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DrawMorseLink

PD Presentation:

X8192 X20,9,21,10 X14,5,15,6 X18,8,19,7 X10,4,11,3 X22,12,7,11 X16,13,17,14 X12,17,13,18 X6,15,1,16 X4,21,5,22 X2,20,3,19

Gauss Code:

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

Jones Polynomial:

q-15/2 - 5q-13/2 + 11q-11/2 - 17q-9/2 + 23q-7/2 - 27q-5/2 + 25q-3/2 - 23q-1/2 + 16q1/2 - 9q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-22 + 3q-20 - 2q-18 + 2q-14 - 6q-12 + 5q-10 + 4q-6 + 5q-4 - 3q-2 + 5 - 4q2 + 2q6 - 2q8 + q10

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-3z3 - a-3z5 - 2a-2z2 + 5a-2z4 - 4a-2z6 + 2a-1z - 7a-1z3 + 11a-1z5 - 8a-1z7 - 2z2 - 2z4 + 11z6 - 10z8 + az-1 + 5az - 16az3 + 16az5 + az7 - 8az9 - a2 + 2a2z2 - 23a2z4 + 39a2z6 - 16a2z8 - 3a2z10 + a3z-1 + 5a3z - 12a3z3 + 26a3z7 - 17a3z9 + 2a4z2 - 29a4z4 + 47a4z6 - 16a4z8 - 3a4z10 + 2a5z - 7a5z3 + 5a5z5 + 12a5z7 - 9a5z9 - 12a6z4 + 22a6z6 - 10a6z8 - 3a7z3 + 9a7z5 - 5a7z7 + a8z4 - 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                 10 3     j = 0               13 6       j = -2             13 11         j = -4           14 12           j = -6         10 14             j = -8       7 13               j = -10     4 10                 j = -12   1 7                   j = -14   4                     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, 184]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 184]]
Out[4]=	PD[X[8, 1, 9, 2], X[20, 9, 21, 10], X[14, 5, 15, 6], X[18, 8, 19, 7], > X[10, 4, 11, 3], X[22, 12, 7, 11], X[16, 13, 17, 14], X[12, 17, 13, 18], > X[6, 15, 1, 16], X[4, 21, 5, 22], X[2, 20, 3, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -10, 3, -9}, > {4, -1, 2, -5, 6, -8, 7, -3, 9, -7, 8, -4, 11, -2, 10, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 5 11 17 23 27 25 23 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 16 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 3 2 2 6 5 4 5 3 2 6 8 10 5 - q + --- - --- + --- - --- + --- + -- + -- - -- - 4 q + 2 q - 2 q + q 20 18 14 12 10 6 4 2 q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 184]][a, z]
Out[8]=	3 3 5 a a z 3 5 2 z 3 3 3 5 3 z -(-) + -- - - + 2 a z - 3 a z + a z - ---- + 4 a z - a z - a z - -- + z z a a a 5 3 5 5 5 7 3 7 > 3 a z + 2 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 184]][a, z]
Out[9]=	3 2 2 a a 2 z 3 5 2 2 z 2 2 -a + - + -- + --- + 5 a z + 5 a z + 2 a z - 2 z - ---- + 2 a z + z z a 2 a 3 3 4 2 z 7 z 3 3 3 5 3 7 3 4 > 2 a z + -- - ---- - 16 a z - 12 a z - 7 a z - 3 a z - 2 z + 3 a a 4 5 5 5 z 2 4 4 4 6 4 8 4 z 11 z 5 > ---- - 23 a z - 29 a z - 12 a z + a z - -- + ----- + 16 a z + 2 3 a a a 6 5 5 7 5 6 4 z 2 6 4 6 6 6 8 6 > 5 a z + 9 a z + 11 z - ---- + 39 a z + 47 a z + 22 a z - a z - 2 a 7 8 z 7 3 7 5 7 7 7 8 2 8 4 8 > ---- + a z + 26 a z + 12 a z - 5 a z - 10 z - 16 a z - 16 a z - a 6 8 9 3 9 5 9 2 10 4 10 > 10 a z - 8 a z - 17 a z - 9 a z - 3 a z - 3 a z
In[10]:=	Kh[L][q, t]
Out[10]=	11 1 4 1 7 4 10 7 13 13 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 10 14 14 12 13 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 6 t + 10 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 L11a184

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