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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X2536 X20,13,15,14 X3,11,4,10 X9,1,10,4 X7,17,8,16 X15,5,16,8 X18,11,19,12 X12,19,13,20 X14,17,9,18

Gauss Code:

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

Jones Polynomial:

- q-13/2 + q-11/2 - 2q-9/2 + q-7/2 - 3q-5/2 - 2q-1/2 - q1/2 - q5/2

A2 (sl(3)) Invariant:

q-20 + q-18 + 3q-16 + 4q-14 + 5q-12 + 8q-10 + 9q-8 + 11q-6 + 10q-4 + 9q-2 + 8 + 5q2 + 4q4 + 2q6 + q8

HOMFLY-PT Polynomial:

- a-1z-3 - 4a-1z-1 - 4a-1z - a-1z3 + 3az-3 + 11az-1 + 14az + 7az3 + az5 - 3a3z-3 - 10a3z-1 - 13a3z - 6a3z3 - a3z5 + a5z-3 + 3a5z-1 + 3a5z + a5z3

Kauffman Polynomial:

a-1z-3 - 6a-1z-1 + 13a-1z - 15a-1z3 + 7a-1z5 - a-1z7 - 3z-2 + 13 - 17z2 + 8z4 - z6 + 3az-3 - 14az-1 + 28az - 25az3 + 9az5 - az7 - 6a2z-2 + 24a2 - 33a2z2 + 15a2z4 - 2a2z6 + 3a3z-3 - 12a3z-1 + 21a3z - 18a3z3 + 7a3z5 - a3z7 - 3a4z-2 + 11a4 - 16a4z2 + 10a4z4 - 2a4z6 + a5z-3 - 3a5z-1 + 3a5z - 4a5z3 + 4a5z5 - a5z7 - a6 + 3a6z4 - a6z6 + a7z-1 - 3a7z + 4a7z3 - 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 = 6                     1 j = 4                     1 j = 2                 1     j = 0             3         j = -2           1 4 1       j = -4         3             j = -6       1 4 1           j = -8     1 1 1             j = -10     1                 j = -12 1 1                   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]=	4
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 97]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 97]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[20, 13, 15, 14], X[3, 11, 4, 10], > X[9, 1, 10, 4], X[7, 17, 8, 16], X[15, 5, 16, 8], X[18, 11, 19, 12], > X[12, 19, 13, 20], X[14, 17, 9, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, 8, -9, 3, -10}, > {-7, 6, 10, -8, 9, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) -(11/2) 2 -(7/2) 3 2 5/2 -q + q - ---- + q - ---- - ------- - Sqrt[q] - q 9/2 5/2 Sqrt[q] q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 3 4 5 8 9 11 10 9 2 4 8 + q + q + --- + --- + --- + --- + -- + -- + -- + -- + 5 q + 4 q + 16 14 12 10 8 6 4 2 q q q q q q q q 6 8 > 2 q + q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 97]][a, z]
Out[8]=	3 5 3 5 1 3 a 3 a a 4 11 a 10 a 3 a 4 z -(----) + --- - ---- + -- - --- + ---- - ----- + ---- - --- + 14 a z - 3 3 3 3 a z z z z a a z z z z 3 3 5 z 3 3 3 5 3 5 3 5 > 13 a z + 3 a z - -- + 7 a z - 6 a z + a z + a z - a z a
In[9]:=	Kauffman[Link[10, NonAlternating, 97]][a, z]
Out[9]=	3 5 2 4 2 4 6 1 3 a 3 a a 3 6 a 3 a 6 13 + 24 a + 11 a - a + ---- + --- + ---- + -- - -- - ---- - ---- - --- - 3 3 3 3 2 2 2 a z a z z z z z z z 3 5 7 14 a 12 a 3 a a 13 z 3 5 7 > ---- - ----- - ---- + -- + ---- + 28 a z + 21 a z + 3 a z - 3 a z - z z z z a 3 2 2 2 4 2 15 z 3 3 3 5 3 > 17 z - 33 a z - 16 a z - ----- - 25 a z - 18 a z - 4 a z + a 5 7 3 4 2 4 4 4 6 4 7 z 5 3 5 > 4 a z + 8 z + 15 a z + 10 a z + 3 a z + ---- + 9 a z + 7 a z + a 7 5 5 7 5 6 2 6 4 6 6 6 z 7 3 7 5 7 > 4 a z - a z - z - 2 a z - 2 a z - a z - -- - a z - a z - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 1 1 1 1 1 1 1 3 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 6 12 5 10 4 8 4 8 3 6 3 8 2 q q t q t q t q t q t q t q t q t 4 3 1 1 t 2 2 4 4 6 4 > ----- + ----- + ---- + ---- + -- + q t + q t + q t 6 2 4 2 6 2 2 q t q t q t q t q

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