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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X11,19,12,18 X17,9,18,8 X7,17,8,16 X15,5,16,14 X19,15,20,22 X13,20,14,21 X21,12,22,13 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-5 - q-4 + 2q-3 + 2 - 2q + 3q2 - 2q3 + 2q4 - q5

A2 (sl(3)) Invariant:

q-16 + 2q-14 + 2q-12 + 2q-10 + 3q-8 + 4q-6 + 2q-4 + 4q-2 + 3 + 2q2 + 2q4 + q8 - q10 + q14 - q16

HOMFLY-PT Polynomial:

- a-4z2 - a-2 - 2a-2z2 + z-2 + 6 + 9z2 + 6z4 + z6 - 2a2z-2 - 8a2 - 9a2z2 - 2a2z4 + a4z-2 + 3a4 + a4z2

Kauffman Polynomial:

a-5z - 3a-5z3 + a-5z5 - a-4 + 5a-4z2 - 7a-4z4 + 2a-4z6 + a-3z - 2a-3z3 - 2a-3z5 + a-3z7 - a-2 + 5a-2z2 - 7a-2z4 + 2a-2z6 - 3a-1z + 9a-1z3 - 6a-1z5 + a-1z7 - z-2 + 8 - 24z2 + 32z4 - 15z6 + 2z8 + 2az-1 - 10az + 10az3 + 5az5 - 6az7 + az9 - 2a2z-2 + 14a2 - 40a2z2 + 48a2z4 - 22a2z6 + 3a2z8 + 2a3z-1 - 7a3z + 2a3z3 + 8a3z5 - 6a3z7 + a3z9 - a4z-2 + 7a4 - 16a4z2 + 16a4z4 - 7a4z6 + a4z8

Khovanov Homology:

trqj r = -6 r = -5 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                     1   j = 7                   1 1   j = 5               1 3 1     j = 3             1 2 2       j = 1           1 3 2         j = -1         1 2 3           j = -3       1 2 1             j = -5     1 1 2               j = -7   1 2                   j = -9                         j = -11 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, 359]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 359]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[11, 19, 12, 18], X[17, 9, 18, 8], > X[7, 17, 8, 16], X[15, 5, 16, 14], X[19, 15, 20, 22], X[13, 20, 14, 21], > X[21, 12, 22, 13], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {-6, 5, -4, 3, -7, 8, -9, 7}, > {10, -1, -5, 4, 11, -2, -3, 9, -8, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-5 -4 2 2 3 4 5 2 + q - q + -- - 2 q + 3 q - 2 q + 2 q - q 3 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 2 2 3 4 2 4 2 4 8 10 14 3 + q + --- + --- + --- + -- + -- + -- + -- + 2 q + 2 q + q - q + q - 14 12 10 8 6 4 2 q q q q q q q 16 > q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 359]][a, z]
Out[8]=	2 4 2 2 -2 2 4 -2 2 a a 2 z 2 z 2 2 4 2 6 - a - 8 a + 3 a + z - ---- + -- + 9 z - -- - ---- - 9 a z + a z + 2 2 4 2 z z a a 4 2 4 6 > 6 z - 2 a z + z
In[9]:=	Kauffman[Link[11, NonAlternating, 359]][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 5 z 5 z 2 2 4 2 3 z 2 z > 10 a z - 7 a z - 24 z + ---- + ---- - 40 a z - 16 a z - ---- - ---- + 4 2 5 3 a a a a 3 4 4 5 9 z 3 3 3 4 7 z 7 z 2 4 4 4 z > ---- + 10 a z + 2 a z + 32 z - ---- - ---- + 48 a z + 16 a z + -- - a 4 2 5 a a a 5 5 6 6 2 z 6 z 5 3 5 6 2 z 2 z 2 6 4 6 > ---- - ---- + 5 a z + 8 a z - 15 z + ---- + ---- - 22 a z - 7 a z + 3 a 4 2 a a a 7 7 z z 7 3 7 8 2 8 4 8 9 3 9 > -- + -- - 6 a z - 6 a z + 2 z + 3 a z + a z + a z + a z 3 a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 1 1 2 1 1 1 2 2 - + 3 q + q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ----- + q 11 6 7 5 7 4 5 4 5 3 3 3 5 2 3 2 q t q t q t q t q t q t q t q t 1 1 2 q 3 5 3 2 5 2 5 3 > ---- + ---- + --- + - + 2 q t + 2 q t + q t + 2 q t + 3 q t + q t + 2 3 q t t q t q t 7 3 7 4 9 4 11 5 > q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n359

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