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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X16,8,5,7 X18,9,19,10 X22,15,17,16 X20,13,21,14 X12,19,13,20 X14,21,15,22 X8,17,9,18 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

q-10 - 2q-9 + 5q-8 - 8q-7 + 11q-6 - 12q-5 + 13q-4 - 10q-3 + 9q-2 - 5q-1 + 3 - q

A2 (sl(3)) Invariant:

q-30 + q-28 + q-26 + 3q-24 + 3q-20 + q-18 + 2q-16 + 5q-14 + q-12 + 6q-10 + q-8 + 2q-6 + q-4 - q-2 + 1 - q2

HOMFLY-PT Polynomial:

- a2 - 4a2z2 - 4a2z4 - a2z6 + a4z-2 + 9a4 + 17a4z2 + 14a4z4 + 6a4z6 + a4z8 - 2a6z-2 - 12a6 - 17a6z2 - 10a6z4 - 2a6z6 + a8z-2 + 4a8 + 4a8z2 + a8z4

Kauffman Polynomial:

- az + 4az3 - 4az5 + az7 + 2a2 - 9a2z2 + 17a2z4 - 13a2z6 + 3a2z8 - 3a3z + 9a3z3 + a3z5 - 9a3z7 + 3a3z9 - a4z-2 + 12a4 - 36a4z2 + 51a4z4 - 33a4z6 + 5a4z8 + a4z10 + 2a5z-1 - 12a5z + 11a5z3 + 7a5z5 - 17a5z7 + 6a5z9 - 2a6z-2 + 15a6 - 35a6z2 + 43a6z4 - 29a6z6 + 6a6z8 + a6z10 + 2a7z-1 - 10a7z + 12a7z3 - 5a7z5 - 3a7z7 + 3a7z9 - a8z-2 + 5a8 - 6a8z2 + 6a8z4 - 6a8z6 + 4a8z8 + 4a9z3 - 5a9z5 + 4a9z7 - 2a10z4 + 3a10z6 - 2a11z3 + 2a11z5 + a12 - 2a12z2 + a12z4

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 3                       1 j = 1                     2   j = -1                   3 1   j = -3                 6 2     j = -5               6 5       j = -7             7 4         j = -9           5 6           j = -11         6 7             j = -13       2 5               j = -15     3 6                 j = -17   1 4                   j = -19   1                     j = -21 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, Alternating, 477]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 477]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[16, 8, 5, 7], X[18, 9, 19, 10], > X[22, 15, 17, 16], X[20, 13, 21, 14], X[12, 19, 13, 20], X[14, 21, 15, 22], > X[8, 17, 9, 18], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {9, -4, 7, -6, 8, -5}, > {10, -1, 3, -9, 4, -2, 11, -7, 6, -8, 5, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-10 2 5 8 11 12 13 10 9 5 3 + q - -- + -- - -- + -- - -- + -- - -- + -- - - - q 9 8 7 6 5 4 3 2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 -26 3 3 -18 2 5 -12 6 -8 2 1 + q + q + q + --- + --- + q + --- + --- + q + --- + q + -- + 24 20 16 14 10 6 q q q q q q -4 -2 2 > q - q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 477]][a, z]
Out[8]=	4 6 8 2 4 6 8 a 2 a a 2 2 4 2 6 2 -a + 9 a - 12 a + 4 a + -- - ---- + -- - 4 a z + 17 a z - 17 a z + 2 2 2 z z z 8 2 2 4 4 4 6 4 8 4 2 6 4 6 > 4 a z - 4 a z + 14 a z - 10 a z + a z - a z + 6 a z - 6 6 4 8 > 2 a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 477]][a, z]
Out[9]=	4 6 8 5 7 2 4 6 8 12 a 2 a a 2 a 2 a 2 a + 12 a + 15 a + 5 a + a - -- - ---- - -- + ---- + ---- - a z - 2 2 2 z z z z z 3 5 7 2 2 4 2 6 2 8 2 > 3 a z - 12 a z - 10 a z - 9 a z - 36 a z - 35 a z - 6 a z - 12 2 3 3 3 5 3 7 3 9 3 11 3 > 2 a z + 4 a z + 9 a z + 11 a z + 12 a z + 4 a z - 2 a z + 2 4 4 4 6 4 8 4 10 4 12 4 5 > 17 a z + 51 a z + 43 a z + 6 a z - 2 a z + a z - 4 a z + 3 5 5 5 7 5 9 5 11 5 2 6 4 6 > a z + 7 a z - 5 a z - 5 a z + 2 a z - 13 a z - 33 a z - 6 6 8 6 10 6 7 3 7 5 7 7 7 > 29 a z - 6 a z + 3 a z + a z - 9 a z - 17 a z - 3 a z + 9 7 2 8 4 8 6 8 8 8 3 9 5 9 > 4 a z + 3 a z + 5 a z + 6 a z + 4 a z + 3 a z + 6 a z + 7 9 4 10 6 10 > 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 6 1 1 1 4 3 6 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 7 17 7 17 6 15 6 15 5 13 5 q q q t q t q t q t q t q t q t 5 6 7 5 6 7 4 6 2 t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + --- + 13 4 11 4 11 3 9 3 9 2 7 2 7 5 3 q t q t q t q t q t q t q t q t q 2 3 t t 2 3 3 > --- + -- + 2 q t + q t q q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a477

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