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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 2q-11/2 - 6q-9/2 + 9q-7/2 - 12q-5/2 + 13q-3/2 - 12q-1/2 + 10q1/2 - 7q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant:

q-22 + 2q-20 + 2q-16 + 4q-14 - 2q-12 + q-10 - q-8 - 3q-6 + q-4 - 2q-2 + 4 + 3q6 - 2q8 + q12

HOMFLY-PT Polynomial:

- a-3z - a-1z-1 + 2a-1z3 + 2az-1 + az - az5 - a3z-1 + a3z + 3a3z3 - a5z-1 - 3a5z + a7z-1

Kauffman Polynomial:

- a-3z + 2a-3z3 - a-3z5 - 2a-2z2 + 5a-2z4 - 3a-2z6 - a-1z-1 + 4a-1z - 6a-1z3 + 9a-1z5 - 5a-1z7 - 1 + z2 + 2z4 + 3z6 - 4z8 - 2az-1 + 13az - 26az3 + 26az5 - 10az7 - az9 - 3a2 + 6a2z2 - 7a2z4 + 10a2z6 - 7a2z8 - a3z-1 + 9a3z - 20a3z3 + 20a3z5 - 8a3z7 - a3z9 - 2a4 + 3a4z2 - a4z4 + 2a4z6 - 3a4z8 + a5z-1 - 2a5z + a5z3 + 3a5z5 - 3a5z7 - a6 + 3a6z4 - 2a6z6 + a7z-1 - 3a7z + 3a7z3 - 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                   2   j = 4                 5 1   j = 2               5 2     j = 0             7 5       j = -2           7 6         j = -4         5 6           j = -6       4 7             j = -8     2 5               j = -10   1 5                 j = -12   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]=	2
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 12]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 12]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 8, 15, 7], X[18, 11, 19, 12], > X[20, 16, 5, 15], X[16, 20, 17, 19], X[12, 17, 13, 18], X[8, 14, 9, 13], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 3, -8, 10, -2, 4, -7, 8, -3, 5, -6, 7, -4, > 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 2 6 9 12 13 12 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 10 Sqrt[q] - 7 q + 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 7/2 > 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 2 4 2 -10 -8 3 -4 2 6 8 4 + q + --- + --- + --- - --- + q - q - -- + q - -- + 3 q - 2 q + 20 16 14 12 6 2 q q q q q q 12 > q
In[8]:=	HOMFLYPT[Link[10, Alternating, 12]][a, z]
Out[8]=	3 5 7 3 1 2 a a a a z 3 5 2 z 3 3 5 -(---) + --- - -- - -- + -- - -- + a z + a z - 3 a z + ---- + 3 a z - a z a z z z z z 3 a a
In[9]:=	Kauffman[Link[10, Alternating, 12]][a, z]
Out[9]=	3 5 7 2 4 6 1 2 a a a a z 4 z 3 -1 - 3 a - 2 a - a - --- - --- - -- + -- + -- - -- + --- + 13 a z + 9 a z - a z z z z z 3 a a 2 3 3 5 7 2 2 z 2 2 4 2 2 z 6 z 3 > 2 a z - 3 a z + z - ---- + 6 a z + 3 a z + ---- - ---- - 26 a z - 2 3 a a a 4 5 3 3 5 3 7 3 4 5 z 2 4 4 4 6 4 z > 20 a z + a z + 3 a z + 2 z + ---- - 7 a z - a z + 3 a z - -- + 2 3 a a 5 6 9 z 5 3 5 5 5 7 5 6 3 z 2 6 > ---- + 26 a z + 20 a z + 3 a z - a z + 3 z - ---- + 10 a z + a 2 a 7 4 6 6 6 5 z 7 3 7 5 7 8 2 8 > 2 a z - 2 a z - ---- - 10 a z - 8 a z - 3 a z - 4 z - 7 a z - a 4 8 9 3 9 > 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 1 1 1 5 2 5 4 7 7 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 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 5 6 7 2 2 2 4 2 4 3 6 3 > ----- + ---- + ---- + 5 t + 5 q t + 2 q t + 5 q t + q t + 2 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 L10a12

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