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DrawMorseLink

PD Presentation:

X8192 X12,3,13,4 X20,15,21,16 X16,9,17,10 X10,19,11,20 X22,13,7,14 X14,21,15,22 X18,5,19,6 X2738 X4,11,5,12 X6,17,1,18

Gauss Code:

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

Jones Polynomial:

q-25/2 - 3q-23/2 + 6q-21/2 - 9q-19/2 + 12q-17/2 - 14q-15/2 + 13q-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 + q-36 - q-34 - 2q-32 + 4q-30 + q-26 + 3q-24 + 3q-20 + 2q-16 + 2q-14 - 3q-12 + 2q-10 + q-8 - 2q-6 + q-4

HOMFLY-PT Polynomial:

- a3z3 - a5z - 3a5z3 - a7z-1 - 5a7z - 5a7z3 - a9z - 3a9z3 + 2a11z-1 + 4a11z - a13z-1

Kauffman Polynomial:

- a3z3 - 3a4z4 - a5z + 4a5z3 - 6a5z5 - a6z2 + 11a6z4 - 9a6z6 - a7z-1 + 6a7z - 17a7z3 + 27a7z5 - 12a7z7 + a8 + 2a8z2 - 15a8z4 + 28a8z6 - 11a8z8 + 2a9z - 14a9z3 + 6a9z5 + 15a9z7 - 7a9z9 - 3a10 + 25a10z2 - 64a10z4 + 50a10z6 - 7a10z8 - 2a10z10 + 2a11z-1 - 7a11z + 21a11z3 - 51a11z5 + 42a11z7 - 10a11z9 - 5a12 + 29a12z2 - 44a12z4 + 18a12z6 + 3a12z8 - 2a12z10 + a13z-1 - 2a13z + 13a13z3 - 24a13z5 + 15a13z7 - 3a13z9 - 2a14 + 7a14z2 - 9a14z4 + 5a14z6 - 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 6         j = -14           8 7           j = -16         5 7             j = -18       4 7               j = -20     2 5                 j = -22   1 4                   j = -24   2                     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, 236]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 236]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[20, 15, 21, 16], X[16, 9, 17, 10], > X[10, 19, 11, 20], X[22, 13, 7, 14], X[14, 21, 15, 22], X[18, 5, 19, 6], > X[2, 7, 3, 8], X[4, 11, 5, 12], X[6, 17, 1, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 8, -11}, > {9, -1, 4, -5, 10, -2, 6, -7, 3, -4, 11, -8, 5, -3, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) 3 6 9 12 14 13 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 -36 -34 2 4 -26 3 3 2 2 3 -q - --- + q - q - --- + --- + q + --- + --- + --- + --- - --- + 38 32 30 24 20 16 14 12 q q q q q q q q 2 -8 2 -4 > --- + q - -- + q 10 6 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 236]][a, z]
Out[8]=	7 11 13 a 2 a a 5 7 9 11 3 3 5 3 -(--) + ----- - --- - a z - 5 a z - a z + 4 a z - a z - 3 a z - z z z 7 3 9 3 > 5 a z - 3 a z
In[9]:=	Kauffman[Link[11, Alternating, 236]][a, z]
Out[9]=	7 11 13 8 10 12 14 a 2 a a 5 7 9 a - 3 a - 5 a - 2 a - -- + ----- + --- - a z + 6 a z + 2 a z - z z z 11 13 6 2 8 2 10 2 12 2 14 2 > 7 a z - 2 a z - a z + 2 a z + 25 a z + 29 a z + 7 a z - 3 3 5 3 7 3 9 3 11 3 13 3 4 4 > a z + 4 a z - 17 a z - 14 a z + 21 a z + 13 a z - 3 a z + 6 4 8 4 10 4 12 4 14 4 5 5 > 11 a z - 15 a z - 64 a z - 44 a z - 9 a z - 6 a z + 7 5 9 5 11 5 13 5 6 6 8 6 > 27 a z + 6 a z - 51 a z - 24 a z - 9 a z + 28 a z + 10 6 12 6 14 6 7 7 9 7 11 7 > 50 a z + 18 a z + 5 a z - 12 a z + 15 a z + 42 a z + 13 7 8 8 10 8 12 8 14 8 9 9 11 9 > 15 a z - 11 a z - 7 a z + 3 a z - a z - 7 a z - 10 a z - 13 9 10 10 12 10 > 3 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 -2 1 2 1 4 2 5 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 5 7 8 7 6 6 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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