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DrawMorseLink

PD Presentation:

X12,1,13,2 X9,19,10,18 X14,6,15,5 X6,12,7,11 X22,15,11,16 X20,8,21,7 X3948 X16,21,17,22 X4,18,5,17 X10,13,1,14 X19,3,20,2

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 3q-3/2 - 5q-1/2 + 7q1/2 - 9q3/2 + 8q5/2 - 8q7/2 + 6q9/2 - 4q11/2 + q13/2

A2 (sl(3)) Invariant:

q-8 - q-4 - 2 + q2 + q4 + 2q6 + 4q8 - q10 + 2q12 - q14 + q16 + 2q18 + q22 - q24

HOMFLY-PT Polynomial:

a-7z - a-5z-1 - 4a-5z - 4a-5z3 - a-5z5 + a-3z-1 + 8a-3z + 9a-3z3 + 5a-3z5 + a-3z7 - 5a-1z - 7a-1z3 - 2a-1z5 + 2az + az3

Kauffman Polynomial:

- a-8z2 + a-7z - 4a-7z3 - a-6z2 + a-6z4 - 2a-6z6 - a-5z-1 + 6a-5z - 14a-5z3 + 14a-5z5 - 5a-5z7 + a-4 + 3a-4z2 - 12a-4z4 + 15a-4z6 - 5a-4z8 - a-3z-1 + 7a-3z - 18a-3z3 + 17a-3z5 - 2a-3z9 + 8a-2z2 - 29a-2z4 + 30a-2z6 - 8a-2z8 + 4a-1z - 13a-1z3 + 7a-1z5 + 4a-1z7 - 2a-1z9 + 5z2 - 16z4 + 13z6 - 3z8 + 2az - 5az3 + 4az5 - az7

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 14                   1 j = 12                 3   j = 10               4 2   j = 8             4 2     j = 6           4 4       j = 4         5 4         j = 2       3 5           j = 0     2 4             j = -2   1 3               j = -4   2                 j = -6 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, NonAlternating, 250]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 250]]
Out[4]=	PD[X[12, 1, 13, 2], X[9, 19, 10, 18], X[14, 6, 15, 5], X[6, 12, 7, 11], > X[22, 15, 11, 16], X[20, 8, 21, 7], X[3, 9, 4, 8], X[16, 21, 17, 22], > X[4, 18, 5, 17], X[10, 13, 1, 14], X[19, 3, 20, 2]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -7, -9, 3, -4, 6, 7, -2, -10}, > {4, -1, 10, -3, 5, -8, 9, 2, -11, -6, 8, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 3 5 3/2 5/2 7/2 9/2 -q + ---- - ------- + 7 Sqrt[q] - 9 q + 8 q - 8 q + 6 q - 3/2 Sqrt[q] q 11/2 13/2 > 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -4 2 4 6 8 10 12 14 16 18 -2 + q - q + q + q + 2 q + 4 q - q + 2 q - q + q + 2 q + 22 24 > q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 250]][a, z]
Out[8]=	3 3 3 1 1 z 4 z 8 z 5 z 4 z 9 z 7 z 3 -(----) + ---- + -- - --- + --- - --- + 2 a z - ---- + ---- - ---- + a z - 5 3 7 5 3 a 5 3 a a z a z a a a a a 5 5 5 7 z 5 z 2 z z > -- + ---- - ---- + -- 5 3 a 3 a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 250]][a, z]
Out[9]=	2 2 2 -4 1 1 z 6 z 7 z 4 z 2 z z 3 z a - ---- - ---- + -- + --- + --- + --- + 2 a z + 5 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 4 4 4 8 z 4 z 14 z 18 z 13 z 3 4 z 12 z 29 z > ---- - ---- - ----- - ----- - ----- - 5 a z - 16 z + -- - ----- - ----- + 2 7 5 3 a 6 4 2 a a a a a a a 5 5 5 6 6 6 7 14 z 17 z 7 z 5 6 2 z 15 z 30 z 5 z > ----- + ----- + ---- + 4 a z + 13 z - ---- + ----- + ----- - ---- + 5 3 a 6 4 2 5 a a a a a a 7 8 8 9 9 4 z 7 8 5 z 8 z 2 z 2 z > ---- - a z - 3 z - ---- - ---- - ---- - ---- a 4 2 3 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 2 1 2 3 4 3 q 4 6 5 q + 5 q + ----- + ----- + ----- + -- + ----- + - + ---- + 4 q t + 4 q t + 6 4 4 3 2 3 2 2 2 t t q t q t q t t q t 6 2 8 2 8 3 10 3 10 4 12 4 14 5 > 4 q t + 4 q t + 2 q t + 4 q t + 2 q t + 3 q t + q t

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