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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-23/2 - 3q-21/2 + 7q-19/2 - 11q-17/2 + 13q-15/2 - 15q-13/2 + 14q-11/2 - 12q-9/2 + 8q-7/2 - 5q-5/2 + 2q-3/2 - q-1/2

A2 (sl(3)) Invariant:

- q-34 + q-32 - 2q-30 - q-28 + q-26 - 2q-24 + 4q-22 + 3q-18 + 3q-16 - q-14 + 3q-12 - 2q-10 + q-8 + q-6 + q-2

HOMFLY-PT Polynomial:

- 4a3z - 4a3z3 - a3z5 - 2a5z-1 - a5z + 4a5z3 + 4a5z5 + a5z7 + 3a7z-1 + 6a7z + 6a7z3 + 4a7z5 + a7z7 - a9z-1 - 3a9z - 3a9z3 - a9z5

Kauffman Polynomial:

4a3z - 8a3z3 + 5a3z5 - a3z7 + 2a4z2 - 9a4z4 + 8a4z6 - 2a4z8 + 2a5z-1 - 5a5z - a5z5 + 5a5z7 - 2a5z9 - 3a6 + 8a6z2 - 14a6z4 + 13a6z6 - 2a6z8 - a6z10 + 3a7z-1 - 14a7z + 24a7z3 - 21a7z5 + 17a7z7 - 6a7z9 - 3a8 + 14a8z2 - 21a8z4 + 22a8z6 - 7a8z8 - a8z10 + a9z-1 - 2a9z + 3a9z3 + a9z5 + 3a9z7 - 4a9z9 - a10 + 3a10z2 - 8a10z4 + 11a10z6 - 7a10z8 + 3a11z - 11a11z3 + 13a11z5 - 8a11z7 - 4a12z2 + 7a12z4 - 6a12z6 + 2a13z3 - 3a13z5 + a14z2 - a14z4

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 0                       1 j = -2                     1   j = -4                   4 1   j = -6                 5 2     j = -8               7 3       j = -10             7 5         j = -12           8 7           j = -14         6 8             j = -16       5 7               j = -18     2 6                 j = -20   1 5                   j = -22   2                     j = -24 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, 270]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 270]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[4, 9, 5, 10], > X[14, 5, 15, 6], X[18, 8, 19, 7], X[20, 15, 21, 16], X[22, 17, 9, 18], > X[16, 21, 17, 22], X[6, 20, 7, 19], X[8, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -4, 5, -10, 6, -11}, > {4, -1, 2, -3, 11, -5, 7, -9, 8, -6, 10, -7, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) 3 7 11 13 15 14 12 8 5 q - ----- + ----- - ----- + ----- - ----- + ----- - ---- + ---- - ---- + 21/2 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 q q q q q q q q q 2 1 > ---- - ------- 3/2 Sqrt[q] q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 -32 2 -28 -26 2 4 3 3 -14 3 2 -q + q - --- - q + q - --- + --- + --- + --- - q + --- - --- + 30 24 22 18 16 12 10 q q q q q q q -8 -6 -2 > q + q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 270]][a, z]
Out[8]=	5 7 9 -2 a 3 a a 3 5 7 9 3 3 5 3 ----- + ---- - -- - 4 a z - a z + 6 a z - 3 a z - 4 a z + 4 a z + z z z 7 3 9 3 3 5 5 5 7 5 9 5 5 7 7 7 > 6 a z - 3 a z - a z + 4 a z + 4 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 270]][a, z]
Out[9]=	5 7 9 6 8 10 2 a 3 a a 3 5 7 9 -3 a - 3 a - a + ---- + ---- + -- + 4 a z - 5 a z - 14 a z - 2 a z + z z z 11 4 2 6 2 8 2 10 2 12 2 14 2 > 3 a z + 2 a z + 8 a z + 14 a z + 3 a z - 4 a z + a z - 3 3 7 3 9 3 11 3 13 3 4 4 6 4 > 8 a z + 24 a z + 3 a z - 11 a z + 2 a z - 9 a z - 14 a z - 8 4 10 4 12 4 14 4 3 5 5 5 7 5 > 21 a z - 8 a z + 7 a z - a z + 5 a z - a z - 21 a z + 9 5 11 5 13 5 4 6 6 6 8 6 10 6 > a z + 13 a z - 3 a z + 8 a z + 13 a z + 22 a z + 11 a z - 12 6 3 7 5 7 7 7 9 7 11 7 4 8 > 6 a z - a z + 5 a z + 17 a z + 3 a z - 8 a z - 2 a z - 6 8 8 8 10 8 5 9 7 9 9 9 6 10 8 10 > 2 a z - 7 a z - 7 a z - 2 a z - 6 a z - 4 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 1 2 1 5 2 6 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 24 9 22 8 20 8 20 7 18 7 18 6 16 6 q q q t q t q t q t q t q t q t 7 6 8 8 7 7 5 7 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 16 5 14 5 14 4 12 4 12 3 10 3 10 2 8 2 q t q t q t q t q t q t q t q t 3 5 t t 2 > ---- + ---- + -- + -- + t 8 6 4 2 q t q t q q

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