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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X18,11,19,12 X22,19,7,20 X20,15,21,16 X16,21,17,22 X12,17,13,18 X6718 X4,13,5,14

Gauss Code:

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

Jones Polynomial:

q-29/2 - 3q-27/2 + 6q-25/2 - 10q-23/2 + 13q-21/2 - 14q-19/2 + 14q-17/2 - 12q-15/2 + 8q-13/2 - 6q-11/2 + 2q-9/2 - q-7/2

A2 (sl(3)) Invariant:

- q-44 - q-42 + q-40 - q-38 + 2q-36 - q-34 - q-32 + q-30 - 3q-28 + 4q-26 + 3q-22 + 3q-20 + 3q-16 - q-14 + q-12

HOMFLY-PT Polynomial:

- a7z-1 - 5a7z - 8a7z3 - 5a7z5 - a7z7 - 7a9z - 15a9z3 - 10a9z5 - 2a9z7 + 2a11z-1 + 12a11z + 12a11z3 + 3a11z5 - a13z-1 - 3a13z - a13z3

Kauffman Polynomial:

- a7z-1 + 5a7z - 8a7z3 + 5a7z5 - a7z7 + a8 - a8z2 - 5a8z4 + 7a8z6 - 2a8z8 - 5a9z + 10a9z3 - 13a9z5 + 11a9z7 - 3a9z9 - 3a10 + 21a10z2 - 41a10z4 + 30a10z6 - 5a10z8 - a10z10 + 2a11z-1 - 13a11z + 22a11z3 - 25a11z5 + 22a11z7 - 7a11z9 - 5a12 + 30a12z2 - 49a12z4 + 37a12z6 - 9a12z8 - a12z10 + a13z-1 - 3a13z + 2a13z5 + 4a13z7 - 4a13z9 - 2a14 + 5a14z2 - 7a14z4 + 9a14z6 - 6a14z8 - a15z - a15z3 + 6a15z5 - 6a15z7 - 2a16z2 + 5a16z4 - 5a16z6 - a17z + 3a17z3 - 3a17z5 + a18z2 - a18z4

Khovanov Homology:

trqj r = -11 r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -6                       1 j = -8                     2 1 j = -10                   4     j = -12                 4 2     j = -14               8 4       j = -16             6 4         j = -18           8 8           j = -20         6 7             j = -22       4 7               j = -24     2 6                 j = -26   1 4                   j = -28   2                     j = -30 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, 163]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 163]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[14, 5, 15, 6], > X[18, 11, 19, 12], X[22, 19, 7, 20], X[20, 15, 21, 16], X[16, 21, 17, 22], > X[12, 17, 13, 18], X[6, 7, 1, 8], X[4, 13, 5, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 4, -10}, > {10, -1, 2, -3, 5, -9, 11, -4, 7, -8, 9, -5, 6, -7, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(29/2) 3 6 10 13 14 14 12 8 q - ----- + ----- - ----- + ----- - ----- + ----- - ----- + ----- - 27/2 25/2 23/2 21/2 19/2 17/2 15/2 13/2 q q q q q q q q 6 2 -(7/2) > ----- + ---- - q 11/2 9/2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-44 -42 -40 -38 2 -34 -32 -30 3 4 3 3 -q - q + q - q + --- - q - q + q - --- + --- + --- + --- + 36 28 26 22 20 q q q q q 3 -14 -12 > --- - q + q 16 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 163]][a, z]
Out[8]=	7 11 13 a 2 a a 7 9 11 13 7 3 -(--) + ----- - --- - 5 a z - 7 a z + 12 a z - 3 a z - 8 a z - z z z 9 3 11 3 13 3 7 5 9 5 11 5 7 7 > 15 a z + 12 a z - a z - 5 a z - 10 a z + 3 a z - a z - 9 7 > 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 163]][a, z]
Out[9]=	7 11 13 8 10 12 14 a 2 a a 7 9 11 a - 3 a - 5 a - 2 a - -- + ----- + --- + 5 a z - 5 a z - 13 a z - z z z 13 15 17 8 2 10 2 12 2 14 2 > 3 a z - a z - a z - a z + 21 a z + 30 a z + 5 a z - 16 2 18 2 7 3 9 3 11 3 15 3 17 3 > 2 a z + a z - 8 a z + 10 a z + 22 a z - a z + 3 a z - 8 4 10 4 12 4 14 4 16 4 18 4 7 5 > 5 a z - 41 a z - 49 a z - 7 a z + 5 a z - a z + 5 a z - 9 5 11 5 13 5 15 5 17 5 8 6 > 13 a z - 25 a z + 2 a z + 6 a z - 3 a z + 7 a z + 10 6 12 6 14 6 16 6 7 7 9 7 > 30 a z + 37 a z + 9 a z - 5 a z - a z + 11 a z + 11 7 13 7 15 7 8 8 10 8 12 8 > 22 a z + 4 a z - 6 a z - 2 a z - 5 a z - 9 a z - 14 8 9 9 11 9 13 9 10 10 12 10 > 6 a z - 3 a z - 7 a z - 4 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 2 1 4 2 6 4 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 30 11 28 10 26 10 26 9 24 9 24 8 22 8 q t q t q t q t q t q t q t 7 6 7 8 8 6 4 8 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 22 7 20 7 20 6 18 6 18 5 16 5 16 4 14 4 q t q t q t q t q t q t q t q t 4 4 2 4 2 > ------ + ------ + ------ + ------ + ---- 14 3 12 3 12 2 10 2 8 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a163

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