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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 3q-5/2 - 6q-3/2 + 10q-1/2 - 14q1/2 + 16q3/2 - 16q5/2 + 13q7/2 - 11q9/2 + 6q11/2 - 3q13/2 + q15/2

A2 (sl(3)) Invariant:

q-10 - q-8 + q-6 - 2q-2 + 3 - 3q2 + q4 + 5q10 + 4q14 + q16 - q18 + q20 - q22

HOMFLY-PT Polynomial:

- a-5z-1 + 2a-5z + 3a-5z3 + a-5z5 + a-3z-1 - a-3z - 5a-3z3 - 4a-3z5 - a-3z7 - a-1z - 5a-1z3 - 4a-1z5 - a-1z7 + 2az + 3az3 + az5

Kauffman Polynomial:

- 2a-8z2 + 3a-8z4 - a-8z6 + 2a-7z - 7a-7z3 + 9a-7z5 - 3a-7z7 - a-6z2 - 2a-6z4 + 9a-6z6 - 4a-6z8 - a-5z-1 + 3a-5z + 3a-5z3 - 6a-5z5 + 9a-5z7 - 4a-5z9 + a-4 + 3a-4z2 - 9a-4z4 + 8a-4z6 - 2a-4z10 - a-3z-1 - 2a-3z + 21a-3z3 - 40a-3z5 + 29a-3z7 - 9a-3z9 + 9a-2z2 - 26a-2z4 + 16a-2z6 - 2a-2z8 - 2a-2z10 - a-1z + 2a-1z3 - 13a-1z5 + 12a-1z7 - 5a-1z9 + 6z2 - 16z4 + 15z6 - 6z8 + 2az - 7az3 + 11az5 - 5az7 - a2z2 + 6a2z4 - 3a2z6 + 2a3z3 - 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                 7 2     j = 8               7 5       j = 6             9 6         j = 4           7 7           j = 2         7 9             j = 0       4 8               j = -2     2 6                 j = -4   1 4                   j = -6   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, 288]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 288]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[16, 9, 17, 10], X[20, 12, 21, 11], > X[22, 15, 9, 16], X[14, 6, 15, 5], X[4, 14, 5, 13], X[6, 20, 7, 19], > X[18, 8, 19, 7], X[8, 18, 1, 17], X[2, 21, 3, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 2, -7, 6, -8, 9, -10}, > {3, -1, 4, -2, 7, -6, 5, -3, 10, -9, 8, -4, 11, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 3 6 10 3/2 5/2 7/2 -q + ---- - ---- + ------- - 14 Sqrt[q] + 16 q - 16 q + 13 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 15/2 > 11 q + 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -8 -6 2 2 4 10 14 16 18 20 22 3 + q - q + q - -- - 3 q + q + 5 q + 4 q + q - q + q - q 2 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 288]][a, z]
Out[8]=	3 3 3 5 1 1 2 z z z 3 z 5 z 5 z 3 z -(----) + ---- + --- - -- - - + 2 a z + ---- - ---- - ---- + 3 a z + -- - 5 3 5 3 a 5 3 a 5 a z a z a a a a a 5 5 7 7 4 z 4 z 5 z z > ---- - ---- + a z - -- - -- 3 a 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 288]][a, z]
Out[9]=	2 2 2 -4 1 1 2 z 3 z 2 z z 2 2 z z 3 z a - ---- - ---- + --- + --- - --- - - + 2 a z + 6 z - ---- - -- + ---- + 5 3 7 5 3 a 8 6 4 a z a z a a a a a a 2 3 3 3 3 9 z 2 2 7 z 3 z 21 z 2 z 3 3 3 4 > ---- - a z - ---- + ---- + ----- + ---- - 7 a z + 2 a z - 16 z + 2 7 5 3 a a a a a 4 4 4 4 5 5 5 5 3 z 2 z 9 z 26 z 2 4 9 z 6 z 40 z 13 z > ---- - ---- - ---- - ----- + 6 a z + ---- - ---- - ----- - ----- + 8 6 4 2 7 5 3 a a a a a a a a 6 6 6 6 7 5 3 5 6 z 9 z 8 z 16 z 2 6 3 z > 11 a z - a z + 15 z - -- + ---- + ---- + ----- - 3 a z - ---- + 8 6 4 2 7 a a a a a 7 7 7 8 8 9 9 9 9 z 29 z 12 z 7 8 4 z 2 z 4 z 9 z 5 z > ---- + ----- + ----- - 5 a z - 6 z - ---- - ---- - ---- - ---- - ---- - 5 3 a 6 2 5 3 a a a a a a a 10 10 2 z 2 z > ----- - ----- 4 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 4 2 4 6 2 4 8 + 7 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 9 q t + 7 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 2 6 2 6 3 8 3 8 4 10 4 10 5 > 7 q t + 9 q t + 6 q t + 7 q t + 5 q t + 7 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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