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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X12,8,13,7 X13,20,14,21 X19,9,20,8 X9,19,10,18 X15,22,16,17 X17,16,18,5 X21,14,22,15 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

2q-3 - q-2 + 3q-1 - 2 + 3q - 2q2 + 2q3 - q4

A2 (sl(3)) Invariant:

q-14 + q-12 + 3q-10 + 4q-8 + 5q-6 + 5q-4 + 3q-2 + 3 + q2 + q4 + q6 - q12

HOMFLY-PT Polynomial:

- a-2 - 3a-2z2 - a-2z4 + z-2 + 4 + 7z2 + 5z4 + z6 - 2a2z-2 - 4a2 - 4a2z2 - a2z4 + a4z-2 + a4

Kauffman Polynomial:

- a-3z + 6a-3z3 - 5a-3z5 + a-3z7 + 2a-2 - 9a-2z2 + 17a-2z4 - 11a-2z6 + 2a-2z8 - 3a-1z + 5a-1z3 + a-1z5 - 4a-1z7 + a-1z9 - z-2 + 7 - 21z2 + 29z4 - 17z6 + 3z8 + 2az-1 - 4az - az3 + 6az5 - 5az7 + az9 - 2a2z-2 + 7a2 - 13a2z2 + 12a2z4 - 6a2z6 + a2z8 + 2a3z-1 - 2a3z - a4z-2 + 3a4 - a4z2

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 9               1 j = 7             1   j = 5           1 1   j = 3         2 1     j = 1     1 1 1       j = -1     3 2         j = -3 1 1 2           j = -5 2 1             j = -7 2

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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 367]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 367]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[13, 20, 14, 21], > X[19, 9, 20, 8], X[9, 19, 10, 18], X[15, 22, 16, 17], X[17, 16, 18, 5], > X[21, 14, 22, 15], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {-8, 6, -5, 4, -9, 7}, > {10, -1, 3, 5, -6, -2, 11, -3, -4, 9, -7, 8}]
In[6]:=	Jones[L][q]
Out[6]=	2 -2 3 2 3 4 -2 + -- - q + - + 3 q - 2 q + 2 q - q 3 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 3 4 5 5 3 2 4 6 12 3 + q + q + --- + -- + -- + -- + -- + q + q + q - q 10 8 6 4 2 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 367]][a, z]
Out[8]=	2 4 2 4 -2 2 4 -2 2 a a 2 3 z 2 2 4 z 4 - a - 4 a + a + z - ---- + -- + 7 z - ---- - 4 a z + 5 z - -- - 2 2 2 2 z z a a 2 4 6 > a z + z
In[9]:=	Kauffman[Link[11, NonAlternating, 367]][a, z]
Out[9]=	2 4 3 2 2 4 -2 2 a a 2 a 2 a z 3 z 7 + -- + 7 a + 3 a - z - ---- - -- + --- + ---- - -- - --- - 4 a z - 2 2 2 z z 3 a a z z a 2 3 3 3 2 9 z 2 2 4 2 6 z 5 z 3 4 > 2 a z - 21 z - ---- - 13 a z - a z + ---- + ---- - a z + 29 z + 2 3 a a a 4 5 5 6 7 17 z 2 4 5 z z 5 6 11 z 2 6 z > ----- + 12 a z - ---- + -- + 6 a z - 17 z - ----- - 6 a z + -- - 2 3 a 2 3 a a a a 7 8 9 4 z 7 8 2 z 2 8 z 9 > ---- - 5 a z + 3 z + ---- + a z + -- + a z a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 2 2 1 1 1 2 t 2 3 2 -- + - + q + ----- + ----- + ----- + ---- + ---- + --- + q t + q t + 2 q t + 3 q 7 2 5 2 3 2 5 3 q q q t q t q t q t q t 3 3 5 3 5 4 7 4 9 5 > q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n367

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