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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 3q-9/2 - 8q-7/2 + 14q-5/2 - 19q-3/2 + 23q-1/2 - 24q1/2 + 20q3/2 - 17q5/2 + 10q7/2 - 4q9/2 + q11/2

A2 (sl(3)) Invariant:

q-18 + q-16 - q-14 + 3q-12 - 4q-8 + 3q-6 - 4q-4 + q-2 + 2 - q2 + 7q4 - 2q6 + 5q8 + q10 - 4q12 + 2q14 - q16

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-6z4 - 4a-5z5 - 3a-4z2 + 7a-4z4 - 10a-4z6 + 7a-3z - 18a-3z3 + 26a-3z5 - 17a-3z7 - a-2 - a-2z2 - 8a-2z4 + 26a-2z6 - 17a-2z8 - 2a-1z-1 + 21a-1z - 49a-1z3 + 50a-1z5 - 5a-1z7 - 9a-1z9 - 2 + 10z2 - 46z4 + 67z6 - 23z8 - 2z10 - 4az-1 + 26az - 49az3 + 24az5 + 20az7 - 13az9 - 3a2 + 14a2z2 - 42a2z4 + 41a2z6 - 9a2z8 - 2a2z10 - 3a3z-1 + 16a3z - 24a3z3 + 8a3z5 + 7a3z7 - 4a3z9 - a4 + 6a4z2 - 12a4z4 + 10a4z6 - 3a4z8 - a5z-1 + 4a5z - 6a5z3 + 4a5z5 - 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                   7 1   j = 6                 10 3     j = 4               10 7       j = 2             14 10         j = 0           11 12           j = -2         8 12             j = -4       6 11               j = -6     2 8                 j = -8   1 6                   j = -10   2                     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, 44]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 44]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[20, 8, 21, 7], X[22, 15, 5, 16], > X[16, 21, 17, 22], X[14, 10, 15, 9], X[18, 13, 19, 14], X[12, 19, 13, 20], > X[8, 18, 9, 17], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -9, 6, -2, 11, -8, 7, -6, 4, -5, 9, -7, > 8, -3, 5, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 3 8 14 19 23 3/2 -q + ---- - ---- + ---- - ---- + ------- - 24 Sqrt[q] + 20 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 17 q + 10 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 -14 3 4 3 4 -2 2 4 6 8 2 + q + q - q + --- - -- + -- - -- + q - q + 7 q - 2 q + 5 q + 12 8 6 4 q q q q 10 12 14 16 > q - 4 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 44]][a, z]
Out[8]=	3 5 3 3 -2 4 a 3 a a 3 z 8 z 3 5 2 z 9 z --- + --- - ---- + -- + --- - --- + 9 a z - 5 a z + a z + ---- - ---- + a z z z z 3 a 3 a a a 5 5 7 3 3 3 z 4 z 5 z > 8 a z - 3 a z + -- - ---- + 3 a z - -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 44]][a, z]
Out[9]=	3 5 -2 2 4 2 4 a 3 a a 7 z 21 z 3 -2 - a - 3 a - a - --- - --- - ---- - -- + --- + ---- + 26 a z + 16 a z + a z z z z 3 a a 2 2 3 3 5 2 3 z z 2 2 4 2 18 z 49 z 3 > 4 a z + 10 z - ---- - -- + 14 a z + 6 a z - ----- - ----- - 49 a z - 4 2 3 a a a a 4 4 4 3 3 5 3 4 z 7 z 8 z 2 4 4 4 > 24 a z - 6 a z - 46 z - -- + ---- - ---- - 42 a z - 12 a z - 6 4 2 a a a 5 5 5 6 4 z 26 z 50 z 5 3 5 5 5 6 10 z > ---- + ----- + ----- + 24 a z + 8 a z + 4 a z + 67 z - ----- + 5 3 a 4 a a a 6 7 7 26 z 2 6 4 6 17 z 5 z 7 3 7 5 7 > ----- + 41 a z + 10 a z - ----- - ---- + 20 a z + 7 a z - a z - 2 3 a a a 8 9 8 17 z 2 8 4 8 9 z 9 3 9 10 > 23 z - ----- - 9 a z - 3 a z - ---- - 13 a z - 4 a z - 2 z - 2 a a 2 10 > 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 6 2 8 6 11 12 + 14 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 8 11 12 2 4 4 2 6 2 6 3 > ----- + -- + ---- + 10 q t + 10 q t + 7 q t + 10 q t + 3 q t + 2 2 t 2 q t q t 8 3 8 4 10 4 12 5 > 7 q t + q t + 3 q t + q t

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