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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X12,3,13,4 X18,10,19,9 X20,13,7,14 X14,19,15,20 X10,16,11,15 X16,6,17,5 X2738 X4,11,5,12 X6,18,1,17

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 3q-9/2 - 6q-7/2 + 8q-5/2 - 11q-3/2 + 11q-1/2 - 11q1/2 + 9q3/2 - 6q5/2 + 3q7/2 - q9/2

A2 (sl(3)) Invariant:

q-18 + q-16 - q-14 + 2q-12 + 3q-6 - q-4 + 2q-2 + 2q4 - 2q6 + 2q8 - q12 + q14

HOMFLY-PT Polynomial:

- a-3z - a-3z3 - a-1z-1 - a-1z + a-1z3 + a-1z5 + 2az-1 + 6az + 6az3 + 2az5 - 2a3z-1 - 6a3z - 3a3z3 + a5z-1 + a5z

Kauffman Polynomial:

- a-5z3 - 3a-4z4 - a-3z + 4a-3z3 - 6a-3z5 - 5a-2z2 + 13a-2z4 - 9a-2z6 - a-1z-1 + 5a-1z - 13a-1z3 + 21a-1z5 - 10a-1z7 - 2z2 + 13z6 - 7z8 - 2az-1 + 17az - 44az3 + 42az5 - 8az7 - 2az9 - a2 + 7a2z2 - 30a2z4 + 34a2z6 - 10a2z8 - 2a3z-1 + 15a3z - 32a3z3 + 19a3z5 + a3z7 - 2a3z9 + 4a4z2 - 14a4z4 + 12a4z6 - 3a4z8 - a5z-1 + 4a5z - 6a5z3 + 4a5z5 - a5z7

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 = 10                     1 j = 8                   2   j = 6                 4 1   j = 4               5 2     j = 2             6 4       j = 0           6 6         j = -2         5 5           j = -4       4 7             j = -6     2 4               j = -8   1 4                 j = -10   2                   j = -12 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, 80]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 80]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[18, 10, 19, 9], X[20, 13, 7, 14], > X[14, 19, 15, 20], X[10, 16, 11, 15], X[16, 6, 17, 5], X[2, 7, 3, 8], > X[4, 11, 5, 12], X[6, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9, 7, -10}, > {8, -1, 3, -6, 9, -2, 4, -5, 6, -7, 10, -3, 5, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 3 6 8 11 11 3/2 -q + ---- - ---- + ---- - ---- + ------- - 11 Sqrt[q] + 9 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 > 6 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 -14 2 3 -4 2 4 6 8 12 14 q + q - q + --- + -- - q + -- + 2 q - 2 q + 2 q - q + q 12 6 2 q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 80]][a, z]
Out[8]=	3 5 3 3 1 2 a 2 a a z z 3 5 z z 3 -(---) + --- - ---- + -- - -- - - + 6 a z - 6 a z + a z - -- + -- + 6 a z - a z z z z 3 a 3 a a a 5 3 3 z 5 > 3 a z + -- + 2 a z a
In[9]:=	Kauffman[Link[10, Alternating, 80]][a, z]
Out[9]=	3 5 2 1 2 a 2 a a z 5 z 3 5 2 -a - --- - --- - ---- - -- - -- + --- + 17 a z + 15 a z + 4 a z - 2 z - a z z z z 3 a a 2 3 3 3 5 z 2 2 4 2 z 4 z 13 z 3 3 3 > ---- + 7 a z + 4 a z - -- + ---- - ----- - 44 a z - 32 a z - 2 5 3 a a a a 4 4 5 5 5 3 3 z 13 z 2 4 4 4 6 z 21 z 5 > 6 a z - ---- + ----- - 30 a z - 14 a z - ---- + ----- + 42 a z + 4 2 3 a a a a 6 7 3 5 5 5 6 9 z 2 6 4 6 10 z 7 > 19 a z + 4 a z + 13 z - ---- + 34 a z + 12 a z - ----- - 8 a z + 2 a a 3 7 5 7 8 2 8 4 8 9 3 9 > a z - a z - 7 z - 10 a z - 3 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 4 2 4 4 7 6 + 6 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 5 6 5 2 4 4 2 6 2 6 3 8 3 > ----- + - + ---- + 4 q t + 5 q t + 2 q t + 4 q t + q t + 2 q t + 2 2 t 2 q t q t 10 4 > q t

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