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DrawMorseLink

PD Presentation:

X6172 X20,7,21,8 X4,21,1,22 X14,10,15,9 X8493 X12,5,13,6 X22,13,5,14 X18,16,19,15 X16,11,17,12 X10,17,11,18 X2,20,3,19

Gauss Code:

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

Jones Polynomial:

q-15/2 - 4q-13/2 + 9q-11/2 - 16q-9/2 + 23q-7/2 - 27q-5/2 + 27q-3/2 - 25q-1/2 + 18q1/2 - 12q3/2 + 5q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-24 + 3q-20 - 2q-18 + 3q-14 - 6q-12 + 3q-10 - q-8 + 5q-4 - 3q-2 + 8 - 2q2 + 4q6 - 3q8 + q10

HOMFLY-PT Polynomial:

- a-1z-1 - a-1z - a-1z3 - a-1z5 + az-1 + 3az + 5az3 + 3az5 + az7 - 4a3z - 6a3z3 - 3a3z5 + 3a5z + 3a5z3 - a7z

Kauffman Polynomial:

- a-3z5 + 3a-2z4 - 5a-2z6 - a-1z-1 + 2a-1z - 7a-1z3 + 16a-1z5 - 12a-1z7 + 1 + 3z2 - 9z4 + 21z6 - 15z8 - az-1 + 6az - 26az3 + 38az5 - 7az7 - 9az9 + 9a2z2 - 41a2z4 + 64a2z6 - 28a2z8 - 2a2z10 + 8a3z - 35a3z3 + 34a3z5 + 9a3z7 - 15a3z9 + 10a4z2 - 41a4z4 + 53a4z6 - 20a4z8 - 2a4z10 + 6a5z - 23a5z3 + 22a5z5 - 6a5z9 + 3a6z2 - 10a6z4 + 14a6z6 - 7a6z8 + 2a7z - 7a7z3 + 9a7z5 - 4a7z7 - a8z2 + 2a8z4 - 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                     4   j = 4                   8 1   j = 2                 10 4     j = 0               15 8       j = -2             14 12         j = -4           13 13           j = -6         10 14             j = -8       6 13               j = -10     3 10                 j = -12   1 6                   j = -14   3                     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, 11]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 11]]
Out[4]=	PD[X[6, 1, 7, 2], X[20, 7, 21, 8], X[4, 21, 1, 22], X[14, 10, 15, 9], > X[8, 4, 9, 3], X[12, 5, 13, 6], X[22, 13, 5, 14], X[18, 16, 19, 15], > X[16, 11, 17, 12], X[10, 17, 11, 18], X[2, 20, 3, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {6, -1, 2, -5, 4, -10, 9, -6, 7, -4, 8, -9, 10, -8, > 11, -2, 3, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 4 9 16 23 27 27 25 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 18 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 > 12 q + 5 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 3 2 3 6 3 -8 5 3 2 6 8 8 - q + --- - --- + --- - --- + --- - q + -- - -- - 2 q + 4 q - 3 q + 20 18 14 12 10 4 2 q q q q q q q 10 > q
In[8]:=	HOMFLYPT[Link[11, Alternating, 11]][a, z]
Out[8]=	3 1 a z 3 5 7 z 3 3 3 -(---) + - - - + 3 a z - 4 a z + 3 a z - a z - -- + 5 a z - 6 a z + a z z a a 5 5 3 z 5 3 5 7 > 3 a z - -- + 3 a z - 3 a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 11]][a, z]
Out[9]=	1 a 2 z 3 5 7 2 2 2 1 - --- - - + --- + 6 a z + 8 a z + 6 a z + 2 a z + 3 z + 9 a z + a z z a 3 4 2 6 2 8 2 7 z 3 3 3 5 3 > 10 a z + 3 a z - a z - ---- - 26 a z - 35 a z - 23 a z - a 4 5 7 3 4 3 z 2 4 4 4 6 4 8 4 z > 7 a z - 9 z + ---- - 41 a z - 41 a z - 10 a z + 2 a z - -- + 2 3 a a 5 6 16 z 5 3 5 5 5 7 5 6 5 z 2 6 > ----- + 38 a z + 34 a z + 22 a z + 9 a z + 21 z - ---- + 64 a z + a 2 a 7 4 6 6 6 8 6 12 z 7 3 7 7 7 8 > 53 a z + 14 a z - a z - ----- - 7 a z + 9 a z - 4 a z - 15 z - a 2 8 4 8 6 8 9 3 9 5 9 2 10 > 28 a z - 20 a z - 7 a z - 9 a z - 15 a z - 6 a z - 2 a z - 4 10 > 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	12 1 3 1 6 3 10 6 13 15 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 13 13 14 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 8 t + 10 q t + 4 q t + 8 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 + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a11

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