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DrawMorseLink

PD Presentation:

X6172 X14,3,15,4 X18,8,19,7 X22,20,5,19 X20,9,21,10 X8,21,9,22 X12,18,13,17 X16,12,17,11 X10,16,11,15 X2536 X4,13,1,14

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 2q-9/2 - 6q-7/2 + 10q-5/2 - 15q-3/2 + 18q-1/2 - 19q1/2 + 17q3/2 - 14q5/2 + 9q7/2 - 4q9/2 + q11/2

A2 (sl(3)) Invariant:

q-18 + 2q-16 + q-14 + 3q-12 + q-10 - 3q-8 + 2q-6 - 3q-4 + q-2 + 3 - q2 + 5q4 - 3q6 + 2q8 - 3q12 + 2q14 - q16

HOMFLY-PT Polynomial:

a-3z-1 + 2a-3z + 2a-3z3 + a-3z5 - 4a-1z-1 - 9a-1z - 8a-1z3 - 4a-1z5 - a-1z7 + 6az-1 + 12az + 10az3 + 3az5 - 5a3z-1 - 8a3z - 3a3z3 + 2a5z-1 + a5z

Kauffman Polynomial:

- a-6z4 + a-5z3 - 4a-5z5 + a-4 - 3a-4z2 + 8a-4z4 - 9a-4z6 - a-3z-1 + 4a-3z - 11a-3z3 + 19a-3z5 - 13a-3z7 + 3a-2 - 9a-2z2 + 5a-2z4 + 12a-2z6 - 11a-2z8 - 4a-1z-1 + 19a-1z - 40a-1z3 + 42a-1z5 - 9a-1z7 - 5a-1z9 + 3 - 4z2 - 18z4 + 38z6 - 15z8 - z10 - 6az-1 + 29az - 50az3 + 30az5 + 6az7 - 7az9 + a2 + 3a2z2 - 21a2z4 + 24a2z6 - 6a2z8 - a2z10 - 5a3z-1 + 21a3z - 31a3z3 + 16a3z5 + a3z7 - 2a3z9 + a4 + a4z2 - 7a4z4 + 7a4z6 - 2a4z8 - 2a5z-1 + 7a5z - 9a5z3 + 5a5z5 - a5z7

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 = 12                       1 j = 10                     3   j = 8                   6 1   j = 6                 8 3     j = 4               9 6       j = 2             10 8         j = 0           9 10           j = -2         6 9             j = -4       4 9               j = -6     2 6                 j = -8   1 5                   j = -10   1                     j = -12 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, 121]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 121]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[18, 8, 19, 7], X[22, 20, 5, 19], > X[20, 9, 21, 10], X[8, 21, 9, 22], X[12, 18, 13, 17], X[16, 12, 17, 11], > X[10, 16, 11, 15], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -6, 5, -9, 8, -7, 11, -2, 9, -8, 7, -3, > 4, -5, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 2 6 10 15 18 3/2 -q + ---- - ---- + ---- - ---- + ------- - 19 Sqrt[q] + 17 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 14 q + 9 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 -14 3 -10 3 2 3 -2 2 4 6 3 + q + --- + q + --- + q - -- + -- - -- + q - q + 5 q - 3 q + 16 12 8 6 4 q q q q q 8 12 14 16 > 2 q - 3 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 121]][a, z]
Out[8]=	3 5 3 1 4 6 a 5 a 2 a 2 z 9 z 3 5 2 z ---- - --- + --- - ---- + ---- + --- - --- + 12 a z - 8 a z + a z + ---- - 3 a z z z z 3 a 3 a z a a 3 5 5 7 8 z 3 3 3 z 4 z 5 z > ---- + 10 a z - 3 a z + -- - ---- + 3 a z - -- a 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 121]][a, z]
Out[9]=	3 5 -4 3 2 4 1 4 6 a 5 a 2 a 4 z 19 z 3 + a + -- + a + a - ---- - --- - --- - ---- - ---- + --- + ---- + 29 a z + 2 3 a z z z z 3 a a a z a 2 2 3 3 3 5 2 3 z 9 z 2 2 4 2 z 11 z > 21 a z + 7 a z - 4 z - ---- - ---- + 3 a z + a z + -- - ----- - 4 2 5 3 a a a a 3 4 4 4 40 z 3 3 3 5 3 4 z 8 z 5 z > ----- - 50 a z - 31 a z - 9 a z - 18 z - -- + ---- + ---- - a 6 4 2 a a a 5 5 5 2 4 4 4 4 z 19 z 42 z 5 3 5 5 5 > 21 a z - 7 a z - ---- + ----- + ----- + 30 a z + 16 a z + 5 a z + 5 3 a a a 6 6 7 7 6 9 z 12 z 2 6 4 6 13 z 9 z 7 3 7 > 38 z - ---- + ----- + 24 a z + 7 a z - ----- - ---- + 6 a z + a z - 4 2 3 a a a a 8 9 5 7 8 11 z 2 8 4 8 5 z 9 3 9 10 > a z - 15 z - ----- - 6 a z - 2 a z - ---- - 7 a z - 2 a z - z - 2 a a 2 10 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 5 2 6 4 9 10 + 10 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 6 9 9 2 4 4 2 6 2 6 3 > ----- + - + ---- + 8 q t + 9 q t + 6 q t + 8 q t + 3 q t + 2 2 t 2 q t q t 8 3 8 4 10 4 12 5 > 6 q t + q t + 3 q t + q t

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