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DrawMorseLink

PD Presentation:

X12,1,13,2 X2,13,3,14 X14,3,15,4 X16,5,17,6 X10,11,1,12 X4,15,5,16 X22,17,11,18 X18,10,19,9 X20,8,21,7 X8,20,9,19 X6,22,7,21

Gauss Code:

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

Jones Polynomial:

q-19/2 - 2q-17/2 + 3q-15/2 - 4q-13/2 + 4q-11/2 - 5q-9/2 + 4q-7/2 - 4q-5/2 + 3q-3/2 - 2q-1/2 + q1/2 - q3/2

A2 (sl(3)) Invariant:

- q-28 + 2q-16 + q-14 + 2q-12 + q-10 + q-2 + 1 + q2 + q4

HOMFLY-PT Polynomial:

- az-1 - 6az - 5az3 - az5 + a3z-1 + 7a3z + 11a3z3 + 6a3z5 + a3z7 + a5z + 6a5z3 + 5a5z5 + a5z7 - 3a7z - 4a7z3 - a7z5

Kauffman Polynomial:

az-1 - 9az + 24az3 - 22az5 + 8az7 - az9 - a2 - 2a2z2 + 13a2z4 - 16a2z6 + 7a2z8 - a2z10 + a3z-1 - 11a3z + 37a3z3 - 44a3z5 + 20a3z7 - 3a3z9 + 3a4z2 - 7a4z4 - a4z6 + 4a4z8 - a4z10 + 2a5z - 5a5z3 - 5a5z5 + 8a5z7 - 2a5z9 - 8a6z4 + 11a6z6 - 3a6z8 + 2a7z - 9a7z3 + 13a7z5 - 4a7z7 - 2a8z2 + 9a8z4 - 4a8z6 - 2a9z + 7a9z3 - 4a9z5 + 2a10z2 - 3a10z4 - 2a11z3 - a12z2

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 = 4                       1 j = 2                         j = 0                   2 1   j = -2                 1       j = -4               3 2       j = -6             2 2         j = -8           3 2           j = -10         2 3             j = -12       2 2               j = -14     1 2                 j = -16   1 2                   j = -18   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, Alternating, 372]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 372]]
Out[4]=	PD[X[12, 1, 13, 2], X[2, 13, 3, 14], X[14, 3, 15, 4], X[16, 5, 17, 6], > X[10, 11, 1, 12], X[4, 15, 5, 16], X[22, 17, 11, 18], X[18, 10, 19, 9], > X[20, 8, 21, 7], X[8, 20, 9, 19], X[6, 22, 7, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -6, 4, -11, 9, -10, 8, -5}, > {5, -1, 2, -3, 6, -4, 7, -8, 10, -9, 11, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 2 3 4 4 5 4 4 3 q - ----- + ----- - ----- + ----- - ---- + ---- - ---- + ---- - 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q 2 3/2 > ------- + Sqrt[q] - q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 2 -14 2 -10 -2 2 4 1 - q + --- + q + --- + q + q + q + q 16 12 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 372]][a, z]
Out[8]=	3 a a 3 5 7 3 3 3 5 3 -(-) + -- - 6 a z + 7 a z + a z - 3 a z - 5 a z + 11 a z + 6 a z - z z 7 3 5 3 5 5 5 7 5 3 7 5 7 > 4 a z - a z + 6 a z + 5 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 372]][a, z]
Out[9]=	3 2 a a 3 5 7 9 2 2 4 2 -a + - + -- - 9 a z - 11 a z + 2 a z + 2 a z - 2 a z - 2 a z + 3 a z - z z 8 2 10 2 12 2 3 3 3 5 3 7 3 > 2 a z + 2 a z - a z + 24 a z + 37 a z - 5 a z - 9 a z + 9 3 11 3 2 4 4 4 6 4 8 4 10 4 > 7 a z - 2 a z + 13 a z - 7 a z - 8 a z + 9 a z - 3 a z - 5 3 5 5 5 7 5 9 5 2 6 4 6 > 22 a z - 44 a z - 5 a z + 13 a z - 4 a z - 16 a z - a z + 6 6 8 6 7 3 7 5 7 7 7 2 8 > 11 a z - 4 a z + 8 a z + 20 a z + 8 a z - 4 a z + 7 a z + 4 8 6 8 9 3 9 5 9 2 10 4 10 > 4 a z - 3 a z - a z - 3 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 1 1 1 2 1 2 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 20 7 18 6 16 6 16 5 14 5 14 4 12 4 q q q t q t q t q t q t q t q t 2 2 3 3 2 2 2 t t 2 3 > ------ + ------ + ------ + ----- + ---- + ---- + --- + -- + 2 t + t + 12 3 10 3 10 2 8 2 8 6 4 2 q t q t q t q t q t q t q q 4 4 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a372

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