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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 3q-3/2 - 7q-1/2 + 10q1/2 - 14q3/2 + 16q5/2 - 16q7/2 + 14q9/2 - 11q11/2 + 6q13/2 - 3q15/2 + q17/2

A2 (sl(3)) Invariant:

q-8 + 3q-2 - 1 + 2q2 + q4 - 2q6 + 3q8 - 3q10 + 2q12 + 4q18 - q20 + q22 - q26

HOMFLY-PT Polynomial:

2a-7z + a-7z3 - a-5z-1 - 5a-5z - 6a-5z3 - 2a-5z5 + 2a-3z-1 + 7a-3z + 7a-3z3 + 4a-3z5 + a-3z7 - 2a-1z-1 - 5a-1z - 6a-1z3 - 2a-1z5 + az-1 + 2az + az3

Kauffman Polynomial:

a-10z2 - a-10z4 - a-9z + 3a-9z3 - 3a-9z5 - a-8z2 + 4a-8z4 - 5a-8z6 + a-7z - 6a-7z3 + 9a-7z5 - 7a-7z7 + 2a-6z2 - 6a-6z4 + 10a-6z6 - 7a-6z8 - a-5z-1 + 9a-5z - 26a-5z3 + 25a-5z5 - 3a-5z7 - 4a-5z9 + 11a-4z2 - 34a-4z4 + 39a-4z6 - 12a-4z8 - a-4z10 - 2a-3z-1 + 15a-3z - 34a-3z3 + 20a-3z5 + 10a-3z7 - 7a-3z9 - a-2 + 11a-2z2 - 35a-2z4 + 35a-2z6 - 8a-2z8 - a-2z10 - 2a-1z-1 + 12a-1z - 23a-1z3 + 11a-1z5 + 5a-1z7 - 3a-1z9 + 4z2 - 12z4 + 11z6 - 3z8 - az-1 + 4az - 6az3 + 4az5 - az7

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 18                       1 j = 16                     2   j = 14                   4 1   j = 12                 7 2     j = 10               7 4       j = 8             9 7         j = 6           8 8           j = 4         6 8             j = 2       5 9               j = 0     2 5                 j = -2   1 5                   j = -4   2                     j = -6 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, 156]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 156]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[22, 10, 7, 9], X[14, 6, 15, 5], > X[18, 14, 19, 13], X[20, 17, 21, 18], X[16, 21, 17, 22], X[12, 20, 13, 19], > X[2, 7, 3, 8], X[4, 12, 5, 11], X[6, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 4, -11}, > {9, -1, 3, -2, 10, -8, 5, -4, 11, -7, 6, -5, 8, -6, 7, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 3 7 3/2 5/2 7/2 -q + ---- - ------- + 10 Sqrt[q] - 14 q + 16 q - 16 q + 3/2 Sqrt[q] q 9/2 11/2 13/2 15/2 17/2 > 14 q - 11 q + 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 3 2 4 6 8 10 12 18 20 22 -1 + q + -- + 2 q + q - 2 q + 3 q - 3 q + 2 q + 4 q - q + q - 2 q 26 > q
In[8]:=	HOMFLYPT[Link[11, Alternating, 156]][a, z]
Out[8]=	3 3 3 1 2 2 a 2 z 5 z 7 z 5 z z 6 z 7 z -(----) + ---- - --- + - + --- - --- + --- - --- + 2 a z + -- - ---- + ---- - 5 3 a z z 7 5 3 a 7 5 3 a z a z a a a a a a 3 5 5 5 7 6 z 3 2 z 4 z 2 z z > ---- + a z - ---- + ---- - ---- + -- a 5 3 a 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 156]][a, z]
Out[9]=	-2 1 2 2 a z z 9 z 15 z 12 z 2 -a - ---- - ---- - --- - - - -- + -- + --- + ---- + ---- + 4 a z + 4 z + 5 3 a z z 9 7 5 3 a a z a z a a a a 2 2 2 2 2 3 3 3 3 3 z z 2 z 11 z 11 z 3 z 6 z 26 z 34 z 23 z > --- - -- + ---- + ----- + ----- + ---- - ---- - ----- - ----- - ----- - 10 8 6 4 2 9 7 5 3 a a a a a a a a a a 4 4 4 4 4 5 5 5 3 4 z 4 z 6 z 34 z 35 z 3 z 9 z 25 z > 6 a z - 12 z - --- + ---- - ---- - ----- - ----- - ---- + ---- + ----- + 10 8 6 4 2 9 7 5 a a a a a a a a 5 5 6 6 6 6 7 20 z 11 z 5 6 5 z 10 z 39 z 35 z 7 z > ----- + ----- + 4 a z + 11 z - ---- + ----- + ----- + ----- - ---- - 3 a 8 6 4 2 7 a a a a a a 7 7 7 8 8 8 9 9 3 z 10 z 5 z 7 8 7 z 12 z 8 z 4 z 7 z > ---- + ----- + ---- - a z - 3 z - ---- - ----- - ---- - ---- - ---- - 5 3 a 6 4 2 5 3 a a a a a a a 9 10 10 3 z z z > ---- - --- - --- a 4 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 2 1 2 5 5 5 q 4 6 9 q + 6 q + ----- + ----- + ----- + -- + ----- + - + ---- + 8 q t + 8 q t + 6 4 4 3 2 3 2 2 2 t t q t q t q t t q t 6 2 8 2 8 3 10 3 10 4 12 4 12 5 > 8 q t + 9 q t + 7 q t + 7 q t + 4 q t + 7 q t + 2 q t + 14 5 14 6 16 6 18 7 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a156

L11a155 L11a157