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DrawMorseLink

PD Presentation:

X6172 X3,10,4,11 X7,19,8,18 X21,15,22,14 X9,20,10,21 X13,9,14,8 X15,17,16,22 X17,5,18,16 X19,12,20,13 X2536 X11,4,12,1

Gauss Code:

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

Jones Polynomial:

3q-4 - 5q-3 + 9q-2 - 11q-1 + 13 - 11q + 10q2 - 6q3 + 3q4 - q5

A2 (sl(3)) Invariant:

q-16 + q-14 + 4q-12 + q-10 + 4q-8 + 3q-6 + q-4 + 6q-2 + 5q2 + q6 + 2q8 - 2q10 + q12 - q14

HOMFLY-PT Polynomial:

- a-2 - 5a-2z2 - 4a-2z4 - a-2z6 + z-2 + 6 + 12z2 + 13z4 + 6z6 + z8 - 2a2z-2 - 8a2 - 10a2z2 - 5a2z4 - a2z6 + a4z-2 + 3a4 + a4z2

Kauffman Polynomial:

a-5z - 2a-5z3 + a-5z5 - a-4 + 3a-4z2 - 6a-4z4 + 3a-4z6 + a-3z - 2a-3z3 - 5a-3z5 + 4a-3z7 - a-2 + a-2z2 - 5a-2z6 + 4a-2z8 - 3a-1z + 11a-1z3 - 10a-1z5 + 2a-1z7 + 2a-1z9 - z-2 + 8 - 24z2 + 37z4 - 26z6 + 9z8 + 2az-1 - 10az + 18az3 - 10az5 + az7 + 2az9 - 2a2z-2 + 14a2 - 36a2z2 + 37a2z4 - 18a2z6 + 5a2z8 + 2a3z-1 - 7a3z + 7a3z3 - 6a3z5 + 3a3z7 - a4z-2 + 7a4 - 14a4z2 + 6a4z4

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                   1 j = 9                 2   j = 7               4 1   j = 5             6 2     j = 3           6 5       j = 1         7 5         j = -1       5 7           j = -3     4 6             j = -5   2 6               j = -7 1 3                 j = -9 3

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, 373]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 373]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 10, 4, 11], X[7, 19, 8, 18], X[21, 15, 22, 14], > X[9, 20, 10, 21], X[13, 9, 14, 8], X[15, 17, 16, 22], X[17, 5, 18, 16], > X[19, 12, 20, 13], X[2, 5, 3, 6], X[11, 4, 12, 1]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {-8, 3, -9, 5, -4, 7}, > {10, -1, -3, 6, -5, 2, -11, 9, -6, 4, -7, 8}]
In[6]:=	Jones[L][q]
Out[6]=	3 5 9 11 2 3 4 5 13 + -- - -- + -- - -- - 11 q + 10 q - 6 q + 3 q - q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 4 -10 4 3 -4 6 2 6 8 10 q + q + --- + q + -- + -- + q + -- + 5 q + q + 2 q - 2 q + 12 8 6 2 q q q q 12 14 > q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 373]][a, z]
Out[8]=	2 4 2 -2 2 4 -2 2 a a 2 5 z 2 2 4 2 6 - a - 8 a + 3 a + z - ---- + -- + 12 z - ---- - 10 a z + a z + 2 2 2 z z a 4 6 4 4 z 2 4 6 z 2 6 8 > 13 z - ---- - 5 a z + 6 z - -- - a z + z 2 2 a a
In[9]:=	Kauffman[Link[11, NonAlternating, 373]][a, z]
Out[9]=	2 4 3 -4 -2 2 4 -2 2 a a 2 a 2 a z z 3 z 8 - a - a + 14 a + 7 a - z - ---- - -- + --- + ---- + -- + -- - --- - 2 2 z z 5 3 a z z a a 2 2 3 3 3 2 3 z z 2 2 4 2 2 z 2 z > 10 a z - 7 a z - 24 z + ---- + -- - 36 a z - 14 a z - ---- - ---- + 4 2 5 3 a a a a 3 4 5 5 11 z 3 3 3 4 6 z 2 4 4 4 z 5 z > ----- + 18 a z + 7 a z + 37 z - ---- + 37 a z + 6 a z + -- - ---- - a 4 5 3 a a a 5 6 6 7 7 10 z 5 3 5 6 3 z 5 z 2 6 4 z 2 z > ----- - 10 a z - 6 a z - 26 z + ---- - ---- - 18 a z + ---- + ---- + a 4 2 3 a a a a 8 9 7 3 7 8 4 z 2 8 2 z 9 > a z + 3 a z + 9 z + ---- + 5 a z + ---- + 2 a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	7 3 1 3 2 6 4 6 5 - + 7 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 5 q t + q 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 9 4 11 5 > 6 q t + 5 q t + 6 q t + 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n373

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