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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-15/2 + 3q-13/2 - 5q-11/2 + 8q-9/2 - 11q-7/2 + 12q-5/2 - 12q-3/2 + 10q-1/2 - 9q1/2 + 5q3/2 - 3q5/2 + q7/2

A2 (sl(3)) Invariant:

q-22 - q-20 - q-14 + 2q-12 - 2q-10 + q-8 + 4q-2 + 1 + 3q2 + q4 + q8 - q10

HOMFLY-PT Polynomial:

- a-1z-1 + a-1z + 3a-1z3 + a-1z5 + az-1 - az - 4az3 - 4az5 - az7 - a3z - 4a3z3 - 4a3z5 - a3z7 + a5z + 3a5z3 + a5z5

Kauffman Polynomial:

4a-2z2 - 8a-2z4 + 5a-2z6 - a-2z8 - a-1z-1 - 2a-1z + 18a-1z3 - 28a-1z5 + 16a-1z7 - 3a-1z9 + 1 + 11z2 - 24z4 + 8z6 + 5z8 - 2z10 - az-1 - 2az + 22az3 - 52az5 + 40az7 - 9az9 + 12a2z2 - 44a2z4 + 35a2z6 - 3a2z8 - 2a2z10 + 2a3z - 9a3z3 + 15a3z7 - 6a3z9 + 4a4z2 - 17a4z4 + 25a4z6 - 9a4z8 + 2a5z - 9a5z3 + 19a5z5 - 9a5z7 + 8a6z4 - 7a6z6 + 3a7z3 - 5a7z5 + a8z2 - 3a8z4 - a9z3

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 8                       1 j = 6                     2   j = 4                   3 1   j = 2                 6 2     j = 0               4 3       j = -2             8 6         j = -4           6 6           j = -6         5 6             j = -8       3 6               j = -10     2 5                 j = -12   1 3                   j = -14   2                     j = -16 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, 286]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 286]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[18, 8, 19, 7], > X[16, 9, 17, 10], X[22, 15, 9, 16], X[4, 21, 5, 22], X[14, 6, 15, 5], > X[20, 14, 21, 13], X[8, 18, 1, 17], X[6, 20, 7, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -7, 8, -11, 4, -10}, > {5, -1, 2, -3, 9, -8, 6, -5, 10, -4, 11, -9, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 5 8 11 12 12 10 -q + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 9 Sqrt[q] + 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 5 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 -14 2 2 -8 4 2 4 8 10 1 + q - q - q + --- - --- + q + -- + 3 q + q + q - q 12 10 2 q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 286]][a, z]
Out[8]=	3 5 1 a z 3 5 3 z 3 3 3 5 3 z -(---) + - + - - a z - a z + a z + ---- - 4 a z - 4 a z + 3 a z + -- - a z z a a a 5 3 5 5 5 7 3 7 > 4 a z - 4 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 286]][a, z]
Out[9]=	2 1 a 2 z 3 5 2 4 z 2 2 1 - --- - - - --- - 2 a z + 2 a z + 2 a z + 11 z + ---- + 12 a z + a z z a 2 a 3 4 2 8 2 18 z 3 3 3 5 3 7 3 9 3 > 4 a z + a z + ----- + 22 a z - 9 a z - 9 a z + 3 a z - a z - a 4 5 4 8 z 2 4 4 4 6 4 8 4 28 z 5 > 24 z - ---- - 44 a z - 17 a z + 8 a z - 3 a z - ----- - 52 a z + 2 a a 6 7 5 5 7 5 6 5 z 2 6 4 6 6 6 16 z > 19 a z - 5 a z + 8 z + ---- + 35 a z + 25 a z - 7 a z + ----- + 2 a a 8 9 7 3 7 5 7 8 z 2 8 4 8 3 z > 40 a z + 15 a z - 9 a z + 5 z - -- - 3 a z - 9 a z - ---- - 2 a a 9 3 9 10 2 10 > 9 a z - 6 a z - 2 z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 8 1 2 1 3 2 5 3 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 4 2 16 6 14 5 12 5 12 4 10 4 10 3 8 3 8 2 q q q t q t q t q t q t q t q t q t 5 6 6 6 t 2 2 2 2 3 4 3 > ----- + ---- + ---- + 4 t + --- + 3 t + 6 q t + 2 q t + 3 q t + 6 2 6 4 2 q t q t q t q 4 4 6 4 8 5 > q t + 2 q t + q t

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