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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X16,11,7,12 X12,15,13,16 X6718 X4,13,5,14

Gauss Code:

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

Jones Polynomial:

- q-21/2 + 2q-19/2 - 3q-17/2 + 4q-15/2 - 4q-13/2 + 3q-11/2 - 3q-9/2 + q-7/2 - q-5/2

A2 (sl(3)) Invariant:

q-32 - q-24 + q-22 + 2q-18 + 2q-16 + q-14 + 2q-12 + q-8

HOMFLY-PT Polynomial:

- a5z-1 - 4a5z - 4a5z3 - a5z5 + a7z-1 - a7z - 3a7z3 - a7z5 + 2a9z + a9z3

Kauffman Polynomial:

a5z-1 - 4a5z + 4a5z3 - a5z5 - a6 + a6z2 + 2a6z4 - a6z6 + a7z-1 - a7z + 2a7z5 - a7z7 - 4a8z2 + 7a8z4 - 3a8z6 + 2a9z - 2a9z3 + a9z5 - a9z7 - 3a10z2 + 3a10z4 - 2a10z6 + a11z3 - 2a11z5 + 2a12z2 - 2a12z4 + a13z - a13z3

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L8a11

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