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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-1/2 + 3q1/2 - 7q3/2 + 11q5/2 - 16q7/2 + 17q9/2 - 17q11/2 + 15q13/2 - 11q15/2 + 6q17/2 - 3q19/2 + q21/2

A2 (sl(3)) Invariant:

q-2 - 1 + 3q4 - 3q6 + 3q8 + 2q10 + 4q14 - 2q16 + 2q18 - q20 - q22 + 4q24 - 2q26 + q30 - q32

HOMFLY-PT Polynomial:

a-9z + a-9z3 - a-7z - a-7z3 - a-7z5 - a-5z-1 - a-5z - 3a-5z3 - 2a-5z5 + a-3z-1 + 2a-3z - a-3z5 + a-1z + a-1z3

Kauffman Polynomial:

- 2a-12z2 + 3a-12z4 - a-12z6 + 2a-11z - 8a-11z3 + 9a-11z5 - 3a-11z7 + a-10z2 - 5a-10z4 + 9a-10z6 - 4a-10z8 + 4a-9z - 11a-9z3 + 10a-9z5 + a-9z7 - 3a-9z9 + 4a-8z2 - 11a-8z4 + 17a-8z6 - 7a-8z8 - a-8z10 + 4a-7z - 13a-7z3 + 10a-7z5 + 4a-7z7 - 6a-7z9 - 6a-6z4 + 14a-6z6 - 8a-6z8 - a-6z10 - a-5z-1 + 7a-5z - 18a-5z3 + 18a-5z5 - 5a-5z7 - 3a-5z9 + a-4 - 3a-4z2 + 2a-4z4 + 4a-4z6 - 5a-4z8 - a-3z-1 + 4a-3z - 6a-3z3 + 8a-3z5 - 5a-3z7 - 2a-2z2 + 5a-2z4 - 3a-2z6 - a-1z + 2a-1z3 - a-1z5

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 22                       1 j = 20                     2   j = 18                   4 1   j = 16                 7 2     j = 14               8 4       j = 12             9 7         j = 10           9 9           j = 8         7 8             j = 6       4 9               j = 4     3 7                 j = 2   1 5                   j = 0   2                     j = -2 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, 265]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 265]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[22, 12, 9, 11], X[18, 8, 19, 7], > X[20, 18, 21, 17], X[16, 22, 17, 21], X[2, 9, 3, 10], X[4, 16, 5, 15], > X[14, 6, 15, 5], X[6, 14, 7, 13], X[8, 20, 1, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -7, 2, -8, 9, -10, 4, -11}, > {7, -1, 3, -2, 10, -9, 8, -6, 5, -4, 11, -5, 6, -3}]
In[6]:=	Jones[L][q]
Out[6]=	1 3/2 5/2 7/2 9/2 11/2 -(-------) + 3 Sqrt[q] - 7 q + 11 q - 16 q + 17 q - 17 q + Sqrt[q] 13/2 15/2 17/2 19/2 21/2 > 15 q - 11 q + 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 4 6 8 10 14 16 18 20 22 -1 + q + 3 q - 3 q + 3 q + 2 q + 4 q - 2 q + 2 q - q - q + 24 26 30 32 > 4 q - 2 q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 265]][a, z]
Out[8]=	3 3 3 3 5 5 5 1 1 z z z 2 z z z z 3 z z z 2 z z -(----) + ---- + -- - -- - -- + --- + - + -- - -- - ---- + -- - -- - ---- - -- 5 3 9 7 5 3 a 9 7 5 a 7 5 3 a z a z a a a a a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 265]][a, z]
Out[9]=	2 2 2 -4 1 1 2 z 4 z 4 z 7 z 4 z z 2 z z 4 z a - ---- - ---- + --- + --- + --- + --- + --- - - - ---- + --- + ---- - 5 3 11 9 7 5 3 a 12 10 8 a z a z a a a a a a a a 2 2 3 3 3 3 3 3 4 4 3 z 2 z 8 z 11 z 13 z 18 z 6 z 2 z 3 z 5 z > ---- - ---- - ---- - ----- - ----- - ----- - ---- + ---- + ---- - ---- - 4 2 11 9 7 5 3 a 12 10 a a a a a a a a a 4 4 4 4 5 5 5 5 5 5 11 z 6 z 2 z 5 z 9 z 10 z 10 z 18 z 8 z z > ----- - ---- + ---- + ---- + ---- + ----- + ----- + ----- + ---- - -- - 8 6 4 2 11 9 7 5 3 a a a a a a a a a a 6 6 6 6 6 6 7 7 7 7 7 z 9 z 17 z 14 z 4 z 3 z 3 z z 4 z 5 z 5 z > --- + ---- + ----- + ----- + ---- - ---- - ---- + -- + ---- - ---- - ---- - 12 10 8 6 4 2 11 9 7 5 3 a a a a a a a a a a a 8 8 8 8 9 9 9 10 10 4 z 7 z 8 z 5 z 3 z 6 z 3 z z z > ---- - ---- - ---- - ---- - ---- - ---- - ---- - --- - --- 10 8 6 4 9 7 5 8 6 a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 2 q 4 6 6 2 8 2 8 3 5 q + 3 q + ----- + - + -- + 7 q t + 4 q t + 9 q t + 7 q t + 8 q t + 2 2 t t q t 10 3 10 4 12 4 12 5 14 5 14 6 > 9 q t + 9 q t + 9 q t + 7 q t + 8 q t + 4 q t + 16 6 16 7 18 7 18 8 20 8 22 9 > 7 q t + 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a265

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