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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9/2 - 3q-7/2 + 7q-5/2 - 12q-3/2 + 14q-1/2 - 17q1/2 + 15q3/2 - 13q5/2 + 9q7/2 - 4q9/2 + q11/2

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

a-3z-1 + 2a-3z + 2a-3z3 + a-3z5 - 4a-1z-1 - 7a-1z - 7a-1z3 - 4a-1z5 - a-1z7 + 4az-1 + 7az + 6az3 + 2az5 - a3z-1 - 2a3z - a3z3

Kauffman Polynomial:

- a-6z4 + a-5z3 - 4a-5z5 + a-4 - 4a-4z2 + 9a-4z4 - 9a-4z6 - a-3z-1 + 4a-3z - 11a-3z3 + 18a-3z5 - 12a-3z7 + 4a-2 - 12a-2z2 + 14a-2z4 + 3a-2z6 - 8a-2z8 - 4a-1z-1 + 13a-1z - 33a-1z3 + 45a-1z5 - 17a-1z7 - 2a-1z9 + 7 - 15z2 + 5z4 + 19z6 - 12z8 - 4az-1 + 13az - 29az3 + 31az5 - 8az7 - 2az9 + 4a2 - 10a2z2 + 4a2z4 + 6a2z6 - 4a2z8 - a3z-1 + 4a3z - 8a3z3 + 8a3z5 - 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                   3   j = 8                 6 1   j = 6               7 3     j = 4             8 6       j = 2           9 7         j = 0         7 10           j = -2       5 7             j = -4     2 7               j = -6   1 5                 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, 20]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 20]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[20, 15, 5, 16], X[16, 7, 17, 8], > X[12, 18, 13, 17], X[14, 10, 15, 9], X[18, 14, 19, 13], X[8, 19, 9, 20], > X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 4, -8, 6, -2, 10, -5, 7, -6, 3, -4, 5, -7, > 8, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 3 7 12 14 3/2 5/2 q - ---- + ---- - ---- + ------- - 17 Sqrt[q] + 15 q - 13 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 > 9 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 3 3 3 4 6 8 12 14 16 6 - q - -- + -- + -- + 5 q - 3 q + q - 3 q + 2 q - q 8 6 2 q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 20]][a, z]
Out[8]=	3 3 3 1 4 4 a a 2 z 7 z 3 2 z 7 z 3 ---- - --- + --- - -- + --- - --- + 7 a z - 2 a z + ---- - ---- + 6 a z - 3 a z z z 3 a 3 a a z a a 5 5 7 3 3 z 4 z 5 z > a z + -- - ---- + 2 a z - -- 3 a a a
In[9]:=	Kauffman[Link[10, Alternating, 20]][a, z]
Out[9]=	3 -4 4 2 4 1 4 4 a a 4 z 13 z 7 + a + -- + 4 a + a - ---- - --- - --- - -- + --- + ---- + 13 a z + 2 3 a z z z 3 a a a z a 2 2 3 3 3 3 2 4 z 12 z 2 2 4 2 z 11 z 33 z > 4 a z - 15 z - ---- - ----- - 10 a z - 3 a z + -- - ----- - ----- - 4 2 5 3 a a a a a 4 4 4 5 3 3 3 4 z 9 z 14 z 2 4 4 4 4 z > 29 a z - 8 a z + 5 z - -- + ---- + ----- + 4 a z + 3 a z - ---- + 6 4 2 5 a a a a 5 5 6 6 18 z 45 z 5 3 5 6 9 z 3 z 2 6 4 6 > ----- + ----- + 31 a z + 8 a z + 19 z - ---- + ---- + 6 a z - a z - 3 a 4 2 a a a 7 7 8 9 12 z 17 z 7 3 7 8 8 z 2 8 2 z 9 > ----- - ----- - 8 a z - 3 a z - 12 z - ---- - 4 a z - ---- - 2 a z 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 5 2 7 5 7 7 10 + 9 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 > 7 q t + 8 q t + 6 q t + 7 q t + 3 q t + 6 q t + q t + 10 4 12 5 > 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a20

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