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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X6718 X18,15,19,16 X16,6,17,5 X4,18,5,17 X22,11,7,12 X20,13,21,14 X14,19,15,20 X12,21,13,22

Gauss Code:

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

Jones Polynomial:

q-21/2 - 2q-19/2 + 3q-17/2 - 5q-15/2 + 6q-13/2 - 7q-11/2 + 7q-9/2 - 6q-7/2 + 4q-5/2 - 3q-3/2 + q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-32 + 2q-24 - q-16 + q-14 + q-10 + 2q-8 + 2q-4 + q-2 + 1 + q2

HOMFLY-PT Polynomial:

- az-1 - 3az - az3 + a3z-1 + 2a3z + 3a3z3 + a3z5 + 2a5z3 + a5z5 + 2a7z + 3a7z3 + a7z5 - 2a9z - a9z3

Kauffman Polynomial:

az-1 - 4az + 4az3 - az5 - a2 + a2z2 + 2a2z4 - a2z6 + a3z-1 - 3a3z + 3a3z3 + a3z5 - a3z7 + a4z2 - 2a4z4 + 2a4z6 - a4z8 + a5z - a5z3 - 4a5z5 + 3a5z7 - a5z9 - 6a6z2 + 11a6z4 - 12a6z6 + 5a6z8 - a6z10 - 3a7z + 19a7z3 - 29a7z5 + 15a7z7 - 3a7z9 - 2a8z2 + 6a8z4 - 7a8z6 + 4a8z8 - a8z10 - 2a9z + 12a9z3 - 15a9z5 + 9a9z7 - 2a9z9 + a10z2 - 5a10z4 + 7a10z6 - 2a10z8 + a11z - 7a11z3 + 8a11z5 - 2a11z7 - 3a12z2 + 4a12z4 - a12z6

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 = 2                       1 j = 0                         j = -2                   3 1   j = -4                 2 1     j = -6               4 2       j = -8             4 3         j = -10           3 3           j = -12         3 4             j = -14       2 3               j = -16     1 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[11, Alternating, 206]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 206]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[6, 7, 1, 8], > X[18, 15, 19, 16], X[16, 6, 17, 5], X[4, 18, 5, 17], X[22, 11, 7, 12], > X[20, 13, 21, 14], X[14, 19, 15, 20], X[12, 21, 13, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -7, 6, -4}, > {4, -1, 2, -3, 8, -11, 9, -10, 5, -6, 7, -5, 10, -9, 11, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 2 3 5 6 7 7 6 4 3 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q 1 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 2 -16 -14 -10 2 2 -2 2 1 - q + --- - q + q + q + -- + -- + q + q 24 8 4 q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 206]][a, z]
Out[8]=	3 a a 3 7 9 3 3 3 5 3 -(-) + -- - 3 a z + 2 a z + 2 a z - 2 a z - a z + 3 a z + 2 a z + z z 7 3 9 3 3 5 5 5 7 5 > 3 a z - a z + a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 206]][a, z]
Out[9]=	3 2 a a 3 5 7 9 11 2 2 -a + - + -- - 4 a z - 3 a z + a z - 3 a z - 2 a z + a z + a z + z z 4 2 6 2 8 2 10 2 12 2 3 3 3 5 3 > a z - 6 a z - 2 a z + a z - 3 a z + 4 a z + 3 a z - a z + 7 3 9 3 11 3 2 4 4 4 6 4 8 4 > 19 a z + 12 a z - 7 a z + 2 a z - 2 a z + 11 a z + 6 a z - 10 4 12 4 5 3 5 5 5 7 5 9 5 > 5 a z + 4 a z - a z + a z - 4 a z - 29 a z - 15 a z + 11 5 2 6 4 6 6 6 8 6 10 6 12 6 > 8 a z - a z + 2 a z - 12 a z - 7 a z + 7 a z - a z - 3 7 5 7 7 7 9 7 11 7 4 8 6 8 > a z + 3 a z + 15 a z + 9 a z - 2 a z - a z + 5 a z + 8 8 10 8 5 9 7 9 9 9 6 10 8 10 > 4 a z - 2 a z - a z - 3 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 3 1 1 1 2 1 3 2 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q t q t q t q t q t q t q t 3 3 4 3 3 4 3 4 2 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q t q t q t q t q t q t q t q t q t 2 t 2 2 > ---- + -- + q t 4 2 q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a206

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