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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 3q-9/2 - 7q-7/2 + 10q-5/2 - 15q-3/2 + 16q-1/2 - 17q1/2 + 16q3/2 - 11q5/2 + 7q7/2 - 4q9/2 + q11/2

A2 (sl(3)) Invariant:

q-18 + q-16 - q-14 + 2q-12 + 2q-10 + q-8 + 5q-6 - q-4 + q-2 - 3q2 + 2q4 - 4q6 + 2q8 + q10 - q12 + 2q14 - q16

HOMFLY-PT Polynomial:

2a-3z3 + a-3z5 - 3a-1z - 6a-1z3 - 4a-1z5 - a-1z7 + 6az + 9az3 + 3az5 - a3z-1 - 6a3z - 3a3z3 + a5z-1 + a5z

Kauffman Polynomial:

- a-6z4 + 3a-5z3 - 4a-5z5 - 3a-4z2 + 7a-4z4 - 7a-4z6 + a-3z + a-3z3 + 6a-3z5 - 8a-3z7 - 11a-2z2 + 14a-2z4 + 2a-2z6 - 7a-2z8 + 8a-1z - 23a-1z3 + 28a-1z5 - 6a-1z7 - 4a-1z9 - 5z2 - 12z4 + 31z6 - 12z8 - z10 + 13az - 36az3 + 24az5 + 8az7 - 7az9 + 6a2z2 - 30a2z4 + 33a2z6 - 8a2z8 - a2z10 - a3z-1 + 10a3z - 21a3z3 + 10a3z5 + 5a3z7 - 3a3z9 + a4 + 3a4z2 - 12a4z4 + 11a4z6 - 3a4z8 - a5z-1 + 4a5z - 6a5z3 + 4a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12                       1 j = 10                     3   j = 8                   4 1   j = 6                 7 3     j = 4               9 4       j = 2             8 7         j = 0           9 10           j = -2         6 7             j = -4       4 9               j = -6     3 6                 j = -8   1 5                   j = -10   2                     j = -12 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, 267]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 267]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[22, 12, 9, 11], X[2, 9, 3, 10], > X[16, 14, 17, 13], X[14, 5, 15, 6], X[4, 15, 5, 16], X[20, 18, 21, 17], > X[18, 7, 19, 8], X[6, 19, 7, 20], X[8, 22, 1, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -7, 6, -10, 9, -11}, > {4, -1, 3, -2, 5, -6, 7, -5, 8, -9, 10, -8, 11, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 3 7 10 15 16 3/2 -q + ---- - ---- + ---- - ---- + ------- - 17 Sqrt[q] + 16 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 11 q + 7 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 -14 2 2 -8 5 -4 -2 2 4 6 q + q - q + --- + --- + q + -- - q + q - 3 q + 2 q - 4 q + 12 10 6 q q q 8 10 12 14 16 > 2 q + q - q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 267]][a, z]
Out[8]=	3 5 3 3 a a 3 z 3 5 2 z 6 z 3 3 3 -(--) + -- - --- + 6 a z - 6 a z + a z + ---- - ---- + 9 a z - 3 a z + z z a 3 a a 5 5 7 z 4 z 5 z > -- - ---- + 3 a z - -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 267]][a, z]
Out[9]=	3 5 2 2 4 a a z 8 z 3 5 2 3 z 11 z a - -- - -- + -- + --- + 13 a z + 10 a z + 4 a z - 5 z - ---- - ----- + z z 3 a 4 2 a a a 3 3 3 2 2 4 2 3 z z 23 z 3 3 3 5 3 > 6 a z + 3 a z + ---- + -- - ----- - 36 a z - 21 a z - 6 a z - 5 3 a a a 4 4 4 5 5 5 4 z 7 z 14 z 2 4 4 4 4 z 6 z 28 z > 12 z - -- + ---- + ----- - 30 a z - 12 a z - ---- + ---- + ----- + 6 4 2 5 3 a a a a a a 6 6 5 3 5 5 5 6 7 z 2 z 2 6 4 6 > 24 a z + 10 a z + 4 a z + 31 z - ---- + ---- + 33 a z + 11 a z - 4 2 a a 7 7 8 8 z 6 z 7 3 7 5 7 8 7 z 2 8 4 8 > ---- - ---- + 8 a z + 5 a z - a z - 12 z - ---- - 8 a z - 3 a z - 3 a 2 a a 9 4 z 9 3 9 10 2 10 > ---- - 7 a z - 3 a z - z - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 5 3 6 4 9 10 + 8 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 6 9 7 2 4 4 2 6 2 6 3 > ----- + - + ---- + 7 q t + 9 q t + 4 q t + 7 q t + 3 q t + 2 2 t 2 q t q t 8 3 8 4 10 4 12 5 > 4 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a267

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