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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 2q-5/2 - 6q-3/2 + 9q-1/2 - 13q1/2 + 14q3/2 - 15q5/2 + 13q7/2 - 9q9/2 + 6q11/2 - 3q13/2 + q15/2

A2 (sl(3)) Invariant:

q-12 + q-10 + 4q-6 + q-4 - q-2 + 4 - q2 + 3q4 + 2q6 + q10 - 5q12 - 2q18 + 2q20 - q24

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-8z2 + 3a-8z4 - a-8z6 + a-7z - 5a-7z3 + 9a-7z5 - 3a-7z7 - 3a-6 + 10a-6z2 - 18a-6z4 + 17a-6z6 - 5a-6z8 + a-5z-1 - a-5z - a-5z3 - 5a-5z5 + 10a-5z7 - 4a-5z9 - 7a-4 + 34a-4z2 - 58a-4z4 + 38a-4z6 - 8a-4z8 - a-4z10 + 2a-3z-1 - 9a-3z + 21a-3z3 - 30a-3z5 + 21a-3z7 - 7a-3z9 - 4a-2 + 23a-2z2 - 37a-2z4 + 23a-2z6 - 6a-2z8 - a-2z10 + a-1z-1 - 7a-1z + 16a-1z3 - 12a-1z5 + 5a-1z7 - 3a-1z9 + 3z4 + z6 - 3z8 + az-1 - 3az + 2az3 + 3az5 - 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                     2   j = 12                   4 1   j = 10                 5 2     j = 8               8 4       j = 6             7 5         j = 4           7 8           j = 2         6 7             j = 0       4 8               j = -2     2 5                 j = -4     4                   j = -6 1 2                     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, 88]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 88]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[18, 8, 19, 7], X[22, 20, 5, 19], > X[20, 14, 21, 13], X[14, 22, 15, 21], X[16, 10, 17, 9], X[10, 16, 11, 15], > X[8, 18, 9, 17], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -9, 7, -8, 11, -2, 5, -6, 8, -7, 9, -3, > 4, -5, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 2 6 9 3/2 5/2 7/2 -q + ---- - ---- + ------- - 13 Sqrt[q] + 14 q - 15 q + 13 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 15/2 > 9 q + 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 4 -4 -2 2 4 6 10 12 18 4 + q + q + -- + q - q - q + 3 q + 2 q + q - 5 q - 2 q + 6 q 20 24 > 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 88]][a, z]
Out[8]=	3 3 5 1 2 1 a a z z z 3 2 z 3 z ---- - ---- + --- - - + -- + -- - -- - -- - 2 a z + a z - ---- - 2 a z + -- + 5 3 a z z z 7 5 3 5 3 a z a z a a a a a 5 z > -- a
In[9]:=	Kauffman[Link[11, Alternating, 88]][a, z]
Out[9]=	3 -3 7 4 2 1 2 1 a a z z 9 z 7 z -- - -- - -- - a + ---- + ---- + --- + - + -- + -- - -- - --- - --- - 3 a z - 6 4 2 5 3 a z z z 7 5 3 a a a a a z a z a a a 2 2 2 2 3 3 3 3 3 z 10 z 34 z 23 z 5 z z 21 z 16 z 3 > 3 a z - -- + ----- + ----- + ----- - ---- - -- + ----- + ----- + 2 a z + 8 6 4 2 7 5 3 a a a a a a a a 4 4 4 4 5 5 3 3 4 3 z 18 z 58 z 37 z 2 4 9 z 5 z > 3 a z + 3 z + ---- - ----- - ----- - ----- + 3 a z + ---- - ---- - 8 6 4 2 7 5 a a a a a a 5 5 6 6 6 6 30 z 12 z 5 3 5 6 z 17 z 38 z 23 z > ----- - ----- + 3 a z - a z + z - -- + ----- + ----- + ----- - 3 a 8 6 4 2 a a a a a 7 7 7 7 8 8 2 6 3 z 10 z 21 z 5 z 7 8 5 z 8 z > 2 a z - ---- + ----- + ----- + ---- - 3 a z - 3 z - ---- - ---- - 7 5 3 a 6 4 a a a a a 8 9 9 9 10 10 6 z 4 z 7 z 3 z z z > ---- - ---- - ---- - ---- - --- - --- 2 5 3 a 4 2 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 2 4 2 4 5 2 4 8 + 6 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 7 q t + 7 q t + 8 4 6 4 6 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 2 6 2 6 3 8 3 8 4 10 4 10 5 > 8 q t + 7 q t + 5 q t + 8 q t + 4 q t + 5 q t + 2 q t + 12 5 12 6 14 6 16 7 > 4 q t + q t + 2 q t + q t

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