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DrawMorseLink

PD Presentation:

X6172 X11,18,12,19 X8493 X2,16,3,15 X16,7,17,8 X9,11,10,22 X4,17,1,18 X19,5,20,10 X5,12,6,13 X21,15,22,14 X13,21,14,20

Gauss Code:

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

Jones Polynomial:

- q-4 + 2q-3 - q-2 + q-1 + 3 - 2q + 4q2 - 4q3 + 4q4 - 3q5 + q6

A2 (sl(3)) Invariant:

- q-12 + 3q-6 + 3q-4 + 6q-2 + 7 + 4q2 + 5q4 + q8 - q10 - q12 + q14 - q16 + q18

HOMFLY-PT Polynomial:

a-4 + 2a-4z2 + a-4z4 + a-2z-2 - 3a-2 - 8a-2z2 - 5a-2z4 - a-2z6 - 2z-2 + 3 + 9z2 + 6z4 + z6 + a2z-2 - a2 - 3a2z2 - a2z4

Kauffman Polynomial:

a-6z2 - 3a-6z4 + a-6z6 - 2a-5z + 7a-5z3 - 11a-5z5 + 3a-5z7 + 2a-4 - 6a-4z2 + 10a-4z4 - 12a-4z6 + 3a-4z8 - 7a-3z + 20a-3z3 - 13a-3z5 - a-3z7 + a-3z9 + a-2z-2 + 6a-2 - 30a-2z2 + 50a-2z4 - 30a-2z6 + 5a-2z8 - 2a-1z-1 - 9a-1z + 19a-1z3 + 3a-1z5 - 10a-1z7 + 2a-1z9 + 2z-2 + 7 - 36z2 + 56z4 - 29z6 + 4z8 - 2az-1 - 5az + 11az3 - 5az7 + az9 + a2z-2 + 2a2 - 13a2z2 + 19a2z4 - 12a2z6 + 2a2z8 - a3z + 5a3z3 - 5a3z5 + a3z7

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                       1 j = 11                     2   j = 9                   2 1   j = 7               1 3 2     j = 5             1 3 2       j = 3           2 3 3         j = 1         2 6 3           j = -1       1 2 5             j = -3     1 2 1               j = -5   1 1 1                 j = -7   1                     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, 293]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 293]]
Out[4]=	PD[X[6, 1, 7, 2], X[11, 18, 12, 19], X[8, 4, 9, 3], X[2, 16, 3, 15], > X[16, 7, 17, 8], X[9, 11, 10, 22], X[4, 17, 1, 18], X[19, 5, 20, 10], > X[5, 12, 6, 13], X[21, 15, 22, 14], X[13, 21, 14, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -7}, {-9, -1, 5, -3, -6, 8}, > {-2, 9, -11, 10, 4, -5, 7, 2, -8, 11, -10, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-4 2 -2 1 2 3 4 5 6 3 - q + -- - q + - - 2 q + 4 q - 4 q + 4 q - 3 q + q 3 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 3 3 6 2 4 8 10 12 14 16 18 7 - q + -- + -- + -- + 4 q + 5 q + q - q - q + q - q + q 6 4 2 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 293]][a, z]
Out[8]=	2 2 2 -4 3 2 2 1 a 2 2 z 8 z 2 2 4 3 + a - -- - a - -- + ----- + -- + 9 z + ---- - ---- - 3 a z + 6 z + 2 2 2 2 2 4 2 a z a z z a a 4 4 6 z 5 z 2 4 6 z > -- - ---- - a z + z - -- 4 2 2 a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 293]][a, z]
Out[9]=	2 2 6 2 2 1 a 2 2 a 2 z 7 z 9 z 7 + -- + -- + 2 a + -- + ----- + -- - --- - --- - --- - --- - --- - 5 a z - 4 2 2 2 2 2 a z z 5 3 a a a z a z z a a 2 2 2 3 3 3 3 2 z 6 z 30 z 2 2 7 z 20 z 19 z > a z - 36 z + -- - ---- - ----- - 13 a z + ---- + ----- + ----- + 6 4 2 5 3 a a a a a a 4 4 4 5 3 3 3 4 3 z 10 z 50 z 2 4 11 z > 11 a z + 5 a z + 56 z - ---- + ----- + ----- + 19 a z - ----- - 6 4 2 5 a a a a 5 5 6 6 6 7 13 z 3 z 3 5 6 z 12 z 30 z 2 6 3 z > ----- + ---- - 5 a z - 29 z + -- - ----- - ----- - 12 a z + ---- - 3 a 6 4 2 5 a a a a a 7 7 8 8 9 9 z 10 z 7 3 7 8 3 z 5 z 2 8 z 2 z > -- - ----- - 5 a z + a z + 4 z + ---- + ---- + 2 a z + -- + ---- + 3 a 4 2 3 a a a a a 9 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 3 1 1 1 1 1 1 2 1 - + 6 q + 2 q + ----- + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 9 5 7 4 5 4 5 3 3 3 5 2 3 2 2 q t q t q t q t q t q t q t q t 1 2 2 q 3 5 3 2 5 2 7 2 > ---- + --- + --- + 3 q t + 3 q t + q t + 3 q t + 3 q t + q t + 3 q t t q t 5 3 7 3 7 4 9 4 9 5 11 5 13 6 > 2 q t + 3 q t + 2 q t + 2 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n293

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