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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-21/2 - 3q-19/2 + 6q-17/2 - 9q-15/2 + 12q-13/2 - 14q-11/2 + 13q-9/2 - 12q-7/2 + 8q-5/2 - 5q-3/2 + 2q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-32 + q-30 - 2q-26 + 2q-24 - 2q-22 - q-20 + q-18 - q-16 + 4q-14 + q-12 + 3q-10 + 3q-8 - 2q-6 + 2q-4 + q2

HOMFLY-PT Polynomial:

- 2az - az3 - 2a3z-1 - 2a3z + a3z3 + a3z5 + 3a5z-1 + 5a5z + 5a5z3 + 2a5z5 - a7z-1 - a7z + a7z3 + a7z5 - a9z - a9z3

Kauffman Polynomial:

- 2az + 3az3 - az5 - a2z2 + 4a2z4 - 2a2z6 - 2a3z-1 + 6a3z - 6a3z3 + 6a3z5 - 3a3z7 + 3a4 - 4a4z2 - a4z4 + 4a4z6 - 3a4z8 - 3a5z-1 + 13a5z - 20a5z3 + 10a5z5 - a5z7 - 2a5z9 + 3a6 - 9a6z2 + 2a6z4 + 3a6z6 - 2a6z8 - a6z10 - a7z-1 + 2a7z + 3a7z3 - 11a7z5 + 11a7z7 - 5a7z9 + a8 - a8z2 - 4a8z4 + 9a8z6 - 3a8z8 - a8z10 - 3a9z + 8a9z3 - 5a9z5 + 6a9z7 - 3a9z9 + 3a10z2 - 8a10z4 + 11a10z6 - 4a10z8 - 6a11z3 + 9a11z5 - 3a11z7 - 2a12z2 + 3a12z4 - 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                     1   j = -2                   4 1   j = -4                 5 2     j = -6               7 3       j = -8             7 6         j = -10           7 6           j = -12         5 7             j = -14       4 7               j = -16     2 5                 j = -18   1 4                   j = -20   2                     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, 191]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 191]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[16, 6, 17, 5], X[20, 11, 21, 12], > X[22, 13, 7, 14], X[12, 21, 13, 22], X[18, 15, 19, 16], X[14, 19, 15, 20], > 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, -6, 5, -8, 7, -3, 9, -7, 8, -4, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 3 6 9 12 14 13 12 8 5 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 2 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -30 2 2 2 -20 -18 -16 4 -12 3 3 -q + q - --- + --- - --- - q + q - q + --- + q + --- + -- - 26 24 22 14 10 8 q q q q q q 2 2 2 > -- + -- + q 6 4 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 191]][a, z]
Out[8]=	3 5 7 -2 a 3 a a 3 5 7 9 3 3 3 ----- + ---- - -- - 2 a z - 2 a z + 5 a z - a z - a z - a z + a z + z z z 5 3 7 3 9 3 3 5 5 5 7 5 > 5 a z + a z - a z + a z + 2 a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 191]][a, z]
Out[9]=	3 5 7 4 6 8 2 a 3 a a 3 5 7 3 a + 3 a + a - ---- - ---- - -- - 2 a z + 6 a z + 13 a z + 2 a z - z z z 9 2 2 4 2 6 2 8 2 10 2 12 2 3 > 3 a z - a z - 4 a z - 9 a z - a z + 3 a z - 2 a z + 3 a z - 3 3 5 3 7 3 9 3 11 3 2 4 4 4 > 6 a z - 20 a z + 3 a z + 8 a z - 6 a z + 4 a z - a z + 6 4 8 4 10 4 12 4 5 3 5 5 5 > 2 a z - 4 a z - 8 a z + 3 a z - a z + 6 a z + 10 a z - 7 5 9 5 11 5 2 6 4 6 6 6 8 6 > 11 a z - 5 a z + 9 a z - 2 a z + 4 a z + 3 a z + 9 a z + 10 6 12 6 3 7 5 7 7 7 9 7 11 7 > 11 a z - a z - 3 a z - a z + 11 a z + 6 a z - 3 a z - 4 8 6 8 8 8 10 8 5 9 7 9 9 9 > 3 a z - 2 a z - 3 a z - 4 a z - 2 a z - 5 a z - 3 a z - 6 10 8 10 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 1 2 1 4 2 5 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q q t q t q t q t q t q t q t 7 5 7 7 6 7 6 7 3 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 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 5 t 2 2 > ---- + t + -- + q t 4 2 q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a191

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