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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q1/2 + 2q3/2 - 4q5/2 + 6q7/2 - 9q9/2 + 10q11/2 - 11q13/2 + 10q15/2 - 8q17/2 + 5q19/2 - 3q21/2 + q23/2

A2 (sl(3)) Invariant:

q2 + q8 - q10 + 2q12 - q14 + 2q16 + 2q18 + 3q22 - q24 + q26 + q32 - q34

HOMFLY-PT Polynomial:

a-9z + 3a-9z3 + a-9z5 - a-7z-1 - 2a-7z - 4a-7z3 - 4a-7z5 - a-7z7 + a-5z-1 + a-5z - 3a-5z3 - 4a-5z5 - a-5z7 + 3a-3z + 4a-3z3 + a-3z5

Kauffman Polynomial:

a-14z2 - a-14z4 - a-13z + 4a-13z3 - 3a-13z5 + 4a-12z4 - 4a-12z6 - 2a-11z + 2a-11z3 + 3a-11z5 - 4a-11z7 + a-10z2 - 6a-10z4 + 7a-10z6 - 4a-10z8 - a-9z + 6a-9z3 - 9a-9z5 + 7a-9z7 - 3a-9z9 + 6a-8z2 - 16a-8z4 + 13a-8z6 - 2a-8z8 - a-8z10 + a-7z-1 - 6a-7z + 17a-7z3 - 25a-7z5 + 19a-7z7 - 5a-7z9 - a-6 + 8a-6z2 - 16a-6z4 + 11a-6z6 - a-6z10 + a-5z-1 - 3a-5z + 2a-5z3 - 5a-5z5 + 7a-5z7 - 2a-5z9 + 4a-4z2 - 11a-4z4 + 9a-4z6 - 2a-4z8 + 3a-3z - 7a-3z3 + 5a-3z5 - a-3z7

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 24                       1 j = 22                     2   j = 20                   3 1   j = 18                 5 2     j = 16               6 4       j = 14             5 4         j = 12           5 6           j = 10         4 5             j = 8       2 5               j = 6     2 4                 j = 4   1 3                   j = 2   1                     j = 0 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, 362]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 362]]
Out[4]=	PD[X[12, 1, 13, 2], X[14, 4, 15, 3], X[22, 14, 11, 13], X[2, 11, 3, 12], > X[4, 22, 5, 21], X[20, 10, 21, 9], X[16, 6, 17, 5], X[18, 8, 19, 7], > X[6, 18, 7, 17], X[8, 20, 9, 19], X[10, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -5, 7, -9, 8, -10, 6, -11}, > {4, -1, 3, -2, 11, -7, 9, -8, 10, -6, 5, -3}]
In[6]:=	Jones[L][q]
Out[6]=	3/2 5/2 7/2 9/2 11/2 13/2 15/2 -Sqrt[q] + 2 q - 4 q + 6 q - 9 q + 10 q - 11 q + 10 q - 17/2 19/2 21/2 23/2 > 8 q + 5 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 8 10 12 14 16 18 22 24 26 32 34 q + q - q + 2 q - q + 2 q + 2 q + 3 q - q + q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 362]][a, z]
Out[8]=	3 3 3 3 5 5 1 1 z 2 z z 3 z 3 z 4 z 3 z 4 z z 4 z -(----) + ---- + -- - --- + -- + --- + ---- - ---- - ---- + ---- + -- - ---- - 7 5 9 7 5 3 9 7 5 3 9 7 a z a z a a a a a a a a a a 5 5 7 7 4 z z z z > ---- + -- - -- - -- 5 3 7 5 a a a a
In[9]:=	Kauffman[Link[11, Alternating, 362]][a, z]
Out[9]=	2 2 2 -6 1 1 z 2 z z 6 z 3 z 3 z z z 6 z -a + ---- + ---- - --- - --- - -- - --- - --- + --- + --- + --- + ---- + 7 5 13 11 9 7 5 3 14 10 8 a z a z a a a a a a a a a 2 2 3 3 3 3 3 3 4 4 8 z 4 z 4 z 2 z 6 z 17 z 2 z 7 z z 4 z > ---- + ---- + ---- + ---- + ---- + ----- + ---- - ---- - --- + ---- - 6 4 13 11 9 7 5 3 14 12 a a a a a a a a a a 4 4 4 4 5 5 5 5 5 5 6 z 16 z 16 z 11 z 3 z 3 z 9 z 25 z 5 z 5 z > ---- - ----- - ----- - ----- - ---- + ---- - ---- - ----- - ---- + ---- - 10 8 6 4 13 11 9 7 5 3 a a a a a a a a a a 6 6 6 6 6 7 7 7 7 7 4 z 7 z 13 z 11 z 9 z 4 z 7 z 19 z 7 z z > ---- + ---- + ----- + ----- + ---- - ---- + ---- + ----- + ---- - -- - 12 10 8 6 4 11 9 7 5 3 a a a a a a a a a a 8 8 8 9 9 9 10 10 4 z 2 z 2 z 3 z 5 z 2 z z z > ---- - ---- - ---- - ---- - ---- - ---- - --- - --- 10 8 4 9 7 5 8 6 a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 4 6 -2 q q 6 8 8 2 10 2 10 3 3 q + 2 q + t + -- + -- + 4 q t + 2 q t + 5 q t + 4 q t + 5 q t + t t 12 3 12 4 14 4 14 5 16 5 16 6 > 5 q t + 6 q t + 5 q t + 4 q t + 6 q t + 4 q t + 18 6 18 7 20 7 20 8 22 8 24 9 > 5 q t + 2 q t + 3 q t + q t + 2 q t + q t

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