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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9/2 - 4q-7/2 + 8q-5/2 - 14q-3/2 + 18q-1/2 - 22q1/2 + 21q3/2 - 19q5/2 + 14q7/2 - 8q9/2 + 4q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-14 + q-12 + q-10 - 2q-8 + 5q-6 + q-2 + 5 - 2q2 + 5q4 - 2q6 + q8 + q10 - 5q12 + 2q14 - q16 - q18 + q20

HOMFLY-PT Polynomial:

- a-5z - a-5z3 + a-3z-1 + 5a-3z + 5a-3z3 + 2a-3z5 - 3a-1z-1 - 9a-1z - 9a-1z3 - 4a-1z5 - a-1z7 + 2az-1 + 5az + 5az3 + 2az5 - a3z - a3z3

Kauffman Polynomial:

a-7z3 - a-7z5 - 3a-6z2 + 6a-6z4 - 4a-6z6 + 2a-5z - 6a-5z3 + 10a-5z5 - 7a-5z7 + a-4 - 8a-4z2 + 9a-4z4 + 3a-4z6 - 7a-4z8 - a-3z-1 + 9a-3z - 27a-3z3 + 33a-3z5 - 11a-3z7 - 4a-3z9 + 3a-2 - 7a-2z2 - 7a-2z4 + 28a-2z6 - 16a-2z8 - a-2z10 - 3a-1z-1 + 16a-1z - 40a-1z3 + 40a-1z5 - 4a-1z7 - 8a-1z9 + 3 - 2z2 - 17z4 + 34z6 - 15z8 - z10 - 2az-1 + 11az - 28az3 + 28az5 - 4az7 - 4az9 - a2z2 - 5a2z4 + 12a2z6 - 6a2z8 + 2a3z - 8a3z3 + 10a3z5 - 4a3z7 - a4z2 + 2a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 14                       1 j = 12                     3   j = 10                   5 1   j = 8                 9 3     j = 6               10 5       j = 4             11 9         j = 2           11 10           j = 0         8 12             j = -2       6 10               j = -4     3 9                 j = -6   1 5                   j = -8   3                     j = -10 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, 161]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 161]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[22, 10, 7, 9], X[2, 7, 3, 8], > X[4, 22, 5, 21], X[14, 5, 15, 6], X[16, 13, 17, 14], X[20, 18, 21, 17], > X[18, 12, 19, 11], X[12, 20, 13, 19], X[6, 15, 1, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -5, 6, -11}, > {4, -1, 3, -2, 9, -10, 7, -6, 11, -7, 8, -9, 10, -8, 5, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 4 8 14 18 3/2 5/2 q - ---- + ---- - ---- + ------- - 22 Sqrt[q] + 21 q - 19 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 14 q - 8 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 -10 2 5 -2 2 4 6 8 10 5 - q + q + q - -- + -- + q - 2 q + 5 q - 2 q + q + q - 8 6 q q 12 14 16 18 20 > 5 q + 2 q - q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 161]][a, z]
Out[8]=	3 3 3 1 3 2 a z 5 z 9 z 3 z 5 z 9 z 3 ---- - --- + --- - -- + --- - --- + 5 a z - a z - -- + ---- - ---- + 5 a z - 3 a z z 5 3 a 5 3 a a z a a a a 5 5 7 3 3 2 z 4 z 5 z > a z + ---- - ---- + 2 a z - -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 161]][a, z]
Out[9]=	-4 3 1 3 2 a 2 z 9 z 16 z 3 2 3 + a + -- - ---- - --- - --- + --- + --- + ---- + 11 a z + 2 a z - 2 z - 2 3 a z z 5 3 a a a z a a 2 2 2 3 3 3 3 3 z 8 z 7 z 2 2 4 2 z 6 z 27 z 40 z 3 > ---- - ---- - ---- - a z - a z + -- - ---- - ----- - ----- - 28 a z - 6 4 2 7 5 3 a a a a a a a 4 4 4 5 5 3 3 4 6 z 9 z 7 z 2 4 4 4 z 10 z > 8 a z - 17 z + ---- + ---- - ---- - 5 a z + 2 a z - -- + ----- + 6 4 2 7 5 a a a a a 5 5 6 6 6 33 z 40 z 5 3 5 6 4 z 3 z 28 z > ----- + ----- + 28 a z + 10 a z + 34 z - ---- + ---- + ----- + 3 a 6 4 2 a a a a 7 7 7 8 2 6 4 6 7 z 11 z 4 z 7 3 7 8 7 z > 12 a z - a z - ---- - ----- - ---- - 4 a z - 4 a z - 15 z - ---- - 5 3 a 4 a a a 8 9 9 10 16 z 2 8 4 z 8 z 9 10 z > ----- - 6 a z - ---- - ---- - 4 a z - z - --- 2 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 5 3 9 6 8 12 + 11 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t q t q t q t q t q t q t q t 10 2 4 4 2 6 2 6 3 8 3 > ---- + 10 q t + 11 q t + 9 q t + 10 q t + 5 q t + 9 q t + 2 q t 8 4 10 4 10 5 12 5 14 6 > 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 L11a161

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