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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X12,8,13,7 X16,10,17,9 X20,17,5,18 X18,13,19,14 X14,19,15,20 X8,16,9,15 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

q-9/2 - 3q-7/2 + 6q-5/2 - 10q-3/2 + 12q-1/2 - 14q1/2 + 12q3/2 - 11q5/2 + 7q7/2 - 3q9/2 + q11/2

A2 (sl(3)) Invariant:

- q-14 - 2q-8 + 3q-6 + 2q-2 + 4 + 5q4 - q6 + 2q8 - 3q12 + q14 - q16

HOMFLY-PT Polynomial:

a-3z-1 + 4a-3z + 3a-3z3 + a-3z5 - 4a-1z-1 - 11a-1z - 11a-1z3 - 5a-1z5 - a-1z7 + 4az-1 + 9az + 7az3 + 2az5 - a3z-1 - 2a3z - a3z3

Kauffman Polynomial:

a-6z2 - a-6z4 + 2a-5z3 - 3a-5z5 + a-4 - 5a-4z2 + 7a-4z4 - 6a-4z6 - a-3z-1 + 8a-3z - 15a-3z3 + 14a-3z5 - 8a-3z7 + 4a-2 - 15a-2z2 + 14a-2z4 - 5a-2z8 - 4a-1z-1 + 21a-1z - 43a-1z3 + 43a-1z5 - 15a-1z7 - a-1z9 + 7 - 18z2 + 12z4 + 10z6 - 8z8 - 4az-1 + 17az - 35az3 + 35az5 - 10az7 - az9 + 4a2 - 12a2z2 + 9a2z4 + 3a2z6 - 3a2z8 - a3z-1 + 4a3z - 9a3z3 + 9a3z5 - 3a3z7 + a4 - 3a4z2 + 3a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12                     1 j = 10                   2   j = 8                 5 1   j = 6               6 2     j = 4             6 5       j = 2           8 6         j = 0         6 8           j = -2       4 6             j = -4     2 6               j = -6   1 4                 j = -8   2                   j = -10 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, 8]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 8]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[16, 10, 17, 9], > X[20, 17, 5, 18], X[18, 13, 19, 14], X[14, 19, 15, 20], X[8, 16, 9, 15], > X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 3, -8, 4, -2, 10, -3, 6, -7, 8, -4, 5, -6, > 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 3 6 10 12 3/2 5/2 q - ---- + ---- - ---- + ------- - 14 Sqrt[q] + 12 q - 11 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 > 7 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 2 3 2 4 6 8 12 14 16 4 - q - -- + -- + -- + 5 q - q + 2 q - 3 q + q - q 8 6 2 q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 8]][a, z]
Out[8]=	3 3 3 1 4 4 a a 4 z 11 z 3 3 z 11 z 3 ---- - --- + --- - -- + --- - ---- + 9 a z - 2 a z + ---- - ----- + 7 a z - 3 a z z z 3 a 3 a a z a a 5 5 7 3 3 z 5 z 5 z > a z + -- - ---- + 2 a z - -- 3 a a a
In[9]:=	Kauffman[Link[10, Alternating, 8]][a, z]
Out[9]=	3 -4 4 2 4 1 4 4 a a 8 z 21 z 7 + a + -- + 4 a + a - ---- - --- - --- - -- + --- + ---- + 17 a z + 2 3 a z z z 3 a a a z a 2 2 2 3 3 3 2 z 5 z 15 z 2 2 4 2 2 z 15 z > 4 a z - 18 z + -- - ---- - ----- - 12 a z - 3 a z + ---- - ----- - 6 4 2 5 3 a a a a a 3 4 4 4 43 z 3 3 3 4 z 7 z 14 z 2 4 4 4 > ----- - 35 a z - 9 a z + 12 z - -- + ---- + ----- + 9 a z + 3 a z - a 6 4 2 a a a 5 5 5 6 3 z 14 z 43 z 5 3 5 6 6 z 2 6 4 6 > ---- + ----- + ----- + 35 a z + 9 a z + 10 z - ---- + 3 a z - a z - 5 3 a 4 a a a 7 7 8 9 8 z 15 z 7 3 7 8 5 z 2 8 z 9 > ---- - ----- - 10 a z - 3 a z - 8 z - ---- - 3 a z - -- - a z 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 4 2 6 4 6 6 8 + 8 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + ---- + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t q t q t 2 4 4 2 6 2 6 3 8 3 8 4 > 6 q t + 6 q t + 5 q t + 6 q t + 2 q t + 5 q t + q t + 10 4 12 5 > 2 q t + q t

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