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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-15/2 - 4q-13/2 + 8q-11/2 - 14q-9/2 + 19q-7/2 - 23q-5/2 + 22q-3/2 - 20q-1/2 + 15q1/2 - 9q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-22 + 2q-20 - q-18 + q-16 + 4q-14 - 3q-12 + 5q-10 + q-6 + 3q-4 - 4q-2 + 4 - 3q2 + 2q6 - 2q8 + q10

HOMFLY-PT Polynomial:

- a-1z - 2a-1z3 - a-1z5 + az + 3az3 + 3az5 + az7 - a3z-1 - 2a3z + 2a3z3 + 3a3z5 + a3z7 + a5z-1 - 2a5z3 - a5z5

Kauffman Polynomial:

a-3z3 - a-3z5 - a-2z2 + 5a-2z4 - 4a-2z6 + a-1z - 5a-1z3 + 12a-1z5 - 8a-1z7 + 3z2 - 9z4 + 16z6 - 10z8 + az3 - 4az5 + 10az7 - 8az9 + 9a2z2 - 29a2z4 + 30a2z6 - 9a2z8 - 3a2z10 - a3z-1 + a3z + 13a3z3 - 36a3z5 + 37a3z7 - 15a3z9 + a4 + 8a4z2 - 30a4z4 + 29a4z6 - 6a4z8 - 3a4z10 - a5z-1 + 2a5z + a5z3 - 9a5z5 + 15a5z7 - 7a5z9 + 3a6z2 - 13a6z4 + 18a6z6 - 7a6z8 - 5a7z3 + 10a7z5 - 4a7z7 + 2a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                       1 j = 6                     3   j = 4                   6 1   j = 2                 9 3     j = 0               11 6       j = -2             12 10         j = -4           11 10           j = -6         8 12             j = -8       6 11               j = -10     3 9                 j = -12   1 5                   j = -14   3                     j = -16 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, 290]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 290]]
Out[4]=	PD[X[10, 1, 11, 2], X[20, 13, 21, 14], X[12, 4, 13, 3], X[2, 19, 3, 20], > X[14, 7, 15, 8], X[16, 5, 17, 6], X[6, 15, 7, 16], X[8, 9, 1, 10], > X[18, 12, 19, 11], X[22, 18, 9, 17], X[4, 22, 5, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -11, 6, -7, 5, -8}, > {8, -1, 9, -3, 2, -5, 7, -6, 10, -9, 4, -2, 11, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 4 8 14 19 23 22 20 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 15 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 9 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 -18 -16 4 3 5 -6 3 4 2 6 4 - q + --- - q + q + --- - --- + --- + q + -- - -- - 3 q + 2 q - 20 14 12 10 4 2 q q q q q q 8 10 > 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 290]][a, z]
Out[8]=	3 5 3 5 a a z 3 2 z 3 3 3 5 3 z -(--) + -- - - + a z - 2 a z - ---- + 3 a z + 2 a z - 2 a z - -- + z z a a a 5 3 5 5 5 7 3 7 > 3 a z + 3 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 290]][a, z]
Out[9]=	3 5 2 4 a a z 3 5 2 z 2 2 4 2 6 2 a - -- - -- + - + a z + 2 a z + 3 z - -- + 9 a z + 8 a z + 3 a z + z z a 2 a 3 3 4 z 5 z 3 3 3 5 3 7 3 4 5 z 2 4 > -- - ---- + a z + 13 a z + a z - 5 a z - 9 z + ---- - 29 a z - 3 a 2 a a 5 5 4 4 6 4 8 4 z 12 z 5 3 5 5 5 > 30 a z - 13 a z + 2 a z - -- + ----- - 4 a z - 36 a z - 9 a z + 3 a a 6 7 7 5 6 4 z 2 6 4 6 6 6 8 6 8 z > 10 a z + 16 z - ---- + 30 a z + 29 a z + 18 a z - a z - ---- + 2 a a 7 3 7 5 7 7 7 8 2 8 4 8 > 10 a z + 37 a z + 15 a z - 4 a z - 10 z - 9 a z - 6 a z - 6 8 9 3 9 5 9 2 10 4 10 > 7 a z - 8 a z - 15 a z - 7 a z - 3 a z - 3 a z
In[10]:=	Kh[L][q, t]
Out[10]=	10 1 3 1 5 3 9 6 11 11 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 8 12 11 10 12 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 6 t + 9 q t + 3 q t + 6 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 3 q t + q t

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