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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 4q-11/2 - 9q-9/2 + 14q-7/2 - 20q-5/2 + 22q-3/2 - 23q-1/2 + 19q1/2 - 14q3/2 + 9q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant:

q-18 - 2q-16 + 3q-14 - 2q-12 + 3q-10 + 4q-8 - q-6 + 7q-4 - 4q-2 + 4 - 2q2 - 2q4 + 2q6 - 3q8 + 2q10 - q12

HOMFLY-PT Polynomial:

a-1z-1 + 3a-1z + 5a-1z3 + 4a-1z5 + a-1z7 - 3az-1 - 8az - 13az3 - 13az5 - 6az7 - az9 + 2a3z-1 + 3a3z + 5a3z3 + 4a3z5 + a3z7

Kauffman Polynomial:

- a-4z2 + 2a-4z4 - a-4z6 - 4a-3z3 + 9a-3z5 - 4a-3z7 - a-2 + 3a-2z2 - 9a-2z4 + 16a-2z6 - 7a-2z8 + a-1z-1 - 3a-1z + 7a-1z3 - 9a-1z5 + 14a-1z7 - 7a-1z9 - 3 + 13z2 - 27z4 + 27z6 - 6z8 - 3z10 + 3az-1 - 9az + 19az3 - 37az5 + 37az7 - 15az9 - 3a2 + 13a2z2 - 34a2z4 + 32a2z6 - 9a2z8 - 3a2z10 + 2a3z-1 - 4a3z + a3z3 - 5a3z5 + 11a3z7 - 8a3z9 + 4a4z2 - 13a4z4 + 18a4z6 - 10a4z8 + 2a5z - 6a5z3 + 13a5z5 - 8a5z7 + 5a6z4 - 4a6z6 + a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                       1 j = 8                     3   j = 6                   6 1   j = 4                 8 3     j = 2               11 6       j = 0             12 8         j = -2           11 12           j = -4         9 11             j = -6       5 11               j = -8     4 9                 j = -10   1 6                   j = -12   3                     j = -14 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, 333]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 333]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[20, 5, 21, 6], X[16, 7, 17, 8], > X[8, 9, 1, 10], X[18, 12, 19, 11], X[6, 15, 7, 16], X[4, 14, 5, 13], > X[22, 18, 9, 17], X[2, 19, 3, 20], X[14, 22, 15, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -8, 3, -7, 4, -5}, > {5, -1, 6, -2, 8, -11, 7, -4, 9, -6, 10, -3, 11, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 4 9 14 20 22 23 -q + ----- - ---- + ---- - ---- + ---- - ------- + 19 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 7/2 9/2 > 14 q + 9 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 3 2 3 4 -6 7 4 2 4 6 4 + q - --- + --- - --- + --- + -- - q + -- - -- - 2 q - 2 q + 2 q - 16 14 12 10 8 4 2 q q q q q q q 8 10 12 > 3 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 333]][a, z]
Out[8]=	3 3 5 1 3 a 2 a 3 z 3 5 z 3 3 3 4 z --- - --- + ---- + --- - 8 a z + 3 a z + ---- - 13 a z + 5 a z + ---- - a z z z a a a 7 5 3 5 z 7 3 7 9 > 13 a z + 4 a z + -- - 6 a z + a z - a z a
In[9]:=	Kauffman[Link[11, Alternating, 333]][a, z]
Out[9]=	3 -2 2 1 3 a 2 a 3 z 3 5 2 -3 - a - 3 a + --- + --- + ---- - --- - 9 a z - 4 a z + 2 a z + 13 z - a z z z a 2 2 3 3 z 3 z 2 2 4 2 4 z 7 z 3 3 3 5 3 > -- + ---- + 13 a z + 4 a z - ---- + ---- + 19 a z + a z - 6 a z + 4 2 3 a a a a 4 4 5 5 7 3 4 2 z 9 z 2 4 4 4 6 4 9 z 9 z > a z - 27 z + ---- - ---- - 34 a z - 13 a z + 5 a z + ---- - ---- - 4 2 3 a a a a 6 6 5 3 5 5 5 7 5 6 z 16 z 2 6 > 37 a z - 5 a z + 13 a z - a z + 27 z - -- + ----- + 32 a z + 4 2 a a 7 7 4 6 6 6 4 z 14 z 7 3 7 5 7 8 > 18 a z - 4 a z - ---- + ----- + 37 a z + 11 a z - 8 a z - 6 z - 3 a a 8 9 7 z 2 8 4 8 7 z 9 3 9 10 2 10 > ---- - 9 a z - 10 a z - ---- - 15 a z - 8 a z - 3 z - 3 a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	12 1 3 1 6 4 9 5 11 12 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 9 11 11 2 2 2 4 2 4 3 > ----- + ---- + ---- + 8 t + 11 q t + 6 q t + 8 q t + 3 q t + 4 2 4 2 q t q t q t 6 3 6 4 8 4 10 5 > 6 q t + q t + 3 q t + q t

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