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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-21/2 + 3q-19/2 - 5q-17/2 + 7q-15/2 - 9q-13/2 + 10q-11/2 - 10q-9/2 + 8q-7/2 - 7q-5/2 + 4q-3/2 - 3q-1/2 + q1/2

A2 (sl(3)) Invariant:

q-30 - q-28 + q-26 - q-24 - q-18 + 3q-16 - q-14 + 3q-12 + q-10 + 2q-8 + 2q-6 + q-2 - 1

HOMFLY-PT Polynomial:

- a3z-1 + 6a3z3 + 5a3z5 + a3z7 + a5z-1 - 5a5z - 17a5z3 - 17a5z5 - 7a5z7 - a5z9 + 3a7z + 7a7z3 + 5a7z5 + a7z7

Kauffman Polynomial:

3a2z2 - 7a2z4 + 5a2z6 - a2z8 - a3z-1 - a3z + 18a3z3 - 30a3z5 + 17a3z7 - 3a3z9 + a4 + 7a4z2 - 16a4z4 + a4z6 + 7a4z8 - 2a4z10 - a5z-1 - 5a5z + 33a5z3 - 61a5z5 + 40a5z7 - 8a5z9 + 9a6z2 - 27a6z4 + 17a6z6 + 2a6z8 - 2a6z10 - 4a7z + 11a7z3 - 17a7z5 + 17a7z7 - 5a7z9 + 3a8z2 - 8a8z4 + 15a8z6 - 6a8z8 - a9z + a9z3 + 9a9z5 - 6a9z7 - a10z2 + 7a10z4 - 6a10z6 - a11z + 4a11z3 - 5a11z5 + a12z2 - 3a12z4 - a13z3

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 2                       1 j = 0                     2   j = -2                   2 1   j = -4                 5 2     j = -6               4 3       j = -8             6 4         j = -10           4 4           j = -12         5 6             j = -14       3 5               j = -16     2 4                 j = -18   1 3                   j = -20   2                     j = -22 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, 254]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 254]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[14, 5, 15, 6], > X[4, 13, 5, 14], X[20, 16, 21, 15], X[16, 8, 17, 7], X[6, 21, 7, 22], > X[18, 9, 19, 10], X[22, 17, 9, 18], X[8, 20, 1, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -5, 4, -8, 7, -11}, > {9, -1, 2, -3, 5, -4, 6, -7, 10, -9, 11, -6, 8, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 3 5 7 9 10 10 8 7 4 -q + ----- - ----- + ----- - ----- + ----- - ---- + ---- - ---- + ---- - 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q 3 > ------- + Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 -26 -24 -18 3 -14 3 -10 2 2 -2 -1 + q - q + q - q - q + --- - q + --- + q + -- + -- + q 16 12 8 6 q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 254]][a, z]
Out[8]=	3 5 a a 5 7 3 3 5 3 7 3 3 5 -(--) + -- - 5 a z + 3 a z + 6 a z - 17 a z + 7 a z + 5 a z - z z 5 5 7 5 3 7 5 7 7 7 5 9 > 17 a z + 5 a z + a z - 7 a z + a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 254]][a, z]
Out[9]=	3 5 4 a a 3 5 7 9 11 2 2 4 2 a - -- - -- - a z - 5 a z - 4 a z - a z - a z + 3 a z + 7 a z + z z 6 2 8 2 10 2 12 2 3 3 5 3 7 3 > 9 a z + 3 a z - a z + a z + 18 a z + 33 a z + 11 a z + 9 3 11 3 13 3 2 4 4 4 6 4 8 4 > a z + 4 a z - a z - 7 a z - 16 a z - 27 a z - 8 a z + 10 4 12 4 3 5 5 5 7 5 9 5 11 5 > 7 a z - 3 a z - 30 a z - 61 a z - 17 a z + 9 a z - 5 a z + 2 6 4 6 6 6 8 6 10 6 3 7 5 7 > 5 a z + a z + 17 a z + 15 a z - 6 a z + 17 a z + 40 a z + 7 7 9 7 2 8 4 8 6 8 8 8 3 9 > 17 a z - 6 a z - a z + 7 a z + 2 a z - 6 a z - 3 a z - 5 9 7 9 4 10 6 10 > 8 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 1 2 1 3 2 4 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 22 8 20 7 18 7 18 6 16 6 16 5 14 5 q q q t q t q t q t q t q t q t 5 5 6 4 4 6 4 4 2 t > ------ + ------ + ------ + ------ + ------ + ----- + ---- + ---- + --- + 14 4 12 4 12 3 10 3 10 2 8 2 8 6 4 q t q t q t q t q t q t q t q t q 2 2 t 2 t 2 3 > --- + 2 t + -- + q t 2 2 q q

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