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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9 - 3q-8 + 7q-7 - 12q-6 + 16q-5 - 17q-4 + 19q-3 - 15q-2 + 12q-1 - 6 + 3q - q2

A2 (sl(3)) Invariant:

q-28 - q-24 + 3q-22 - 2q-20 + 4q-16 + 7q-12 + 3q-10 + 5q-8 + 5q-6 - q-4 + 5q-2 - 1 - q2 + q4 - q6

HOMFLY-PT Polynomial:

- 1 - 2z2 - z4 + a2z-2 + 5a2 + 6a2z2 + 3a2z4 + a2z6 - 2a4z-2 - 4a4 - 2a4z2 + a4z4 + a4z6 + a6z-2 - a6 - 3a6z2 - 2a6z4 + a8 + a8z2

Kauffman Polynomial:

a-1z - 2a-1z3 + a-1z5 - 2 + 5z2 - 6z4 + 3z6 + 3az - 3az3 - 3az5 + 4az7 + a2z-2 - 8a2 + 18a2z2 - 15a2z4 + 2a2z6 + 4a2z8 - 2a3z-1 + 5a3z + 3a3z3 - 10a3z5 + 4a3z7 + 3a3z9 + 2a4z-2 - 8a4 + 16a4z2 - 9a4z4 - 9a4z6 + 8a4z8 + a4z10 - 2a5z-1 - a5z + 12a5z3 - 15a5z5 - a5z7 + 6a5z9 + a6z-2 - 3a6 + 6a6z2 - 15a6z6 + 8a6z8 + a6z10 - 6a7z + 15a7z3 - 17a7z5 + 2a7z7 + 3a7z9 - a8 + 6a8z2 - 3a8z4 - 6a8z6 + 4a8z8 - 2a9z + 7a9z3 - 8a9z5 + 3a9z7 - a10 + 3a10z2 - 3a10z4 + a10z6

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 = 5                       1 j = 3                     2   j = 1                   4 1   j = -1                 8 2     j = -3               9 6       j = -5             10 6         j = -7           7 9           j = -9         9 10             j = -11       5 9               j = -13     2 7                 j = -15   1 5                   j = -17   2                     j = -19 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, 466]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 466]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[14, 8, 5, 7], X[18, 21, 19, 22], > X[16, 9, 17, 10], X[8, 15, 9, 16], X[20, 13, 21, 14], X[12, 19, 13, 20], > X[22, 17, 15, 18], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {6, -5, 9, -4, 8, -7, 4, -9}, > {10, -1, 3, -6, 5, -2, 11, -8, 7, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-9 3 7 12 16 17 19 15 12 2 -6 + q - -- + -- - -- + -- - -- + -- - -- + -- + 3 q - q 8 7 6 5 4 3 2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -24 3 2 4 7 3 5 5 -4 5 2 4 -1 + q - q + --- - --- + --- + --- + --- + -- + -- - q + -- - q + q - 22 20 16 12 10 8 6 2 q q q q q q q q 6 > q
In[8]:=	HOMFLYPT[Link[11, Alternating, 466]][a, z]
Out[8]=	2 4 6 2 4 6 8 a 2 a a 2 2 2 4 2 -1 + 5 a - 4 a - a + a + -- - ---- + -- - 2 z + 6 a z - 2 a z - 2 2 2 z z z 6 2 8 2 4 2 4 4 4 6 4 2 6 4 6 > 3 a z + a z - z + 3 a z + a z - 2 a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 466]][a, z]
Out[9]=	2 4 6 3 5 2 4 6 8 10 a 2 a a 2 a 2 a z -2 - 8 a - 8 a - 3 a - a - a + -- + ---- + -- - ---- - ---- + - + 3 a z + 2 2 2 z z a z z z 3 5 7 9 2 2 2 4 2 6 2 > 5 a z - a z - 6 a z - 2 a z + 5 z + 18 a z + 16 a z + 6 a z + 3 8 2 10 2 2 z 3 3 3 5 3 7 3 > 6 a z + 3 a z - ---- - 3 a z + 3 a z + 12 a z + 15 a z + a 5 9 3 4 2 4 4 4 8 4 10 4 z 5 > 7 a z - 6 z - 15 a z - 9 a z - 3 a z - 3 a z + -- - 3 a z - a 3 5 5 5 7 5 9 5 6 2 6 4 6 > 10 a z - 15 a z - 17 a z - 8 a z + 3 z + 2 a z - 9 a z - 6 6 8 6 10 6 7 3 7 5 7 7 7 > 15 a z - 6 a z + a z + 4 a z + 4 a z - a z + 2 a z + 9 7 2 8 4 8 6 8 8 8 3 9 5 9 > 3 a z + 4 a z + 8 a z + 8 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]=	6 8 1 2 1 5 2 7 5 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 9 9 10 7 9 10 6 9 2 t > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 2 3 2 5 3 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a466

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