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DrawMorseLink

PD Presentation:

X6172 X14,3,15,4 X22,15,5,16 X18,7,19,8 X10,21,11,22 X20,11,21,12 X12,19,13,20 X16,9,17,10 X8,17,9,18 X2536 X4,13,1,14

Gauss Code:

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

Jones Polynomial:

q-25/2 - 2q-23/2 + 5q-21/2 - 8q-19/2 + 11q-17/2 - 13q-15/2 + 12q-13/2 - 13q-11/2 + 9q-9/2 - 6q-7/2 + 3q-5/2 - q-3/2

A2 (sl(3)) Invariant:

- q-40 - 2q-38 - 2q-34 - 3q-32 + 3q-30 + q-26 + 4q-24 + q-22 + 4q-20 + q-18 + 2q-16 + 2q-14 - 3q-12 + 2q-10 + q-8 - 2q-6 + q-4

HOMFLY-PT Polynomial:

- a3z3 - a5z - 3a5z3 - 2a7z-1 - 6a7z - 5a7z3 + 2a9z-1 + 2a9z - 2a9z3 + a11z-1 + 3a11z - a13z-1

Kauffman Polynomial:

- a3z3 - 3a4z4 - a5z + 4a5z3 - 6a5z5 - a6z2 + 11a6z4 - 9a6z6 - 2a7z-1 + 8a7z - 20a7z3 + 28a7z5 - 12a7z7 + 2a8 - a8z2 - 10a8z4 + 24a8z6 - 10a8z8 - 2a9z-1 + 11a9z - 35a9z3 + 28a9z5 + 4a9z7 - 5a9z9 - 4a10 + 26a10z2 - 65a10z4 + 54a10z6 - 11a10z8 - a10z10 + a11z-1 + 2a11z - 2a11z3 - 22a11z5 + 26a11z7 - 7a11z9 - 9a12 + 38a12z2 - 54a12z4 + 27a12z6 - 2a12z8 - a12z10 + a13z-1 + 8a13z3 - 16a13z5 + 10a13z7 - 2a13z9 - 4a14 + 12a14z2 - 13a14z4 + 6a14z6 - a14z8

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 = -2                       1 j = -4                     3 1 j = -6                   3     j = -8                 6 3     j = -10               7 3       j = -12             6 7         j = -14           7 6           j = -16         4 6             j = -18       4 7               j = -20     1 4                 j = -22   1 4                   j = -24   1                     j = -26 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, 120]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 120]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[22, 15, 5, 16], X[18, 7, 19, 8], > X[10, 21, 11, 22], X[20, 11, 21, 12], X[12, 19, 13, 20], X[16, 9, 17, 10], > X[8, 17, 9, 18], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -9, 8, -5, 6, -7, 11, -2, 3, -8, 9, -4, > 7, -6, 5, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) 2 5 8 11 13 12 13 9 q - ----- + ----- - ----- + ----- - ----- + ----- - ----- + ---- - 23/2 21/2 19/2 17/2 15/2 13/2 11/2 9/2 q q q q q q q q 6 3 -(3/2) > ---- + ---- - q 7/2 5/2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-40 2 2 3 3 -26 4 -22 4 -18 2 2 -q - --- - --- - --- + --- + q + --- + q + --- + q + --- + --- - 38 34 32 30 24 20 16 14 q q q q q q q q 3 2 -8 2 -4 > --- + --- + q - -- + q 12 10 6 q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 120]][a, z]
Out[8]=	7 9 11 13 -2 a 2 a a a 5 7 9 11 3 3 5 3 ----- + ---- + --- - --- - a z - 6 a z + 2 a z + 3 a z - a z - 3 a z - z z z z 7 3 9 3 > 5 a z - 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 120]][a, z]
Out[9]=	7 9 11 13 8 10 12 14 2 a 2 a a a 5 7 2 a - 4 a - 9 a - 4 a - ---- - ---- + --- + --- - a z + 8 a z + z z z z 9 11 6 2 8 2 10 2 12 2 14 2 > 11 a z + 2 a z - a z - a z + 26 a z + 38 a z + 12 a z - 3 3 5 3 7 3 9 3 11 3 13 3 4 4 > a z + 4 a z - 20 a z - 35 a z - 2 a z + 8 a z - 3 a z + 6 4 8 4 10 4 12 4 14 4 5 5 > 11 a z - 10 a z - 65 a z - 54 a z - 13 a z - 6 a z + 7 5 9 5 11 5 13 5 6 6 8 6 > 28 a z + 28 a z - 22 a z - 16 a z - 9 a z + 24 a z + 10 6 12 6 14 6 7 7 9 7 11 7 > 54 a z + 27 a z + 6 a z - 12 a z + 4 a z + 26 a z + 13 7 8 8 10 8 12 8 14 8 9 9 11 9 > 10 a z - 10 a z - 11 a z - 2 a z - a z - 5 a z - 7 a z - 13 9 10 10 12 10 > 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 -2 1 1 1 4 1 4 4 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 26 11 24 10 22 10 22 9 20 9 20 8 18 8 q t q t q t q t q t q t q t 7 4 6 7 6 6 7 7 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 18 7 16 7 16 6 14 6 14 5 12 5 12 4 10 4 q t q t q t q t q t q t q t q t 3 6 3 3 3 > ------ + ----- + ----- + ----- + ---- 10 3 8 3 8 2 6 2 4 q t q t q t q t q t

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