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DrawMorseLink

PD Presentation:

X8192 X10,3,11,4 X16,6,17,5 X18,11,19,12 X20,13,21,14 X22,15,7,16 X12,19,13,20 X14,21,15,22 X4,18,5,17 X2738 X6,9,1,10

Gauss Code:

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

Jones Polynomial:

q-23/2 - 2q-21/2 + 4q-19/2 - 6q-17/2 + 8q-15/2 - 9q-13/2 + 8q-11/2 - 8q-9/2 + 5q-7/2 - 4q-5/2 + 2q-3/2 - q-1/2

A2 (sl(3)) Invariant:

- q-34 - q-30 - 2q-28 - q-26 - 2q-24 + 2q-22 + q-20 + 3q-18 + 4q-16 + q-14 + 3q-12 + q-8 + q-2

HOMFLY-PT Polynomial:

- 3a3z - 4a3z3 - a3z5 - 3a5z-1 - 4a5z + 2a5z3 + 4a5z5 + a5z7 + 5a7z-1 + 10a7z + 9a7z3 + 5a7z5 + a7z7 - 2a9z-1 - 4a9z - 4a9z3 - a9z5

Kauffman Polynomial:

3a3z - 7a3z3 + 5a3z5 - a3z7 + 2a4z2 - 10a4z4 + 9a4z6 - 2a4z8 + 3a5z-1 - 10a5z + 9a5z3 - 9a5z5 + 8a5z7 - 2a5z9 - 5a6 + 15a6z2 - 18a6z4 + 9a6z6 + a6z8 - a6z10 + 5a7z-1 - 25a7z + 53a7z3 - 50a7z5 + 25a7z7 - 5a7z9 - 5a8 + 22a8z2 - 25a8z4 + 11a8z6 - a8z10 + 2a9z-1 - 12a9z + 28a9z3 - 28a9z5 + 13a9z7 - 3a9z9 + 2a10z2 - 10a10z4 + 8a10z6 - 3a10z8 - 6a11z3 + 6a11z5 - 3a11z7 + a12 - 5a12z2 + 6a12z4 - 3a12z6 + 3a13z3 - 2a13z5 + 2a14z2 - a14z4

Khovanov Homology:

trqj r = -9 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                     1   j = -4                   3 1   j = -6                 3 2     j = -8               5 2       j = -10             4 4         j = -12           5 4           j = -14         3 4             j = -16       3 5               j = -18     1 3                 j = -20   1 3                   j = -22   1                     j = -24 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, 188]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 188]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[16, 6, 17, 5], X[18, 11, 19, 12], > X[20, 13, 21, 14], X[22, 15, 7, 16], X[12, 19, 13, 20], X[14, 21, 15, 22], > X[4, 18, 5, 17], X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -9, 3, -11}, > {10, -1, 11, -2, 4, -7, 5, -8, 6, -3, 9, -4, 7, -5, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) 2 4 6 8 9 8 8 5 4 q - ----- + ----- - ----- + ----- - ----- + ----- - ---- + ---- - ---- + 21/2 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 q q q q q q q q q 2 1 > ---- - ------- 3/2 Sqrt[q] q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 -30 2 -26 2 2 -20 3 4 -14 3 -8 -q - q - --- - q - --- + --- + q + --- + --- + q + --- + q + 28 24 22 18 16 12 q q q q q q -2 > q
In[8]:=	HOMFLYPT[Link[11, Alternating, 188]][a, z]
Out[8]=	5 7 9 -3 a 5 a 2 a 3 5 7 9 3 3 5 3 ----- + ---- - ---- - 3 a z - 4 a z + 10 a z - 4 a z - 4 a z + 2 a z + z z z 7 3 9 3 3 5 5 5 7 5 9 5 5 7 7 7 > 9 a z - 4 a z - a z + 4 a z + 5 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 188]][a, z]
Out[9]=	5 7 9 6 8 12 3 a 5 a 2 a 3 5 7 -5 a - 5 a + a + ---- + ---- + ---- + 3 a z - 10 a z - 25 a z - z z z 9 4 2 6 2 8 2 10 2 12 2 14 2 > 12 a z + 2 a z + 15 a z + 22 a z + 2 a z - 5 a z + 2 a z - 3 3 5 3 7 3 9 3 11 3 13 3 4 4 > 7 a z + 9 a z + 53 a z + 28 a z - 6 a z + 3 a z - 10 a z - 6 4 8 4 10 4 12 4 14 4 3 5 5 5 > 18 a z - 25 a z - 10 a z + 6 a z - a z + 5 a z - 9 a z - 7 5 9 5 11 5 13 5 4 6 6 6 8 6 > 50 a z - 28 a z + 6 a z - 2 a z + 9 a z + 9 a z + 11 a z + 10 6 12 6 3 7 5 7 7 7 9 7 11 7 > 8 a z - 3 a z - a z + 8 a z + 25 a z + 13 a z - 3 a z - 4 8 6 8 10 8 5 9 7 9 9 9 6 10 8 10 > 2 a z + a z - 3 a z - 2 a z - 5 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 1 1 1 3 1 3 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 24 9 22 8 20 8 20 7 18 7 18 6 16 6 q q q t q t q t q t q t q t q t 5 3 4 5 4 4 4 5 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 16 5 14 5 14 4 12 4 12 3 10 3 10 2 8 2 q t q t q t q t q t q t q t q t 2 3 t t 2 > ---- + ---- + -- + -- + t 8 6 4 2 q t q t q q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a188

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