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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 2q-5/2 - 6q-3/2 + 10q-1/2 - 15q1/2 + 18q3/2 - 18q5/2 + 16q7/2 - 13q9/2 + 8q11/2 - 4q13/2 + q15/2

A2 (sl(3)) Invariant:

q-12 + q-10 + 4q-6 + q-4 - 2q-2 + 4 - 3q2 - 2q8 + 3q10 - 3q12 + 3q14 + 2q16 - 2q18 + 3q20 - q24

HOMFLY-PT Polynomial:

a-7z - a-5z-1 - 3a-5z - 3a-5z3 + 2a-3z-1 + 5a-3z + 4a-3z3 + 2a-3z5 - a-1z-1 - 2a-1z - a-1z3 + a-1z5 - az-1 - 2az - 2az3 + a3z-1 + a3z

Kauffman Polynomial:

- a-8z2 + 2a-8z4 - a-8z6 + 2a-7z - 7a-7z3 + 10a-7z5 - 4a-7z7 + a-6z2 - 5a-6z4 + 13a-6z6 - 6a-6z8 - a-5z-1 + 9a-5z - 24a-5z3 + 24a-5z5 - a-5z7 - 4a-5z9 - a-4 + 7a-4z2 - 26a-4z4 + 34a-4z6 - 12a-4z8 - a-4z10 - 2a-3z-1 + 17a-3z - 36a-3z3 + 23a-3z5 + 4a-3z7 - 7a-3z9 - 3a-2 + 12a-2z2 - 29a-2z4 + 27a-2z6 - 10a-2z8 - a-2z10 - a-1z-1 + 10a-1z - 19a-1z3 + 12a-1z5 - 2a-1z7 - 3a-1z9 - 2 + 7z2 - 7z4 + 5z6 - 4z8 + az-1 - 3az + 3az3 + 2az5 - 3az7 - a2 + 3a2z4 - 2a2z6 + a3z-1 - 3a3z + 3a3z3 - a3z5

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 16                       1 j = 14                     3   j = 12                   5 1   j = 10                 8 3     j = 8               8 5       j = 6             10 8         j = 4           8 8           j = 2         7 10             j = 0       5 10               j = -2     1 5                 j = -4   1 5                   j = -6   1                     j = -8 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, 39]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 39]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[16, 8, 17, 7], X[20, 13, 21, 14], > X[22, 18, 5, 17], X[18, 22, 19, 21], X[14, 19, 15, 20], X[12, 10, 13, 9], > X[8, 16, 9, 15], 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, 8, -2, 11, -8, 4, -7, 9, -3, 5, -6, > 7, -4, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 2 6 10 3/2 5/2 7/2 -q + ---- - ---- + ------- - 15 Sqrt[q] + 18 q - 18 q + 16 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 15/2 > 13 q + 8 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 4 -4 2 2 8 10 12 14 16 4 + q + q + -- + q - -- - 3 q - 2 q + 3 q - 3 q + 3 q + 2 q - 6 2 q q 18 20 24 > 2 q + 3 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 39]][a, z]
Out[8]=	3 3 1 2 1 a a z 3 z 5 z 2 z 3 3 z -(----) + ---- - --- - - + -- + -- - --- + --- - --- - 2 a z + a z - ---- + 5 3 a z z z 7 5 3 a 5 a z a z a a a a 3 3 5 5 4 z z 3 2 z z > ---- - -- - 2 a z + ---- + -- 3 a 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 39]][a, z]
Out[9]=	3 -4 3 2 1 2 1 a a 2 z 9 z 17 z 10 z -2 - a - -- - a - ---- - ---- - --- + - + -- + --- + --- + ---- + ---- - 2 5 3 a z z z 7 5 3 a a a z a z a a a 2 2 2 2 3 3 3 3 2 z z 7 z 12 z 7 z 24 z 36 z > 3 a z - 3 a z + 7 z - -- + -- + ---- + ----- - ---- - ----- - ----- - 8 6 4 2 7 5 3 a a a a a a a 3 4 4 4 4 19 z 3 3 3 4 2 z 5 z 26 z 29 z 2 4 > ----- + 3 a z + 3 a z - 7 z + ---- - ---- - ----- - ----- + 3 a z + a 8 6 4 2 a a a a 5 5 5 5 6 6 10 z 24 z 23 z 12 z 5 3 5 6 z 13 z > ----- + ----- + ----- + ----- + 2 a z - a z + 5 z - -- + ----- + 7 5 3 a 8 6 a a a a a 6 6 7 7 7 7 8 34 z 27 z 2 6 4 z z 4 z 2 z 7 8 6 z > ----- + ----- - 2 a z - ---- - -- + ---- - ---- - 3 a z - 4 z - ---- - 4 2 7 5 3 a 6 a a a a a a 8 8 9 9 9 10 10 12 z 10 z 4 z 7 z 3 z z z > ----- - ----- - ---- - ---- - ---- - --- - --- 4 2 5 3 a 4 2 a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 5 1 5 5 2 10 + 7 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 10 q t + 8 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 4 2 6 2 6 3 8 3 8 4 10 4 > 8 q t + 8 q t + 10 q t + 8 q t + 8 q t + 5 q t + 8 q t + 10 5 12 5 12 6 14 6 16 7 > 3 q t + 5 q t + q t + 3 q t + q t

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