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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X7,14,8,15 X11,19,12,18 X19,5,20,22 X15,21,16,20 X21,17,22,16 X17,13,18,12 X13,8,14,9 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-11/2 + 2q-9/2 - 4q-7/2 + 4q-5/2 - 4q-3/2 + 3q-1/2 - 3q1/2 + q3/2 - q7/2 + q9/2

A2 (sl(3)) Invariant:

q-18 + q-16 + 2q-12 + q-8 + q-6 + 2q-2 + 2q2 + q4 + q8 - q10 - q12 - q14

HOMFLY-PT Polynomial:

a-3z-1 + 3a-3z + a-3z3 - 3a-1z-1 - 8a-1z - 6a-1z3 - a-1z5 + 3az-1 + 6az + 4az3 + az5 - 2a3z-1 - 4a3z - 2a3z3 + a5z-1 + a5z

Kauffman Polynomial:

a-4 - 5a-4z2 + 5a-4z4 - a-4z6 - a-3z-1 + 4a-3z - 8a-3z3 + 6a-3z5 - a-3z7 + 2a-2 - 7a-2z2 + 4a-2z4 - 3a-1z-1 + 13a-1z - 18a-1z3 + 9a-1z5 - a-1z7 + 5z2 - 14z4 + 11z6 - 2z8 - 3az-1 + 14az - 22az3 + 10az5 + 2az7 - az9 - 2a2 + 10a2z2 - 23a2z4 + 19a2z6 - 4a2z8 - 2a3z-1 + 10a3z - 20a3z3 + 12a3z5 + a3z7 - a3z9 + 3a4z2 - 10a4z4 + 9a4z6 - 2a4z8 - a5z-1 + 5a5z - 8a5z3 + 5a5z5 - 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 = 10                       1 j = 8                         j = 6                 1 1 1   j = 4               2 1       j = 2             2 1 1       j = 0           3 4 1         j = -2         2 2 1           j = -4       2 3 1             j = -6     2 2                 j = -8   1 3                   j = -10   1                     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, NonAlternating, 57]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 57]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[7, 14, 8, 15], 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[13, 8, 14, 9], 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]=	-(11/2) 2 4 4 4 3 3/2 7/2 9/2 -q + ---- - ---- + ---- - ---- + ------- - 3 Sqrt[q] + q - q + q 9/2 7/2 5/2 3/2 Sqrt[q] q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 2 -8 -6 2 2 4 8 10 12 14 q + q + --- + q + q + -- + 2 q + q + q - q - q - q 12 2 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 57]][a, z]
Out[8]=	3 5 3 3 1 3 3 a 2 a a 3 z 8 z 3 5 z 6 z ---- - --- + --- - ---- + -- + --- - --- + 6 a z - 4 a z + a z + -- - ---- + 3 a z z z z 3 a 3 a a z a a 5 3 3 3 z 5 > 4 a z - 2 a z - -- + a z a
In[9]:=	Kauffman[Link[11, NonAlternating, 57]][a, z]
Out[9]=	3 5 -4 2 2 1 3 3 a 2 a a 4 z 13 z a + -- - 2 a - ---- - --- - --- - ---- - -- + --- + ---- + 14 a z + 2 3 a z z z z 3 a a a z a 2 2 3 3 3 5 2 5 z 7 z 2 2 4 2 8 z 18 z > 10 a z + 5 a z + 5 z - ---- - ---- + 10 a z + 3 a z - ---- - ----- - 4 2 3 a a a a 4 4 3 3 3 5 3 4 5 z 4 z 2 4 4 4 > 22 a z - 20 a z - 8 a z - 14 z + ---- + ---- - 23 a z - 10 a z + 4 2 a a 5 5 6 6 z 9 z 5 3 5 5 5 6 z 2 6 > ---- + ---- + 10 a z + 12 a z + 5 a z + 11 z - -- + 19 a z + 3 a 4 a a 7 7 4 6 z z 7 3 7 5 7 8 2 8 4 8 > 9 a z - -- - -- + 2 a z + a z - a z - 2 z - 4 a z - 2 a z - 3 a a 9 3 9 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-2 2 1 1 1 3 2 2 2 4 + q + 2 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 q t q t q t q t q t q t q t 3 2 3 1 2 2 4 2 2 4 2 > ----- + ----- + - + ---- + ---- + t + q t + 2 q t + q t + q t + 4 2 2 2 t 4 2 q t q t q t q t 6 2 6 3 6 4 10 5 > q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n57

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