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Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X20,5,9,6 X14,3,15,4 X4,15,5,16 X16,7,17,8 X18,12,19,11 X12,18,13,17 X2,9,3,10 X8,13,1,14 X6,19,7,20

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 3q-17/2 - 8q-15/2 + 11q-13/2 - 14q-11/2 + 15q-9/2 - 14q-7/2 + 11q-5/2 - 7q-3/2 + 3q-1/2 - q1/2

A2 (sl(3)) Invariant:

q-30 + q-28 - q-26 + 4q-24 + q-22 + 4q-18 - 2q-16 + 2q-14 - 2q-12 - q-10 + 2q-8 - 3q-6 + 3q-4 - 1 + q2

HOMFLY-PT Polynomial:

- az - az3 - 2a3z + a3z5 + 2a5z + 4a5z3 + 2a5z5 - a7z-1 - 4a7z - 3a7z3 + a9z-1 + a9z

Kauffman Polynomial:

- az + 2az3 - az5 - 2a2z2 + 5a2z4 - 3a2z6 + 2a3z - 5a3z3 + 8a3z5 - 5a3z7 + 2a4z2 - 3a4z4 + 6a4z6 - 5a4z8 + a5z - 10a5z3 + 17a5z5 - 7a5z7 - 2a5z9 - a6z2 - 5a6z4 + 15a6z6 - 10a6z8 - a7z-1 + 7a7z - 17a7z3 + 21a7z5 - 8a7z7 - 2a7z9 + a8 - 6a8z2 + 7a8z4 + 3a8z6 - 5a8z8 - a9z-1 + 8a9z - 12a9z3 + 12a9z5 - 6a9z7 - a10z2 + 4a10z4 - 3a10z6 - a11z + 2a11z3 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                     1 j = 0                   2   j = -2                 5 1   j = -4               7 3     j = -6             7 4       j = -8           8 7         j = -10         6 7           j = -12       5 8             j = -14     3 6               j = -16     5                 j = -18 1 3                   j = -20 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, 107]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 107]]
Out[4]=	PD[X[10, 1, 11, 2], X[20, 5, 9, 6], X[14, 3, 15, 4], X[4, 15, 5, 16], > X[16, 7, 17, 8], X[18, 12, 19, 11], X[12, 18, 13, 17], X[2, 9, 3, 10], > X[8, 13, 1, 14], X[6, 19, 7, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 3, -4, 2, -10, 5, -9}, > {8, -1, 6, -7, 9, -3, 4, -5, 7, -6, 10, -2}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 3 8 11 14 15 14 11 7 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q 3 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 -26 4 -22 4 2 2 2 -10 2 3 -1 + q + q - q + --- + q + --- - --- + --- - --- - q + -- - -- + 24 18 16 14 12 8 6 q q q q q q q 3 2 > -- + q 4 q
In[8]:=	HOMFLYPT[Link[10, Alternating, 107]][a, z]
Out[8]=	7 9 a a 3 5 7 9 3 5 3 7 3 -(--) + -- - a z - 2 a z + 2 a z - 4 a z + a z - a z + 4 a z - 3 a z + z z 3 5 5 5 > a z + 2 a z
In[9]:=	Kauffman[Link[10, Alternating, 107]][a, z]
Out[9]=	7 9 8 a a 3 5 7 9 11 2 2 a - -- - -- - a z + 2 a z + a z + 7 a z + 8 a z - a z - 2 a z + z z 4 2 6 2 8 2 10 2 3 3 3 5 3 > 2 a z - a z - 6 a z - a z + 2 a z - 5 a z - 10 a z - 7 3 9 3 11 3 2 4 4 4 6 4 8 4 > 17 a z - 12 a z + 2 a z + 5 a z - 3 a z - 5 a z + 7 a z + 10 4 5 3 5 5 5 7 5 9 5 11 5 > 4 a z - a z + 8 a z + 17 a z + 21 a z + 12 a z - a z - 2 6 4 6 6 6 8 6 10 6 3 7 5 7 > 3 a z + 6 a z + 15 a z + 3 a z - 3 a z - 5 a z - 7 a z - 7 7 9 7 4 8 6 8 8 8 5 9 7 9 > 8 a z - 6 a z - 5 a z - 10 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 1 1 3 5 3 6 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 8 18 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 8 6 7 8 7 7 4 7 t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 2 t + -- + 12 4 10 4 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q t q t q 2 2 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a107

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