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DrawMorseLink

PD Presentation:

X6172 X14,3,15,4 X20,12,21,11 X16,8,17,7 X18,10,19,9 X10,20,11,19 X22,16,13,15 X12,18,5,17 X8,22,9,21 X2536 X4,13,1,14

Gauss Code:

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

Jones Polynomial:

q-3 - 2q-2 + 6q-1 - 9 + 14q - 17q2 + 18q3 - 15q4 + 13q5 - 8q6 + 4q7 - q8

A2 (sl(3)) Invariant:

q-10 + q-8 + 3q-4 + 1 + 5q2 - q4 + 5q6 + 2q8 + 3q10 + 5q12 - q14 + 4q16 - q18 - q20 + 2q22 - q24

HOMFLY-PT Polynomial:

- a-6z2 - a-6z4 + a-4z-2 + 3a-4 + 2a-4z2 + 2a-4z4 + a-4z6 - 2a-2z-2 - 5a-2 - 3a-2z2 + a-2z4 + a-2z6 + z-2 - 4z2 - 2z4 + 2a2 + a2z2

Kauffman Polynomial:

- a-9z3 + a-9z5 + 2a-8z2 - 6a-8z4 + 4a-8z6 + 3a-7z3 - 11a-7z5 + 7a-7z7 - 2a-6 + 2a-6z2 + a-6z4 - 9a-6z6 + 7a-6z8 + a-5z + 3a-5z3 - 8a-5z5 + a-5z7 + 4a-5z9 - a-4z-2 + 3a-4 - 5a-4z2 + 12a-4z4 - 18a-4z6 + 9a-4z8 + a-4z10 + 2a-3z-1 - 8a-3z + 13a-3z3 - 5a-3z5 - 6a-3z7 + 6a-3z9 - 2a-2z-2 + 9a-2 - 15a-2z2 + 15a-2z4 - 13a-2z6 + 5a-2z8 + a-2z10 + 2a-1z-1 - 8a-1z + 16a-1z3 - 14a-1z5 + 2a-1z7 + 2a-1z9 - z-2 + 3 - 5z2 + 6z4 - 7z6 + 3z8 + az + 2az3 - 5az5 + 2az7 - 2a2 + 5a2z2 - 4a2z4 + a2z6

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 = 17                       1 j = 15                     3   j = 13                   5 1   j = 11                 8 3     j = 9               9 7       j = 7             9 6         j = 5           8 9           j = 3         6 9             j = 1       4 9               j = -1     2 5                 j = -3     4                   j = -5 1 2                     j = -7 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 444]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 444]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[20, 12, 21, 11], X[16, 8, 17, 7], > X[18, 10, 19, 9], X[10, 20, 11, 19], X[22, 16, 13, 15], X[12, 18, 5, 17], > X[8, 22, 9, 21], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -9, 5, -6, 3, -8}, > {11, -2, 7, -4, 8, -5, 6, -3, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-3 2 6 2 3 4 5 6 7 8 -9 + q - -- + - + 14 q - 17 q + 18 q - 15 q + 13 q - 8 q + 4 q - q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -8 3 2 4 6 8 10 12 14 16 1 + q + q + -- + 5 q - q + 5 q + 2 q + 3 q + 5 q - q + 4 q - 4 q 18 20 22 24 > q - q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 444]][a, z]
Out[8]=	2 2 2 3 5 2 -2 1 2 2 z 2 z 3 z 2 2 4 -- - -- + 2 a + z + ----- - ----- - 4 z - -- + ---- - ---- + a z - 2 z - 4 2 4 2 2 2 6 4 2 a a a z a z a a a 4 4 4 6 6 z 2 z z z z > -- + ---- + -- + -- + -- 6 4 2 4 2 a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 444]][a, z]
Out[9]=	2 3 9 2 -2 1 2 2 2 z 8 z 8 z 3 - -- + -- + -- - 2 a - z - ----- - ----- + ---- + --- + -- - --- - --- + 6 4 2 4 2 2 2 3 a z 5 3 a a a a a z a z a z a a 2 2 2 2 3 3 3 2 2 z 2 z 5 z 15 z 2 2 z 3 z 3 z > a z - 5 z + ---- + ---- - ---- - ----- + 5 a z - -- + ---- + ---- + 8 6 4 2 9 7 5 a a a a a a a 3 3 4 4 4 4 5 13 z 16 z 3 4 6 z z 12 z 15 z 2 4 z > ----- + ----- + 2 a z + 6 z - ---- + -- + ----- + ----- - 4 a z + -- - 3 a 8 6 4 2 9 a a a a a a 5 5 5 5 6 6 6 6 11 z 8 z 5 z 14 z 5 6 4 z 9 z 18 z 13 z > ----- - ---- - ---- - ----- - 5 a z - 7 z + ---- - ---- - ----- - ----- + 7 5 3 a 8 6 4 2 a a a a a a a 7 7 7 7 8 8 8 2 6 7 z z 6 z 2 z 7 8 7 z 9 z 5 z > a z + ---- + -- - ---- + ---- + 2 a z + 3 z + ---- + ---- + ---- + 7 5 3 a 6 4 2 a a a a a a 9 9 9 10 10 4 z 6 z 2 z z z > ---- + ---- + ---- + --- + --- 5 3 a 4 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 2 4 2 5 4 q 3 9 q + 6 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 9 q t + 7 4 5 4 5 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 8 q t + 9 q t + 9 q t + 6 q t + 9 q t + 7 q t + 8 q t + 11 5 13 5 13 6 15 6 17 7 > 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a444

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