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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X14,8,15,7 X11,19,12,18 X19,5,20,22 X15,21,16,20 X21,17,22,16 X17,13,18,12 X8,14,9,13 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-5/2 - 4q1/2 + 5q3/2 - 6q5/2 + 7q7/2 - 6q9/2 + 5q11/2 - 3q13/2 + q15/2

A2 (sl(3)) Invariant:

q-8 + 2q-6 + 3q-4 + 2q-2 + 5 + 2q2 + q6 - 3q8 - q10 - 3q12 - q14 + q16 - q18 + 2q20 - q24

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- 2a-8z2 + 3a-8z4 - a-8z6 + a-7z - 7a-7z3 + 10a-7z5 - 3a-7z7 - a-6 + 3a-6z2 - 5a-6z4 + 9a-6z6 - 3a-6z8 + a-5z - 8a-5z3 + 11a-5z5 - a-5z7 - a-5z9 + 10a-4z2 - 19a-4z4 + 15a-4z6 - 4a-4z8 - 2a-3z-1 + 3a-3z - 3a-3z3 - 2a-3z5 + 3a-3z7 - a-3z9 + 5a-2 - 5a-2z2 - 7a-2z4 + 5a-2z6 - a-2z8 - 5a-1z-1 + 14a-1z - 16a-1z3 + 4a-1z5 + 5 - 10z2 + 4z4 - 3az-1 + 11az - 14az3 + 7az5 - 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                     2   j = 12                   3 1   j = 10                 3 2     j = 8               4 3       j = 6           1 3 3         j = 4           3 4           j = 2       1 3 3             j = 0         4               j = -2     1 1                 j = -4 1                       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, 59]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 59]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 8, 15, 7], X[11, 19, 12, 18], > X[19, 5, 20, 22], X[15, 21, 16, 20], X[21, 17, 22, 16], X[17, 13, 18, 12], > X[8, 14, 9, 13], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, -4, 8, 9, -3, -6, 7, -8, 4, > -5, 6, -7, 5}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 3/2 5/2 7/2 9/2 11/2 13/2 -q - 4 Sqrt[q] + 5 q - 6 q + 7 q - 6 q + 5 q - 3 q + 15/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 2 3 2 2 6 8 10 12 14 16 18 5 + q + -- + -- + -- + 2 q + q - 3 q - q - 3 q - q + q - q + 6 4 2 q q q 20 24 > 2 q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 59]][a, z]
Out[8]=	3 3 3 2 5 3 a z 2 z 5 z 10 z 2 z 3 z 7 z 3 ---- - --- + --- + -- - --- + --- - ---- + 4 a z - ---- + ---- - ---- + a z + 3 a z z 7 5 3 a 5 3 a a z a a a a a 5 5 z z > -- - -- 3 a a
In[9]:=	Kauffman[Link[11, NonAlternating, 59]][a, z]
Out[9]=	-6 5 2 5 3 a z z 3 z 14 z 2 5 - a + -- - ---- - --- - --- + -- + -- + --- + ---- + 11 a z - 10 z - 2 3 a z z 7 5 3 a a a z a a a 2 2 2 2 3 3 3 3 2 z 3 z 10 z 5 z 7 z 8 z 3 z 16 z 3 4 > ---- + ---- + ----- - ---- - ---- - ---- - ---- - ----- - 14 a z + 4 z + 8 6 4 2 7 5 3 a a a a a a a a 4 4 4 4 5 5 5 5 6 3 z 5 z 19 z 7 z 10 z 11 z 2 z 4 z 5 z > ---- - ---- - ----- - ---- + ----- + ----- - ---- + ---- + 7 a z - -- + 8 6 4 2 7 5 3 a 8 a a a a a a a a 6 6 6 7 7 7 8 8 8 9 9 9 z 15 z 5 z 3 z z 3 z 7 3 z 4 z z z z > ---- + ----- + ---- - ---- - -- + ---- - a z - ---- - ---- - -- - -- - -- 6 4 2 7 5 3 6 4 2 5 3 a a a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 1 1 q 2 4 6 4 + 3 q + ----- + ----- + ----- + ---- + -- + 3 q t + 3 q t + q t + 6 4 4 4 2 2 2 t q t q t q t q t 4 2 6 2 6 3 8 3 8 4 10 4 10 5 > 4 q t + 3 q t + 3 q t + 4 q t + 3 q t + 3 q t + 2 q t + 12 5 12 6 14 6 16 7 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n59

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