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Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X5,14,6,15 X17,22,18,9 X19,5,20,4 X21,6,22,7 X7,17,8,16 X8,9,1,10 X13,18,14,19 X15,21,16,20

Gauss Code:

{{1, -2, 3, 6, -4, 7, -8, -9}, {9, -1, 2, -3, -10, 4, -11, 8, -5, 10, -6, 11, -7, 5}}

Jones Polynomial:

- q-19/2 + 3q-17/2 - 6q-15/2 + 9q-13/2 - 10q-11/2 + 10q-9/2 - 10q-7/2 + 6q-5/2 - 4q-3/2 + q-1/2

A2 (sl(3)) Invariant:

q-28 - q-26 + 2q-24 - q-22 - 2q-20 + q-18 - 2q-16 + 4q-14 + q-12 + 3q-10 + 3q-8 - q-6 + 2q-4 - q-2

HOMFLY-PT Polynomial:

- a3z-1 + 2a3z3 + a3z5 + a5z-1 - 5a5z - 9a5z3 - 5a5z5 - a5z7 + 3a7z + 3a7z3 + a7z5

Kauffman Polynomial:

a2z2 - a2z4 - a3z-1 - a3z + 5a3z3 - 4a3z5 + a4 + 2a4z2 - 3a4z4 - a4z8 - a5z-1 - 5a5z + 13a5z3 - 9a5z5 + a5z7 - a5z9 + 5a6z2 - 6a6z4 + 5a6z6 - 4a6z8 - 4a7z + 6a7z3 + 2a7z5 - 3a7z7 - a7z9 + a8z2 + 2a8z4 + 2a8z6 - 3a8z8 - a9z + 6a9z5 - 4a9z7 - 3a10z2 + 6a10z4 - 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 j = 0                   1 j = -2                 3   j = -4               4 2   j = -6             6 2     j = -8           4 4       j = -10         6 6         j = -12       4 5           j = -14     2 5             j = -16   1 4               j = -18   2                 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[11, NonAlternating, 207]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 207]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[5, 14, 6, 15], > X[17, 22, 18, 9], X[19, 5, 20, 4], X[21, 6, 22, 7], X[7, 17, 8, 16], > X[8, 9, 1, 10], X[13, 18, 14, 19], X[15, 21, 16, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, 6, -4, 7, -8, -9}, > {9, -1, 2, -3, -10, 4, -11, 8, -5, 10, -6, 11, -7, 5}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 3 6 9 10 10 10 6 4 1 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -26 2 -22 2 -18 2 4 -12 3 3 -6 q - q + --- - q - --- + q - --- + --- + q + --- + -- - q + 24 20 16 14 10 8 q q q q q q 2 -2 > -- - q 4 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 207]][a, z]
Out[8]=	3 5 a a 5 7 3 3 5 3 7 3 3 5 5 5 -(--) + -- - 5 a z + 3 a z + 2 a z - 9 a z + 3 a z + a z - 5 a z + z z 7 5 5 7 > a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 207]][a, z]
Out[9]=	3 5 4 a a 3 5 7 9 11 2 2 4 2 a - -- - -- - a z - 5 a z - 4 a z - a z - a z + a z + 2 a z + z z 6 2 8 2 10 2 3 3 5 3 7 3 11 3 > 5 a z + a z - 3 a z + 5 a z + 13 a z + 6 a z + 2 a z - 2 4 4 4 6 4 8 4 10 4 3 5 5 5 > a z - 3 a z - 6 a z + 2 a z + 6 a z - 4 a z - 9 a z + 7 5 9 5 11 5 6 6 8 6 10 6 5 7 > 2 a z + 6 a z - a z + 5 a z + 2 a z - 3 a z + a z - 7 7 9 7 4 8 6 8 8 8 5 9 7 9 > 3 a z - 4 a z - a z - 4 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 1 2 1 4 2 5 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 7 16 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 5 6 6 4 4 6 2 4 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + t 12 4 10 4 10 3 8 3 8 2 6 2 6 4 q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n207

L11n206 L11n208