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DrawMorseLink

PD Presentation:

X6172 X12,7,13,8 X4,13,1,14 X5,18,6,19 X8493 X9,21,10,20 X19,11,20,10 X17,14,18,15 X15,22,16,17 X21,16,22,5 X2,12,3,11

Gauss Code:

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

Jones Polynomial:

- q-7 + q-6 + 3q-3 - 2q-2 + 4q-1 - 3 + 3q - 2q2 + q3

A2 (sl(3)) Invariant:

- q-22 - q-20 - q-18 + q-16 + 2q-14 + 3q-12 + 6q-10 + 4q-8 + 5q-6 + 3q-4 + 2q-2 + 2 + q4 + q10

HOMFLY-PT Polynomial:

a-2 + a-2z2 + z-2 - 2z2 - z4 - 2a2z-2 - 4a2 - 3a2z2 - a2z4 + a4z-2 + 5a4 + 5a4z2 + a4z4 - 2a6 - a6z2

Kauffman Polynomial:

- a-2 + 3a-2z2 - 4a-2z4 + a-2z6 - a-1z + 6a-1z3 - 8a-1z5 + 2a-1z7 - z-2 + 3 - 2z2 + 6z4 - 8z6 + 2z8 + 2az-1 - 7az + 10az3 - 3az5 - 3az7 + az9 - 2a2z-2 + 13a2 - 31a2z2 + 35a2z4 - 18a2z6 + 3a2z8 + 2a3z-1 - 13a3z + 13a3z3 + 2a3z5 - 5a3z7 + a3z9 - a4z-2 + 15a4 - 38a4z2 + 39a4z4 - 16a4z6 + 2a4z8 - 10a5z + 18a5z3 - 9a5z5 + a5z7 + 5a6 - 12a6z2 + 14a6z4 - 7a6z6 + a6z8 - 3a7z + 9a7z3 - 6a7z5 + a7z7

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                       1 j = 5                     1   j = 3                   2 1   j = 1               1 2 1     j = -1             1 4 2       j = -3           1 2 3         j = -5         1 4 2           j = -7       1 1 3             j = -9       1 1               j = -11   1 1 1                 j = -13                         j = -15 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, 360]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 360]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 7, 13, 8], X[4, 13, 1, 14], X[5, 18, 6, 19], > X[8, 4, 9, 3], X[9, 21, 10, 20], X[19, 11, 20, 10], X[17, 14, 18, 15], > X[15, 22, 16, 17], X[21, 16, 22, 5], X[2, 12, 3, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {-8, 4, -7, 6, -10, 9}, > {-4, -1, 2, -5, -6, 7, 11, -2, 3, 8, -9, 10}]
In[6]:=	Jones[L][q]
Out[6]=	-7 -6 3 2 4 2 3 -3 - q + q + -- - -- + - + 3 q - 2 q + q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 -18 -16 2 3 6 4 5 3 2 4 10 2 - q - q - q + q + --- + --- + --- + -- + -- + -- + -- + q + q 14 12 10 8 6 4 2 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 360]][a, z]
Out[8]=	2 4 2 -2 2 4 6 -2 2 a a 2 z 2 2 4 2 a - 4 a + 5 a - 2 a + z - ---- + -- - 2 z + -- - 3 a z + 5 a z - 2 2 2 z z a 6 2 4 2 4 4 4 > a z - z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 360]][a, z]
Out[9]=	2 4 3 -2 2 4 6 -2 2 a a 2 a 2 a z 3 - a + 13 a + 15 a + 5 a - z - ---- - -- + --- + ---- - - - 7 a z - 2 2 z z a z z 2 3 5 7 2 3 z 2 2 4 2 6 2 > 13 a z - 10 a z - 3 a z - 2 z + ---- - 31 a z - 38 a z - 12 a z + 2 a 3 4 6 z 3 3 3 5 3 7 3 4 4 z 2 4 > ---- + 10 a z + 13 a z + 18 a z + 9 a z + 6 z - ---- + 35 a z + a 2 a 5 4 4 6 4 8 z 5 3 5 5 5 7 5 6 > 39 a z + 14 a z - ---- - 3 a z + 2 a z - 9 a z - 6 a z - 8 z + a 6 7 z 2 6 4 6 6 6 2 z 7 3 7 5 7 > -- - 18 a z - 16 a z - 7 a z + ---- - 3 a z - 5 a z + a z + 2 a a 7 7 8 2 8 4 8 6 8 9 3 9 > a z + 2 z + 3 a z + 2 a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 1 1 1 1 1 1 1 -- + - + q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 7 11 6 11 5 11 4 9 4 7 4 9 3 q q t q t q t q t q t q t q t 1 1 3 4 1 2 2 1 2 t > ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + --- + 2 q t + 7 3 5 3 7 2 5 2 3 2 5 3 q t q q t q t q t q t q t q t q t 2 3 2 3 3 5 3 7 4 > q t + 2 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n360

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