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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-15/2 - 3q-13/2 + 6q-11/2 - 11q-9/2 + 16q-7/2 - 19q-5/2 + 19q-3/2 - 18q-1/2 + 13q1/2 - 9q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-22 + q-20 - q-18 + 3q-14 - 3q-12 + 3q-10 - q-8 - q-6 + 3q-4 - 2q-2 + 6 + q4 + 2q6 - 2q8 + q10

HOMFLY-PT Polynomial:

- a-1z-1 - a-1z - 2a-1z3 - a-1z5 + az-1 + 2az3 + 3az5 + az7 + 3a3z + 6a3z3 + 4a3z5 + a3z7 - 2a5z - 3a5z3 - a5z5

Kauffman Polynomial:

a-3z3 - a-3z5 + 5a-2z4 - 4a-2z6 - a-1z-1 + 2a-1z - 7a-1z3 + 14a-1z5 - 8a-1z7 + 1 + 2z2 - 10z4 + 16z6 - 9z8 - az-1 - az5 + 6az7 - 6az9 + 3a2z2 - 14a2z4 + 18a2z6 - 7a2z8 - 2a2z10 - 6a3z + 25a3z3 - 34a3z5 + 25a3z7 - 10a3z9 + 4a4z2 - 7a4z4 + 8a4z6 - 2a4z8 - 2a4z10 - 4a5z + 10a5z3 - 9a5z5 + 8a5z7 - 4a5z9 + a6z2 - 5a6z4 + 9a6z6 - 4a6z8 - 7a7z3 + 9a7z5 - 3a7z7 - 2a8z2 + 3a8z4 - a8z6

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 = 8                       1 j = 6                     3   j = 4                   6 1   j = 2                 7 3     j = 0               11 6       j = -2             10 9         j = -4           9 9           j = -6         7 10             j = -8       4 9               j = -10     2 7                 j = -12   1 4                   j = -14   2                     j = -16 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, 101]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 101]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[20, 11, 21, 12], X[8, 19, 9, 20], > X[18, 9, 19, 10], X[10, 17, 11, 18], X[4, 21, 1, 22], X[14, 6, 15, 5], > X[12, 4, 13, 3], X[22, 14, 5, 13], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 9, -7}, {8, -1, 2, -4, 5, -6, 3, -9, 10, -8, 11, -2, 6, -5, > 4, -3, 7, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 6 11 16 19 19 18 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 13 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 9 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 -18 3 3 3 -8 -6 3 2 4 6 6 - q + q - q + --- - --- + --- - q - q + -- - -- + q + 2 q - 14 12 10 4 2 q q q q q 8 10 > 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 101]][a, z]
Out[8]=	3 5 1 a z 3 5 2 z 3 3 3 5 3 z -(---) + - - - + 3 a z - 2 a z - ---- + 2 a z + 6 a z - 3 a z - -- + a z z a a a 5 3 5 5 5 7 3 7 > 3 a z + 4 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 101]][a, z]
Out[9]=	1 a 2 z 3 5 2 2 2 4 2 6 2 1 - --- - - + --- - 6 a z - 4 a z + 2 z + 3 a z + 4 a z + a z - a z z a 3 3 4 8 2 z 7 z 3 3 5 3 7 3 4 5 z > 2 a z + -- - ---- + 25 a z + 10 a z - 7 a z - 10 z + ---- - 3 a 2 a a 5 5 2 4 4 4 6 4 8 4 z 14 z 5 3 5 > 14 a z - 7 a z - 5 a z + 3 a z - -- + ----- - a z - 34 a z - 3 a a 6 5 5 7 5 6 4 z 2 6 4 6 6 6 8 6 > 9 a z + 9 a z + 16 z - ---- + 18 a z + 8 a z + 9 a z - a z - 2 a 7 8 z 7 3 7 5 7 7 7 8 2 8 4 8 > ---- + 6 a z + 25 a z + 8 a z - 3 a z - 9 z - 7 a z - 2 a z - a 6 8 9 3 9 5 9 2 10 4 10 > 4 a z - 6 a z - 10 a z - 4 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 2 1 4 2 7 4 9 11 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 7 10 9 9 10 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 6 t + 7 q t + 3 q t + 6 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a101

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