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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-19/2 - 2q-17/2 + 3q-15/2 - 5q-13/2 + 6q-11/2 - 7q-9/2 + 7q-7/2 - 7q-5/2 + 5q-3/2 - 4q-1/2 + 2q1/2 - q3/2

A2 (sl(3)) Invariant:

- q-30 + q-26 - q-24 + q-22 + q-20 - q-18 + q-16 + q-10 + q-8 + 3q-6 + q-4 + 2 - q2 + q6

HOMFLY-PT Polynomial:

- a-1z - az-1 - az + az3 + a3z-1 + a3z + 2a3z3 + a5z + 2a5z3 + a7z3 - a9z

Kauffman Polynomial:

a-1z - a-1z3 + z2 - 2z4 + az-1 - 3az + 3az3 - 3az5 - a2 + a2z2 + 2a2z4 - 3a2z6 + a3z-1 - 4a3z + 4a3z3 + 3a3z5 - 3a3z7 + 3a4z2 - 4a4z4 + 7a4z6 - 3a4z8 + a5z - 4a5z3 + 3a5z5 + 4a5z7 - 2a5z9 + a6z2 - 8a6z4 + 7a6z6 + a6z8 - a6z10 - 2a7z + 12a7z3 - 26a7z5 + 19a7z7 - 4a7z9 + 4a8z2 - 11a8z4 + 3a8z6 + 3a8z8 - a8z10 - 3a9z + 16a9z3 - 23a9z5 + 12a9z7 - 2a9z9 + 6a10z2 - 11a10z4 + 6a10z6 - a10z8

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 = 4                       1 j = 2                     1   j = 0                   3 1   j = -2                 3 2     j = -4               4 2       j = -6             4 4         j = -8           3 3           j = -10         3 4             j = -12       2 3               j = -14     1 3                 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, 192]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 192]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[16, 6, 17, 5], X[22, 11, 7, 12], > X[20, 13, 21, 14], X[18, 15, 19, 16], X[14, 19, 15, 20], X[12, 21, 13, 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, -8, 5, -7, 6, -3, 9, -6, 7, -5, 8, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 2 3 5 6 7 7 7 5 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 4 3/2 > ------- + 2 Sqrt[q] - q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -26 -24 -22 -20 -18 -16 -10 -8 3 -4 2 - q + q - q + q + q - q + q + q + q + -- + q - 6 q 2 6 > q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 192]][a, z]
Out[8]=	3 a a z 3 5 9 3 3 3 5 3 7 3 -(-) + -- - - - a z + a z + a z - a z + a z + 2 a z + 2 a z + a z z z a
In[9]:=	Kauffman[Link[11, Alternating, 192]][a, z]
Out[9]=	3 2 a a z 3 5 7 9 2 2 2 -a + - + -- + - - 3 a z - 4 a z + a z - 2 a z - 3 a z + z + a z + z z a 3 4 2 6 2 8 2 10 2 z 3 3 3 5 3 > 3 a z + a z + 4 a z + 6 a z - -- + 3 a z + 4 a z - 4 a z + a 7 3 9 3 4 2 4 4 4 6 4 8 4 > 12 a z + 16 a z - 2 z + 2 a z - 4 a z - 8 a z - 11 a z - 10 4 5 3 5 5 5 7 5 9 5 2 6 > 11 a z - 3 a z + 3 a z + 3 a z - 26 a z - 23 a z - 3 a z + 4 6 6 6 8 6 10 6 3 7 5 7 7 7 > 7 a z + 7 a z + 3 a z + 6 a z - 3 a z + 4 a z + 19 a z + 9 7 4 8 6 8 8 8 10 8 5 9 7 9 > 12 a z - 3 a z + a z + 3 a z - a z - 2 a z - 4 a z - 9 9 6 10 8 10 > 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 2 1 3 2 3 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 20 9 18 8 16 8 16 7 14 7 14 6 12 6 q q t q t q t q t q t q t q t 3 3 4 3 3 4 4 4 2 > ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 12 5 10 5 10 4 8 4 8 3 6 3 6 2 4 2 4 q t q t q t q t q t q t q t q t q t 3 2 4 2 > ---- + t + q t + q t 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a192

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