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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X16,5,17,6 X18,10,19,9 X10,20,11,19 X14,18,15,17 X2,11,3,12 X12,3,13,4 X4758 X20,14,7,13 X6,15,1,16

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 3q-11/2 - 8q-9/2 + 11q-7/2 - 15q-5/2 + 16q-3/2 - 15q-1/2 + 12q1/2 - 8q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

q-20 + q-16 + 5q-14 + 4q-10 + q-8 - 2q-6 + 2q-4 - 4q-2 + 3 - 2q2 - q4 + 2q6 - 2q8 + q10

HOMFLY-PT Polynomial:

- a-1z - 2a-1z3 - a-1z5 + az-1 + 3az + 6az3 + 4az5 + az7 - 3a3z-1 - 7a3z - 6a3z3 - 2a3z5 + 2a5z-1 + 2a5z + a5z3

Kauffman Polynomial:

a-3z3 - a-3z5 - 2a-2z2 + 6a-2z4 - 4a-2z6 + a-1z - 6a-1z3 + 12a-1z5 - 7a-1z7 + 1 - 3z2 + 4z4 + 5z6 - 6z8 - az-1 + 3az - 15az3 + 26az5 - 11az7 - 2az9 + 3a2 - 10a2z2 + 3a2z4 + 14a2z6 - 11a2z8 - 3a3z-1 + 12a3z - 23a3z3 + 26a3z5 - 10a3z7 - 2a3z9 + 3a4 - 10a4z2 + 9a4z4 + 2a4z6 - 5a4z8 - 2a5z-1 + 9a5z - 13a5z3 + 12a5z5 - 6a5z7 - a6z2 + 4a6z4 - 3a6z6 - a7z + 2a7z3 - a7z5

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 j = 8                     1 j = 6                   3   j = 4                 5 1   j = 2               7 3     j = 0             8 5       j = -2           9 8         j = -4         6 7           j = -6       5 9             j = -8     3 6               j = -10     5                 j = -12 1 3                   j = -14 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[10, Alternating, 85]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 85]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 5, 17, 6], X[18, 10, 19, 9], X[10, 20, 11, 19], > X[14, 18, 15, 17], X[2, 11, 3, 12], X[12, 3, 13, 4], X[4, 7, 5, 8], > X[20, 14, 7, 13], X[6, 15, 1, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -6, 7, -8, 2, -10}, > {8, -1, 3, -4, 6, -7, 9, -5, 10, -2, 5, -3, 4, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 3 8 11 15 16 15 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 12 Sqrt[q] - 8 q + 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 7/2 > 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -16 5 4 -8 2 2 4 2 4 6 8 10 3 + q + q + --- + --- + q - -- + -- - -- - 2 q - q + 2 q - 2 q + q 14 10 6 4 2 q q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 85]][a, z]
Out[8]=	3 5 3 a 3 a 2 a z 3 5 2 z 3 3 3 - - ---- + ---- - - + 3 a z - 7 a z + 2 a z - ---- + 6 a z - 6 a z + z z z a a 5 5 3 z 5 3 5 7 > a z - -- + 4 a z - 2 a z + a z a
In[9]:=	Kauffman[Link[10, Alternating, 85]][a, z]
Out[9]=	3 5 2 4 a 3 a 2 a z 3 5 7 1 + 3 a + 3 a - - - ---- - ---- + - + 3 a z + 12 a z + 9 a z - a z - z z z a 2 3 3 2 2 z 2 2 4 2 6 2 z 6 z 3 > 3 z - ---- - 10 a z - 10 a z - a z + -- - ---- - 15 a z - 2 3 a a a 4 3 3 5 3 7 3 4 6 z 2 4 4 4 6 4 > 23 a z - 13 a z + 2 a z + 4 z + ---- + 3 a z + 9 a z + 4 a z - 2 a 5 5 6 z 12 z 5 3 5 5 5 7 5 6 4 z > -- + ----- + 26 a z + 26 a z + 12 a z - a z + 5 z - ---- + 3 a 2 a a 7 2 6 4 6 6 6 7 z 7 3 7 5 7 8 > 14 a z + 2 a z - 3 a z - ---- - 11 a z - 10 a z - 6 a z - 6 z - a 2 8 4 8 9 3 9 > 11 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	8 1 1 3 5 3 6 5 9 8 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 6 12 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 6 7 9 2 2 2 4 2 4 3 6 3 > ----- + ---- + ---- + 5 t + 7 q t + 3 q t + 5 q t + q t + 3 q t + 4 2 4 2 q t q t q t 8 4 > q t

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