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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X7,18,8,19 X19,22,20,5 X9,21,10,20 X21,9,22,8 X11,17,12,16 X17,15,18,14 X15,11,16,10 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

- q-5/2 + q-3/2 - q-1/2 - q1/2 + 2q3/2 - 3q5/2 + 3q7/2 - 4q9/2 + 3q11/2 - 2q13/2 + q15/2

A2 (sl(3)) Invariant:

q-8 + q-6 + 2q-4 + q-2 + 2 - 2q4 - q8 + 2q10 + 2q12 + 2q14 + 2q16 - q18 - q22 - q24

HOMFLY-PT Polynomial:

a-7z-1 + a-7z - 4a-5z-1 - 5a-5z - 2a-5z3 + 6a-3z-1 + 10a-3z + 5a-3z3 + a-3z5 - 5a-1z-1 - 9a-1z - 6a-1z3 - a-1z5 + 2az-1 + 3az + az3

Kauffman Polynomial:

a-8 - 4a-8z2 + 4a-8z4 - a-8z6 - a-7z-1 + 4a-7z - 8a-7z3 + 8a-7z5 - 2a-7z7 + 3a-6 - 9a-6z2 + 8a-6z4 + a-6z6 - a-6z8 - 4a-5z-1 + 16a-5z - 24a-5z3 + 18a-5z5 - 4a-5z7 + 3a-4 - 7a-4z2 + 2a-4z4 + 3a-4z6 - a-4z8 - 6a-3z-1 + 27a-3z - 39a-3z3 + 20a-3z5 - 3a-3z7 + a-2 - 10a-2z4 + 7a-2z6 - a-2z8 - 5a-1z-1 + 22a-1z - 33a-1z3 + 16a-1z5 - 2a-1z7 + 1 + 2z2 - 8z4 + 6z6 - z8 - 2az-1 + 7az - 10az3 + 6az5 - az7

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 = 16                       1 j = 14                     1   j = 12                   2 1   j = 10                 2 1     j = 8             1 2 2       j = 6             2 2         j = 4         1 2 2           j = 2       1 2 2             j = 0       1 3               j = -2   1 1                   j = -4                         j = -6 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, NonAlternating, 112]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 112]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[7, 18, 8, 19], X[19, 22, 20, 5], > X[9, 21, 10, 20], X[21, 9, 22, 8], X[11, 17, 12, 16], X[17, 15, 18, 14], > X[15, 11, 16, 10], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -3, 6, -5, 9, -7, -2, 11, 8, -9, 7, -8, 3, > -4, 5, -6, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) -(3/2) 1 3/2 5/2 7/2 9/2 -q + q - ------- - Sqrt[q] + 2 q - 3 q + 3 q - 4 q + Sqrt[q] 11/2 13/2 15/2 > 3 q - 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 2 -2 4 8 10 12 14 16 18 2 + q + q + -- + q - 2 q - q + 2 q + 2 q + 2 q + 2 q - q - 4 q 22 24 > q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 112]][a, z]
Out[8]=	3 3 1 4 6 5 2 a z 5 z 10 z 9 z 2 z 5 z ---- - ---- + ---- - --- + --- + -- - --- + ---- - --- + 3 a z - ---- + ---- - 7 5 3 a z z 7 5 3 a 5 3 a z a z a z a a a a a 3 5 5 6 z 3 z z > ---- + a z + -- - -- a 3 a a
In[9]:=	Kauffman[Link[11, NonAlternating, 112]][a, z]
Out[9]=	-8 3 3 -2 1 4 6 5 2 a 4 z 16 z 27 z 1 + a + -- + -- + a - ---- - ---- - ---- - --- - --- + --- + ---- + ---- + 6 4 7 5 3 a z z 7 5 3 a a a z a z a z a a a 2 2 2 3 3 3 3 22 z 2 4 z 9 z 7 z 8 z 24 z 39 z 33 z > ---- + 7 a z + 2 z - ---- - ---- - ---- - ---- - ----- - ----- - ----- - a 8 6 4 7 5 3 a a a a a a a 4 4 4 4 5 5 5 3 4 4 z 8 z 2 z 10 z 8 z 18 z 20 z > 10 a z - 8 z + ---- + ---- + ---- - ----- + ---- + ----- + ----- + 8 6 4 2 7 5 3 a a a a a a a 5 6 6 6 6 7 7 7 7 16 z 5 6 z z 3 z 7 z 2 z 4 z 3 z 2 z > ----- + 6 a z + 6 z - -- + -- + ---- + ---- - ---- - ---- - ---- - ---- - a 8 6 4 2 7 5 3 a a a a a a a a 8 8 8 7 8 z z z > a z - z - -- - -- - -- 6 4 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 1 1 q 2 4 4 2 3 + 2 q + q + ----- + ----- + ----- + - + -- + 2 q t + 2 q t + 2 q t + 6 4 2 3 2 2 t t q t q t q t 6 2 8 2 6 3 8 3 8 4 10 4 10 5 > 2 q t + q t + 2 q t + 2 q t + 2 q t + 2 q t + q t + 12 5 12 6 14 6 16 7 > 2 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n112

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