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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-15/2 - 3q-13/2 + 6q-11/2 - 12q-9/2 + 16q-7/2 - 20q-5/2 + 20q-3/2 - 18q-1/2 + 14q1/2 - 9q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-22 + q-20 - q-18 + q-16 + 5q-14 - q-12 + 5q-10 - q-6 + 2q-4 - 4q-2 + 4 - 2q2 + 2q6 - 2q8 + q10

HOMFLY-PT Polynomial:

- a-1z - 2a-1z3 - a-1z5 - az + 2az3 + 3az5 + az7 - a3z-1 + 2a3z + 6a3z3 + 4a3z5 + a3z7 + a5z-1 - 2a5z - 3a5z3 - a5z5

Kauffman Polynomial:

a-3z3 - a-3z5 - a-2z2 + 5a-2z4 - 4a-2z6 + a-1z - 6a-1z3 + 13a-1z5 - 8a-1z7 + 2z2 - 7z4 + 14z6 - 9z8 - 2az + az3 + 2az5 + 4az7 - 6az9 + 2a2z2 - 11a2z4 + 18a2z6 - 8a2z8 - 2a2z10 - a3z-1 - a3z + 13a3z3 - 22a3z5 + 21a3z7 - 10a3z9 + a4 - a4z2 - 4a4z4 + 9a4z6 - 3a4z8 - 2a4z10 - a5z-1 + 4a5z - 3a5z3 - a5z5 + 6a5z7 - 4a5z9 - 2a6z2 - 2a6z4 + 8a6z6 - 4a6z8 + 2a7z - 8a7z3 + 9a7z5 - 3a7z7 - 2a8z2 + 3a8z4 - a8z6

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 = 8                       1 j = 6                     3   j = 4                   6 1   j = 2                 8 3     j = 0               10 6       j = -2             11 9         j = -4           9 9           j = -6         7 11             j = -8       5 9               j = -10     2 8                 j = -12   1 4                   j = -14   2                     j = -16 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, 256]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 256]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 3, 13, 4], X[14, 5, 15, 6], X[18, 10, 19, 9], > X[22, 18, 9, 17], X[8, 21, 1, 22], X[20, 15, 21, 16], X[16, 8, 17, 7], > X[6, 20, 7, 19], X[4, 11, 5, 12], X[2, 13, 3, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 2, -10, 3, -9, 8, -6}, > {4, -1, 10, -2, 11, -3, 7, -8, 5, -4, 9, -7, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 6 12 16 20 20 18 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 14 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 9 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 -18 -16 5 -12 5 -6 2 4 2 4 - q + q - q + q + --- - q + --- - q + -- - -- - 2 q + 14 10 4 2 q q q q 6 8 10 > 2 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 256]][a, z]
Out[8]=	3 5 3 a a z 3 5 2 z 3 3 3 5 3 -(--) + -- - - - a z + 2 a z - 2 a z - ---- + 2 a z + 6 a z - 3 a z - z z a a 5 z 5 3 5 5 5 7 3 7 > -- + 3 a z + 4 a z - a z + a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 256]][a, z]
Out[9]=	3 5 2 4 a a z 3 5 7 2 z 2 2 a - -- - -- + - - 2 a z - a z + 4 a z + 2 a z + 2 z - -- + 2 a z - z z a 2 a 3 3 4 2 6 2 8 2 z 6 z 3 3 3 5 3 > a z - 2 a z - 2 a z + -- - ---- + a z + 13 a z - 3 a z - 3 a a 4 5 7 3 4 5 z 2 4 4 4 6 4 8 4 z > 8 a z - 7 z + ---- - 11 a z - 4 a z - 2 a z + 3 a z - -- + 2 3 a a 5 6 13 z 5 3 5 5 5 7 5 6 4 z 2 6 > ----- + 2 a z - 22 a z - a z + 9 a z + 14 z - ---- + 18 a z + a 2 a 7 4 6 6 6 8 6 8 z 7 3 7 5 7 7 7 > 9 a z + 8 a z - a z - ---- + 4 a z + 21 a z + 6 a z - 3 a z - a 8 2 8 4 8 6 8 9 3 9 5 9 > 9 z - 8 a z - 3 a z - 4 a z - 6 a z - 10 a z - 4 a z - 2 10 4 10 > 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 2 1 4 2 8 5 9 10 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 7 11 9 9 11 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 6 t + 8 q t + 3 q t + 6 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a256

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