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Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X12,3,13,4 X5,14,6,15 X16,7,17,8 X15,21,16,20 X18,14,19,13 X21,6,22,7 X22,18,9,17 X4,19,5,20 X2,9,3,10 X8,11,1,12

Gauss Code:

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

Jones Polynomial:

- q-17/2 + q-15/2 - 3q-13/2 + 4q-11/2 - 5q-9/2 + 5q-7/2 - 5q-5/2 + 4q-3/2 - 3q-1/2 + q1/2

A2 (sl(3)) Invariant:

q-26 + q-24 + 2q-22 + 3q-20 + q-18 + 2q-16 - q-12 - q-8 + q-6 + 1 - q2

HOMFLY-PT Polynomial:

az + az3 + a3z-1 - 2a3z - 3a3z3 - a3z5 - 3a5z-1 - 6a5z - 4a5z3 - a5z5 + 2a7z-1 + 3a7z + a7z3

Kauffman Polynomial:

- z2 + az - 3az3 - a2 - a2z6 + a3z-1 + a3z - 5a3z3 + 5a3z5 - 2a3z7 - 3a4 + 11a4z2 - 11a4z4 + 7a4z6 - 2a4z8 + 3a5z-1 - 12a5z + 15a5z3 - 6a5z5 + 3a5z7 - a5z9 - 3a6 + 9a6z2 - 14a6z4 + 12a6z6 - 3a6z8 + 2a7z-1 - 6a7z + 6a7z3 - 5a7z5 + 4a7z7 - a7z9 - a8z2 - 3a8z4 + 4a8z6 - a8z8 + 6a9z - 11a9z3 + 6a9z5 - a9z7

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 j = 2                   1 j = 0                 2   j = -2               3 2   j = -4             2 1     j = -6           3 3       j = -8         2 2         j = -10       2 3           j = -12     1 2             j = -14     2               j = -16 1 1                 j = -18 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, 201]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 201]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 3, 13, 4], X[5, 14, 6, 15], X[16, 7, 17, 8], > X[15, 21, 16, 20], X[18, 14, 19, 13], X[21, 6, 22, 7], X[22, 18, 9, 17], > X[4, 19, 5, 20], X[2, 9, 3, 10], X[8, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -9, -3, 7, 4, -11}, > {10, -1, 11, -2, 6, 3, -5, -4, 8, -6, 9, 5, -7, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) -(15/2) 3 4 5 5 5 4 3 -q + q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q > Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -24 2 3 -18 2 -12 -8 -6 2 1 + q + q + --- + --- + q + --- - q - q + q - q 22 20 16 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 201]][a, z]
Out[8]=	3 5 7 a 3 a 2 a 3 5 7 3 3 3 5 3 -- - ---- + ---- + a z - 2 a z - 6 a z + 3 a z + a z - 3 a z - 4 a z + z z z 7 3 3 5 5 5 > a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 201]][a, z]
Out[9]=	3 5 7 2 4 6 a 3 a 2 a 3 5 7 9 -a - 3 a - 3 a + -- + ---- + ---- + a z + a z - 12 a z - 6 a z + 6 a z - z z z 2 4 2 6 2 8 2 3 3 3 5 3 7 3 > z + 11 a z + 9 a z - a z - 3 a z - 5 a z + 15 a z + 6 a z - 9 3 4 4 6 4 8 4 3 5 5 5 7 5 > 11 a z - 11 a z - 14 a z - 3 a z + 5 a z - 6 a z - 5 a z + 9 5 2 6 4 6 6 6 8 6 3 7 5 7 > 6 a z - a z + 7 a z + 12 a z + 4 a z - 2 a z + 3 a z + 7 7 9 7 4 8 6 8 8 8 5 9 7 9 > 4 a z - a z - 2 a z - 3 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 2 1 2 2 2 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 18 8 16 8 16 7 14 6 12 6 12 5 10 5 q q t q t q t q t q t q t q t 3 2 2 3 3 2 1 3 2 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + q t 10 4 8 4 8 3 6 3 6 2 4 2 4 2 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 L11n201

L11n200 L11n202