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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X22,12,19,11 X10,4,11,3 X20,5,21,6 X18,21,5,22 X12,20,13,19 X14,9,15,10 X2,14,3,13 X8,15,9,16

Gauss Code:

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

Jones Polynomial:

- q-7 + 3q-6 - 7q-5 + 12q-4 - 16q-3 + 21q-2 - 20q-1 + 19 - 14q + 10q2 - 4q3 + q4

A2 (sl(3)) Invariant:

- q-22 + q-18 - 3q-16 + 2q-14 + q-12 - 2q-10 + 6q-8 + q-6 + 5q-4 + 4q-2 + 1 + 7q2 - q4 + 4q6 + 3q8 - 2q10 + q12

HOMFLY-PT Polynomial:

a-2z-2 + 2a-2 + a-2z2 + a-2z4 - 2z-2 - 5 - 4z2 - 2z4 - z6 + a2z-2 + 3a2 + a2z2 - a2z4 - a2z6 + a4 + 3a4z2 + 2a4z4 - a6 - a6z2

Kauffman Polynomial:

a-4z4 + 4a-3z5 + a-2z-2 - 4a-2 + 6a-2z2 - 10a-2z4 + 10a-2z6 - 2a-1z-1 + 3a-1z + 4a-1z3 - 18a-1z5 + 14a-1z7 + 2z-2 - 9 + 15z2 - 5z4 - 16z6 + 13z8 - 2az-1 + 7az - 2az3 - 10az5 - 5az7 + 8az9 + a2z-2 - 6a2 - a2z2 + 34a2z4 - 48a2z6 + 15a2z8 + 2a2z10 + 3a3z - 6a3z3 + 21a3z5 - 31a3z7 + 12a3z9 + 2a4 - 17a4z2 + 41a4z4 - 33a4z6 + 5a4z8 + 2a4z10 - 3a5z + 5a5z3 + 5a5z5 - 11a5z7 + 4a5z9 + 2a6 - 7a6z2 + 13a6z4 - 11a6z6 + 3a6z8 - 2a7z + 5a7z3 - 4a7z5 + a7z7

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 9                       1 j = 7                     3   j = 5                   7 1   j = 3                 7 3     j = 1               12 7       j = -1             12 11         j = -3           9 8           j = -5         7 12             j = -7       5 9               j = -9     2 7                 j = -11   1 5                   j = -13   2                     j = -15 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, 495]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 495]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[22, 12, 19, 11], > X[10, 4, 11, 3], X[20, 5, 21, 6], X[18, 21, 5, 22], X[12, 20, 13, 19], > X[14, 9, 15, 10], X[2, 14, 3, 13], X[8, 15, 9, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -3}, {8, -6, 7, -4}, > {6, -1, 2, -11, 9, -5, 4, -8, 10, -9, 11, -2, 3, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-7 3 7 12 16 21 20 2 3 4 19 - q + -- - -- + -- - -- + -- - -- - 14 q + 10 q - 4 q + q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -18 3 2 -12 2 6 -6 5 4 2 4 1 - q + q - --- + --- + q - --- + -- + q + -- + -- + 7 q - q + 16 14 10 8 4 2 q q q q q q 6 8 10 12 > 4 q + 3 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 495]][a, z]
Out[8]=	2 2 2 2 4 6 2 1 a 2 z 2 2 4 2 -5 + -- + 3 a + a - a - -- + ----- + -- - 4 z + -- + a z + 3 a z - 2 2 2 2 2 2 a z a z z a 4 6 2 4 z 2 4 4 4 6 2 6 > a z - 2 z + -- - a z + 2 a z - z - a z 2 a
In[9]:=	Kauffman[Link[11, Alternating, 495]][a, z]
Out[9]=	2 4 2 4 6 2 1 a 2 2 a 3 z -9 - -- - 6 a + 2 a + 2 a + -- + ----- + -- - --- - --- + --- + 7 a z + 2 2 2 2 2 a z z a a z a z z 2 3 5 7 2 6 z 2 2 4 2 6 2 > 3 a z - 3 a z - 2 a z + 15 z + ---- - a z - 17 a z - 7 a z + 2 a 3 4 4 4 z 3 3 3 5 3 7 3 4 z 10 z > ---- - 2 a z - 6 a z + 5 a z + 5 a z - 5 z + -- - ----- + a 4 2 a a 5 5 2 4 4 4 6 4 4 z 18 z 5 3 5 > 34 a z + 41 a z + 13 a z + ---- - ----- - 10 a z + 21 a z + 3 a a 6 5 5 7 5 6 10 z 2 6 4 6 6 6 > 5 a z - 4 a z - 16 z + ----- - 48 a z - 33 a z - 11 a z + 2 a 7 14 z 7 3 7 5 7 7 7 8 2 8 4 8 > ----- - 5 a z - 31 a z - 11 a z + a z + 13 z + 15 a z + 5 a z + a 6 8 9 3 9 5 9 2 10 4 10 > 3 a z + 8 a z + 12 a z + 4 a z + 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	11 1 2 1 5 2 7 5 9 -- + 12 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 7 3 q t q t q t q t q t q t q t q t 7 12 9 8 12 3 3 2 5 2 > ----- + ----- + ----- + ---- + --- + 7 q t + 7 q t + 3 q t + 7 q t + 5 3 5 2 3 2 3 q t q t q t q t q t 5 3 7 3 9 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a495

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