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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X20,5,21,6 X22,13,9,14 X18,21,19,22 X16,7,17,8 X14,17,15,18 X8,9,1,10 X6,15,7,16 X4,19,5,20

Gauss Code:

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

Jones Polynomial:

q-29/2 - 4q-27/2 + 7q-25/2 - 11q-23/2 + 14q-21/2 - 15q-19/2 + 15q-17/2 - 12q-15/2 + 8q-13/2 - 6q-11/2 + 2q-9/2 - q-7/2

A2 (sl(3)) Invariant:

- q-44 + 2q-40 + 3q-36 - q-34 - q-32 - 4q-28 + 3q-26 - q-24 + 3q-22 + 3q-20 + 3q-16 - q-14 + q-12

HOMFLY-PT Polynomial:

- a7z-1 - 5a7z - 8a7z3 - 5a7z5 - a7z7 + a9z-1 - 6a9z - 15a9z3 - 10a9z5 - 2a9z7 + 9a11z + 11a11z3 + 3a11z5 - 2a13z - a13z3

Kauffman Polynomial:

- a7z-1 + 5a7z - 8a7z3 + 5a7z5 - a7z7 + a8 - a8z2 - 5a8z4 + 7a8z6 - 2a8z8 - a9z-1 - 3a9z + 9a9z3 - 13a9z5 + 11a9z7 - 3a9z9 + 16a10z2 - 38a10z4 + 29a10z6 - 5a10z8 - a10z10 - 7a11z + 12a11z3 - 18a11z5 + 20a11z7 - 7a11z9 + 18a12z2 - 42a12z4 + 37a12z6 - 10a12z8 - a12z10 - a13z - 7a13z3 + 12a13z5 - 4a13z9 - a14z4 + 8a14z6 - 7a14z8 - 2a15z + a15z3 + 8a15z5 - 8a15z7 - a16z2 + 7a16z4 - 7a16z6 + 3a17z3 - 4a17z5 - 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             7 4         j = -18           8 8           j = -20         6 7             j = -22       5 8               j = -24     3 7                 j = -26   1 4                   j = -28   3                     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, 332]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 332]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[20, 5, 21, 6], > X[22, 13, 9, 14], X[18, 21, 19, 22], X[16, 7, 17, 8], X[14, 17, 15, 18], > X[8, 9, 1, 10], X[6, 15, 7, 16], X[4, 19, 5, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 4, -10, 7, -9}, > {9, -1, 2, -3, 5, -8, 10, -7, 8, -6, 11, -4, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(29/2) 4 7 11 14 15 15 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 2 3 -34 -32 4 3 -24 3 3 3 -14 -q + --- + --- - q - q - --- + --- - q + --- + --- + --- - q + 40 36 28 26 22 20 16 q q q q q q q -12 > q
In[8]:=	HOMFLYPT[Link[11, Alternating, 332]][a, z]
Out[8]=	7 9 a a 7 9 11 13 7 3 9 3 -(--) + -- - 5 a z - 6 a z + 9 a z - 2 a z - 8 a z - 15 a z + z z 11 3 13 3 7 5 9 5 11 5 7 7 9 7 > 11 a z - a z - 5 a z - 10 a z + 3 a z - a z - 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 332]][a, z]
Out[9]=	7 9 8 a a 7 9 11 13 15 8 2 a - -- - -- + 5 a z - 3 a z - 7 a z - a z - 2 a z - a z + z z 10 2 12 2 16 2 7 3 9 3 11 3 13 3 > 16 a z + 18 a z - a z - 8 a z + 9 a z + 12 a z - 7 a z + 15 3 17 3 8 4 10 4 12 4 14 4 16 4 > a z + 3 a z - 5 a z - 38 a z - 42 a z - a z + 7 a z - 18 4 7 5 9 5 11 5 13 5 15 5 17 5 > a z + 5 a z - 13 a z - 18 a z + 12 a z + 8 a z - 4 a z + 8 6 10 6 12 6 14 6 16 6 7 7 9 7 > 7 a z + 29 a z + 37 a z + 8 a z - 7 a z - a z + 11 a z + 11 7 15 7 8 8 10 8 12 8 14 8 > 20 a z - 8 a z - 2 a z - 5 a z - 10 a z - 7 a z - 9 9 11 9 13 9 10 10 12 10 > 3 a z - 7 a z - 4 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 3 1 4 3 7 5 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 8 6 7 8 8 7 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

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