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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-27/2 - 4q-25/2 + 8q-23/2 - 13q-21/2 + 17q-19/2 - 19q-17/2 + 19q-15/2 - 17q-13/2 + 11q-11/2 - 7q-9/2 + 3q-7/2 - q-5/2

A2 (sl(3)) Invariant:

- q-42 - q-40 + 2q-38 - 2q-36 + q-34 + 3q-32 - 2q-30 + 4q-28 - q-26 + q-24 + 2q-22 - 2q-20 + 5q-18 - q-16 + 2q-12 - 2q-10 + q-8

HOMFLY-PT Polynomial:

- 2a5z3 - a5z5 - 5a7z - 8a7z3 - 3a7z5 - 2a9z-1 - 6a9z - 8a9z3 - 3a9z5 + 3a11z-1 + 8a11z + 4a11z3 - a13z-1 - a13z

Kauffman Polynomial:

2a5z3 - a5z5 + 5a6z4 - 3a6z6 + 5a7z - 13a7z3 + 14a7z5 - 6a7z7 + 3a8z2 - 14a8z4 + 15a8z6 - 7a8z8 + 2a9z-1 - 8a9z + 8a9z3 - 14a9z5 + 12a9z7 - 6a9z9 - 3a10 + 20a10z2 - 42a10z4 + 31a10z6 - 8a10z8 - 2a10z10 + 3a11z-1 - 14a11z + 27a11z3 - 34a11z5 + 28a11z7 - 11a11z9 - 3a12 + 19a12z2 - 30a12z4 + 27a12z6 - 7a12z8 - 2a12z10 + a13z-1 - a13z - 2a13z3 + 5a13z5 + 6a13z7 - 5a13z9 - a14 + a14z2 - 5a14z4 + 13a14z6 - 6a14z8 - 6a15z3 + 10a15z5 - 4a15z7 - a16z2 + 2a16z4 - a16z6

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 = -4                       1 j = -6                     3 1 j = -8                   4     j = -10                 7 3     j = -12               10 4       j = -14             9 7         j = -16           10 10           j = -18         7 9             j = -20       6 10               j = -22     3 8                 j = -24   1 5                   j = -26   3                     j = -28 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, 306]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 306]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[4, 9, 5, 10], > X[16, 5, 17, 6], X[22, 17, 9, 18], X[18, 13, 19, 14], X[14, 21, 15, 22], > X[20, 7, 21, 8], X[6, 15, 7, 16], X[8, 19, 1, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -4, 5, -10, 9, -11}, > {4, -1, 2, -3, 7, -8, 10, -5, 6, -7, 11, -9, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(27/2) 4 8 13 17 19 19 17 11 q - ----- + ----- - ----- + ----- - ----- + ----- - ----- + ----- - 25/2 23/2 21/2 19/2 17/2 15/2 13/2 11/2 q q q q q q q q 7 3 -(5/2) > ---- + ---- - q 9/2 7/2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 -40 2 2 -34 3 2 4 -26 -24 2 2 -q - q + --- - --- + q + --- - --- + --- - q + q + --- - --- + 38 36 32 30 28 22 20 q q q q q q q 5 -16 2 2 -8 > --- - q + --- - --- + q 18 12 10 q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 306]][a, z]
Out[8]=	9 11 13 -2 a 3 a a 7 9 11 13 5 3 7 3 ----- + ----- - --- - 5 a z - 6 a z + 8 a z - a z - 2 a z - 8 a z - z z z 9 3 11 3 5 5 7 5 9 5 > 8 a z + 4 a z - a z - 3 a z - 3 a z
In[9]:=	Kauffman[Link[11, Alternating, 306]][a, z]
Out[9]=	9 11 13 10 12 14 2 a 3 a a 7 9 11 -3 a - 3 a - a + ---- + ----- + --- + 5 a z - 8 a z - 14 a z - z z z 13 8 2 10 2 12 2 14 2 16 2 5 3 > a z + 3 a z + 20 a z + 19 a z + a z - a z + 2 a z - 7 3 9 3 11 3 13 3 15 3 6 4 8 4 > 13 a z + 8 a z + 27 a z - 2 a z - 6 a z + 5 a z - 14 a z - 10 4 12 4 14 4 16 4 5 5 7 5 9 5 > 42 a z - 30 a z - 5 a z + 2 a z - a z + 14 a z - 14 a z - 11 5 13 5 15 5 6 6 8 6 10 6 > 34 a z + 5 a z + 10 a z - 3 a z + 15 a z + 31 a z + 12 6 14 6 16 6 7 7 9 7 11 7 > 27 a z + 13 a z - a z - 6 a z + 12 a z + 28 a z + 13 7 15 7 8 8 10 8 12 8 14 8 9 9 > 6 a z - 4 a z - 7 a z - 8 a z - 7 a z - 6 a z - 6 a z - 11 9 13 9 10 10 12 10 > 11 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 3 1 5 3 8 6 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 28 11 26 10 24 10 24 9 22 9 22 8 20 8 q t q t q t q t q t q t q t 10 7 9 10 10 9 7 10 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 20 7 18 7 18 6 16 6 16 5 14 5 14 4 12 4 q t q t q t q t q t q t q t q t 4 7 3 4 3 > ------ + ------ + ------ + ----- + ---- 12 3 10 3 10 2 8 2 6 q t q t q t q t q t

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