2 8 6 4 4 q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n237

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