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DrawMorseLink

PD Presentation:

X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X18,12,19,11 X19,22,20,9 X15,20,16,21 X21,16,22,17 X12,18,13,17 X2536 X9,1,10,4

Gauss Code:

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

Jones Polynomial:

q-4 - 4q-3 + 7q-2 - 8q-1 + 10 - 8q + 8q2 - 4q3 + 2q4

A2 (sl(3)) Invariant:

q-12 - 2q-10 + q-8 - q-4 + 4q-2 + 1 + 6q2 + 3q4 + 4q6 + 5q8 + q10 + 3q12 + q14

HOMFLY-PT Polynomial:

a-4z-2 + a-4 - 2a-2z-2 - 3a-2 + a-2z4 + z-2 + 2 - 2z2 - 3z4 - z6 + a2z2 + a2z4

Kauffman Polynomial:

- a-4z-2 + 3a-4 - 5a-4z2 + 3a-4z4 + 2a-3z-1 - 3a-3z + a-3z5 + a-3z7 - 2a-2z-2 + 5a-2 - 3a-2z2 - 4a-2z4 + 3a-2z6 + a-2z8 + 2a-1z-1 - 3a-1z - 6a-1z5 + 6a-1z7 - z-2 + 3 + 5z2 - 18z4 + 10z6 + z8 - 3az3 - 3az5 + 5az7 + 3a2z2 - 10a2z4 + 7a2z6 - 3a3z3 + 4a3z5 + a4z4

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 9                 2 j = 7               3 1 j = 5             5 1   j = 3           3 3     j = 1         7 5       j = -1       4 6         j = -3     3 4           j = -5   1 4             j = -7   3               j = -9 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 260]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 260]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 11, 4, 10], X[7, 15, 8, 14], X[13, 5, 14, 8], > X[18, 12, 19, 11], X[19, 22, 20, 9], X[15, 20, 16, 21], X[21, 16, 22, 17], > X[12, 18, 13, 17], X[2, 5, 3, 6], X[9, 1, 10, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -3, 4}, > {-11, 2, 5, -9, -4, 3, -7, 8, 9, -5, -6, 7, -8, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-4 4 7 8 2 3 4 10 + q - -- + -- - - - 8 q + 8 q - 4 q + 2 q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 2 -8 -4 4 2 4 6 8 10 12 14 1 + q - --- + q - q + -- + 6 q + 3 q + 4 q + 5 q + q + 3 q + q 10 2 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 260]][a, z]
Out[8]=	4 -4 3 -2 1 2 2 2 2 4 z 2 4 6 2 + a - -- + z + ----- - ----- - 2 z + a z - 3 z + -- + a z - z 2 4 2 2 2 2 a a z a z a
In[9]:=	Kauffman[Link[11, NonAlternating, 260]][a, z]
Out[9]=	2 3 5 -2 1 2 2 2 3 z 3 z 2 5 z 3 + -- + -- - z - ----- - ----- + ---- + --- - --- - --- + 5 z - ---- - 4 2 4 2 2 2 3 a z 3 a 4 a a a z a z a z a a 2 4 4 3 z 2 2 3 3 3 4 3 z 4 z 2 4 > ---- + 3 a z - 3 a z - 3 a z - 18 z + ---- - ---- - 10 a z + 2 4 2 a a a 5 5 6 7 7 4 4 z 6 z 5 3 5 6 3 z 2 6 z 6 z > a z + -- - ---- - 3 a z + 4 a z + 10 z + ---- + 7 a z + -- + ---- + 3 a 2 3 a a a a 8 7 8 z > 5 a z + z + -- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	6 1 3 1 4 3 4 4 3 - + 7 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 5 q t + 3 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 > 3 q t + 5 q t + q t + 3 q t + q t + 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n260

L11n259 L11n261