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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-11 + 3q-10 - 6q-9 + 10q-8 - 12q-7 + 15q-6 - 13q-5 + 12q-4 - 8q-3 + 5q-2 - 2q-1 + 1

A2 (sl(3)) Invariant:

- q-34 - q-32 + q-30 - q-28 + 3q-26 + 3q-24 + 2q-22 + 6q-20 + q-18 + 5q-16 + 2q-14 + q-12 + 4q-10 - q-8 + 2q-6 + 1

HOMFLY-PT Polynomial:

a2 + 3a2z2 + a2z4 + a4z-2 + 3a4 + 2a4z2 - 2a4z4 - a4z6 - 2a6z-2 - 8a6 - 11a6z2 - 8a6z4 - 2a6z6 + a8z-2 + 6a8 + 9a8z2 + 3a8z4 - 2a10 - a10z2

Kauffman Polynomial:

- a2 + 4a2z2 - 4a2z4 + a2z6 + 3a3z3 - 6a3z5 + 2a3z7 - a4z-2 + 5a4 - 9a4z2 + 9a4z4 - 9a4z6 + 3a4z8 + 2a5z-1 - 7a5z + 2a5z3 + 6a5z5 - 8a5z7 + 3a5z9 - 2a6z-2 + 14a6 - 40a6z2 + 51a6z4 - 28a6z6 + 5a6z8 + a6z10 + 2a7z-1 - 10a7z + 7a7z3 + 14a7z5 - 16a7z7 + 6a7z9 - a8z-2 + 10a8 - 29a8z2 + 42a8z4 - 25a8z6 + 6a8z8 + a8z10 - 3a9z + 8a9z3 - 4a9z5 - 2a9z7 + 3a9z9 + a10z2 - 2a10z4 - 4a10z6 + 4a10z8 + a11z - 2a11z3 - 5a11z5 + 4a11z7 - a12 + 3a12z2 - 6a12z4 + 3a12z6 + a13z - 2a13z3 + a13z5

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 1                       1 j = -1                     1   j = -3                   4 1   j = -5                 5 2     j = -7               7 3       j = -9             7 6         j = -11           8 6           j = -13         5 8             j = -15       5 7               j = -17     2 6                 j = -19   1 4                   j = -21   2                     j = -23 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, 468]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 468]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[18, 11, 19, 12], X[16, 8, 17, 7], > X[8, 16, 9, 15], X[22, 17, 15, 18], X[20, 13, 21, 14], X[12, 19, 13, 20], > X[14, 21, 5, 22], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {5, -4, 6, -3, 8, -7, 9, -6}, > {10, -1, 4, -5, 11, -2, 3, -8, 7, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-11 3 6 10 12 15 13 12 8 5 2 1 - q + --- - -- + -- - -- + -- - -- + -- - -- + -- - - 10 9 8 7 6 5 4 3 2 q q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 -32 -30 -28 3 3 2 6 -18 5 2 1 - q - q + q - q + --- + --- + --- + --- + q + --- + --- + 26 24 22 20 16 14 q q q q q q -12 4 -8 2 > q + --- - q + -- 10 6 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 468]][a, z]
Out[8]=	4 6 8 2 4 6 8 10 a 2 a a 2 2 4 2 a + 3 a - 8 a + 6 a - 2 a + -- - ---- + -- + 3 a z + 2 a z - 2 2 2 z z z 6 2 8 2 10 2 2 4 4 4 6 4 8 4 4 6 > 11 a z + 9 a z - a z + a z - 2 a z - 8 a z + 3 a z - a z - 6 6 > 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 468]][a, z]
Out[9]=	4 6 8 5 7 2 4 6 8 12 a 2 a a 2 a 2 a 5 -a + 5 a + 14 a + 10 a - a - -- - ---- - -- + ---- + ---- - 7 a z - 2 2 2 z z z z z 7 9 11 13 2 2 4 2 6 2 > 10 a z - 3 a z + a z + a z + 4 a z - 9 a z - 40 a z - 8 2 10 2 12 2 3 3 5 3 7 3 9 3 > 29 a z + a z + 3 a z + 3 a z + 2 a z + 7 a z + 8 a z - 11 3 13 3 2 4 4 4 6 4 8 4 10 4 > 2 a z - 2 a z - 4 a z + 9 a z + 51 a z + 42 a z - 2 a z - 12 4 3 5 5 5 7 5 9 5 11 5 13 5 > 6 a z - 6 a z + 6 a z + 14 a z - 4 a z - 5 a z + a z + 2 6 4 6 6 6 8 6 10 6 12 6 3 7 > a z - 9 a z - 28 a z - 25 a z - 4 a z + 3 a z + 2 a z - 5 7 7 7 9 7 11 7 4 8 6 8 8 8 > 8 a z - 16 a z - 2 a z + 4 a z + 3 a z + 5 a z + 6 a z + 10 8 5 9 7 9 9 9 6 10 8 10 > 4 a z + 3 a z + 6 a z + 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 1 2 1 4 2 6 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 23 9 21 8 19 8 19 7 17 7 17 6 15 6 q q q t q t q t q t q t q t q t 7 5 8 8 6 7 6 7 3 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 15 5 13 5 13 4 11 4 11 3 9 3 9 2 7 2 7 q t q t q t q t q t q t q t q t q t 5 t t 2 > ---- + -- + - + q t 5 3 q q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a468

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