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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X18,15,19,16 X16,10,17,9 X8,18,9,17 X4,19,1,20 X12,6,13,5 X10,4,11,3 X20,12,5,11 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 4q-5/2 - 8q-3/2 + 11q-1/2 - 16q1/2 + 16q3/2 - 15q5/2 + 12q7/2 - 8q9/2 + 4q11/2 - q13/2

A2 (sl(3)) Invariant:

q-10 - 2q-8 + 2q-6 + 6 - q2 + 4q4 - q6 + 2q10 - 3q12 + 2q14 - q16 - q18 + q20

HOMFLY-PT Polynomial:

- a-5z - a-5z3 + 3a-3z + 5a-3z3 + 2a-3z5 - a-1z-1 - 3a-1z - 6a-1z3 - 4a-1z5 - a-1z7 + az-1 + az + 2az3 + az5

Kauffman Polynomial:

a-7z3 - a-7z5 - 2a-6z2 + 6a-6z4 - 4a-6z6 + 2a-5z - 6a-5z3 + 12a-5z5 - 7a-5z7 - 3a-4z2 + 4a-4z4 + 5a-4z6 - 6a-4z8 + 6a-3z - 23a-3z3 + 31a-3z5 - 12a-3z7 - 2a-3z9 - a-2z2 - 7a-2z4 + 19a-2z6 - 12a-2z8 - a-1z-1 + 6a-1z - 25a-1z3 + 32a-1z5 - 12a-1z7 - 2a-1z9 + 1 - z2 + z4 + 6z6 - 6z8 - az-1 + 2az - 8az3 + 13az5 - 7az7 - a2z2 + 6a2z4 - 4a2z6 + a3z3 - a3z5

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 14                     1 j = 12                   3   j = 10                 5 1   j = 8               7 3     j = 6             8 5       j = 4           8 7         j = 2         8 8           j = 0       5 10             j = -2     3 6               j = -4   1 5                 j = -6   3                   j = -8 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, 21]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 21]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[18, 15, 19, 16], X[16, 10, 17, 9], > X[8, 18, 9, 17], X[4, 19, 1, 20], X[12, 6, 13, 5], X[10, 4, 11, 3], > X[20, 12, 5, 11], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 8, -6}, {7, -1, 2, -5, 4, -8, 9, -7, 10, -2, 3, -4, 5, -3, > 6, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 4 8 11 3/2 5/2 7/2 -q + ---- - ---- + ------- - 16 Sqrt[q] + 16 q - 15 q + 12 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 > 8 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 2 2 2 4 6 10 12 14 16 18 20 6 + q - -- + -- - q + 4 q - q + 2 q - 3 q + 2 q - q - q + q 8 6 q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 21]][a, z]
Out[8]=	3 3 3 5 5 1 a z 3 z 3 z z 5 z 6 z 3 2 z 4 z -(---) + - - -- + --- - --- + a z - -- + ---- - ---- + 2 a z + ---- - ---- + a z z 5 3 a 5 3 a 3 a a a a a a 7 5 z > a z - -- a
In[9]:=	Kauffman[Link[10, Alternating, 21]][a, z]
Out[9]=	2 2 2 3 1 a 2 z 6 z 6 z 2 2 z 3 z z 2 2 z 1 - --- - - + --- + --- + --- + 2 a z - z - ---- - ---- - -- - a z + -- - a z z 5 3 a 6 4 2 7 a a a a a a 3 3 3 4 4 4 6 z 23 z 25 z 3 3 3 4 6 z 4 z 7 z 2 4 > ---- - ----- - ----- - 8 a z + a z + z + ---- + ---- - ---- + 6 a z - 5 3 a 6 4 2 a a a a a 5 5 5 5 6 6 6 z 12 z 31 z 32 z 5 3 5 6 4 z 5 z 19 z > -- + ----- + ----- + ----- + 13 a z - a z + 6 z - ---- + ---- + ----- - 7 5 3 a 6 4 2 a a a a a a 7 7 7 8 8 9 9 2 6 7 z 12 z 12 z 7 8 6 z 12 z 2 z 2 z > 4 a z - ---- - ----- - ----- - 7 a z - 6 z - ---- - ----- - ---- - ---- 5 3 a 4 2 3 a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 5 3 5 6 2 10 + 8 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 8 q t + 8 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 4 2 6 2 6 3 8 3 8 4 10 4 > 8 q t + 7 q t + 8 q t + 5 q t + 7 q t + 3 q t + 5 q t + 10 5 12 5 14 6 > q t + 3 q t + q t

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