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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-23/2 + q-21/2 - 4q-19/2 + 6q-17/2 - 7q-15/2 + 9q-13/2 - 9q-11/2 + 7q-9/2 - 6q-7/2 + 3q-5/2 - q-3/2

A2 (sl(3)) Invariant:

q-36 + q-34 + q-32 + 4q-30 + q-28 - q-26 + q-24 - 2q-22 + 2q-14 - q-12 + 2q-10 + q-8 - 2q-6 + q-4

HOMFLY-PT Polynomial:

- a3z3 - 3a5z - 3a5z3 - 3a7z - 3a7z3 - a9z-1 - a9z3 + a11z-1 + a11z

Kauffman Polynomial:

- a3z3 - 3a4z4 - 3a5z + 6a5z3 - 6a5z5 - 3a6z2 + 9a6z4 - 7a6z6 + 3a7z - 3a7z3 + 9a7z5 - 6a7z7 - 6a8z2 + 12a8z4 - 3a8z8 + a9z-1 - 2a9z - 2a9z3 + 11a9z5 - 3a9z7 - a9z9 - a10 - 7a10z2 + 10a10z6 - 4a10z8 + a11z-1 - 4a11z3 + 2a11z5 + 2a11z7 - a11z9 - 4a12z2 + 3a12z6 - a12z8 + 8a13z - 12a13z3 + 6a13z5 - a13z7

Khovanov Homology:

trqj r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -2                     1 j = -4                   3 1 j = -6                 3     j = -8               4 3     j = -10             5 3       j = -12           4 4         j = -14         3 5           j = -16       3 4             j = -18     1 3               j = -20     3                 j = -22 1 1                   j = -24 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, 119]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 119]]
Out[4]=	PD[X[12, 1, 13, 2], X[20, 7, 11, 8], X[14, 3, 15, 4], X[6, 15, 7, 16], > X[16, 5, 17, 6], X[4, 17, 5, 18], X[18, 9, 19, 10], X[2, 11, 3, 12], > X[10, 13, 1, 14], X[8, 19, 9, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 3, -6, 5, -4, 2, -10, 7, -9}, > {8, -1, 9, -3, 4, -5, 6, -7, 10, -2}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) -(21/2) 4 6 7 9 9 7 6 -q + q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + 19/2 17/2 15/2 13/2 11/2 9/2 7/2 q q q q q q q 3 -(3/2) > ---- - q 5/2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-36 -34 -32 4 -28 -26 -24 2 2 -12 2 -8 q + q + q + --- + q - q + q - --- + --- - q + --- + q - 30 22 14 10 q q q q 2 -4 > -- + q 6 q
In[8]:=	HOMFLYPT[Link[10, Alternating, 119]][a, z]
Out[8]=	9 11 a a 5 7 11 3 3 5 3 7 3 9 3 -(--) + --- - 3 a z - 3 a z + a z - a z - 3 a z - 3 a z - a z z z
In[9]:=	Kauffman[Link[10, Alternating, 119]][a, z]
Out[9]=	9 11 10 a a 5 7 9 13 6 2 8 2 -a + -- + --- - 3 a z + 3 a z - 2 a z + 8 a z - 3 a z - 6 a z - z z 10 2 12 2 3 3 5 3 7 3 9 3 11 3 > 7 a z - 4 a z - a z + 6 a z - 3 a z - 2 a z - 4 a z - 13 3 4 4 6 4 8 4 5 5 7 5 9 5 > 12 a z - 3 a z + 9 a z + 12 a z - 6 a z + 9 a z + 11 a z + 11 5 13 5 6 6 10 6 12 6 7 7 9 7 > 2 a z + 6 a z - 7 a z + 10 a z + 3 a z - 6 a z - 3 a z + 11 7 13 7 8 8 10 8 12 8 9 9 11 9 > 2 a z - a z - 3 a z - 4 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 -2 1 1 1 3 1 3 3 q + q + ------- + ------- + ------ + ------ + ------ + ------ + ------ + 24 10 22 10 22 9 20 8 18 8 18 7 16 7 q t q t q t q t q t q t q t 4 3 5 4 4 5 3 4 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 16 6 14 6 14 5 12 5 12 4 10 4 10 3 8 3 q t q t q t q t q t q t q t q t 3 3 3 > ----- + ----- + ---- 8 2 6 2 4 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a119

L10a118 L10a120