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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 4q-11/2 - 8q-9/2 + 12q-7/2 - 16q-5/2 + 16q-3/2 - 16q-1/2 + 12q1/2 - 8q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

q-20 - q-18 - q-16 + 2q-14 - 3q-12 + 3q-10 + 2q-8 + q-6 + 5q-4 - 2q-2 + 4 - 2q2 - q4 + 2q6 - 2q8 + q10

HOMFLY-PT Polynomial:

- a-1z - 2a-1z3 - a-1z5 - az-1 + 2az + 6az3 + 4az5 + az7 + a3z-1 - 3a3z - 5a3z3 - 2a3z5 + a5z + a5z3

Kauffman Polynomial:

a-3z3 - a-3z5 - 2a-2z2 + 6a-2z4 - 4a-2z6 + 2a-1z - 7a-1z3 + 12a-1z5 - 7a-1z7 - 4z2 + 6z4 + 4z6 - 6z8 + az-1 + 5az - 24az3 + 33az5 - 13az7 - 2az9 - a2 - 4a2z2 + 16a2z6 - 12a2z8 + a3z-1 + 5a3z - 24a3z3 + 33a3z5 - 13a3z7 - 2a3z9 - 4a4z2 + 6a4z4 + 4a4z6 - 6a4z8 + 2a5z - 7a5z3 + 12a5z5 - 7a5z7 - 2a6z2 + 6a6z4 - 4a6z6 + a7z3 - 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 = 8                     1 j = 6                   3   j = 4                 5 1   j = 2               7 3     j = 0             9 5       j = -2           8 8         j = -4         8 8           j = -6       5 9             j = -8     3 7               j = -10   1 5                 j = -12   3                   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]=	2
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 51]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 51]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[20, 10, 7, 9], X[2, 7, 3, 8], > X[4, 15, 5, 16], X[12, 5, 13, 6], X[16, 12, 17, 11], X[6, 18, 1, 17], > X[14, 19, 15, 20], X[18, 13, 19, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -5, 6, -8}, > {4, -1, 3, -2, 7, -6, 10, -9, 5, -7, 8, -10, 9, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 4 8 12 16 16 16 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 12 Sqrt[q] - 8 q + 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 7/2 > 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 -16 2 3 3 2 -6 5 2 2 4 4 + q - q - q + --- - --- + --- + -- + q + -- - -- - 2 q - q + 14 12 10 8 4 2 q q q q q q 6 8 10 > 2 q - 2 q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 51]][a, z]
Out[8]=	3 3 5 a a z 3 5 2 z 3 3 3 5 3 z -(-) + -- - - + 2 a z - 3 a z + a z - ---- + 6 a z - 5 a z + a z - -- + z z a a a 5 3 5 7 > 4 a z - 2 a z + a z
In[9]:=	Kauffman[Link[10, Alternating, 51]][a, z]
Out[9]=	3 2 2 a a 2 z 3 5 2 2 z 2 2 -a + - + -- + --- + 5 a z + 5 a z + 2 a z - 4 z - ---- - 4 a z - z z a 2 a 3 3 4 2 6 2 z 7 z 3 3 3 5 3 7 3 > 4 a z - 2 a z + -- - ---- - 24 a z - 24 a z - 7 a z + a z + 3 a a 4 5 5 4 6 z 4 4 6 4 z 12 z 5 3 5 > 6 z + ---- + 6 a z + 6 a z - -- + ----- + 33 a z + 33 a z + 2 3 a a a 6 7 5 5 7 5 6 4 z 2 6 4 6 6 6 7 z > 12 a z - a z + 4 z - ---- + 16 a z + 4 a z - 4 a z - ---- - 2 a a 7 3 7 5 7 8 2 8 4 8 9 3 9 > 13 a z - 13 a z - 7 a z - 6 z - 12 a z - 6 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	8 1 3 1 5 3 7 5 9 9 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 8 8 8 2 2 2 4 2 4 3 6 3 > ----- + ---- + ---- + 5 t + 7 q t + 3 q t + 5 q t + q t + 3 q t + 4 2 4 2 q t q t q t 8 4 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a51

L10a50 L10a52