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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X8493 X13,22,14,5 X21,14,22,15 X11,18,12,19 X9,20,10,21 X19,10,20,11 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

q-19/2 - q-17/2 + q-15/2 + q-13/2 - 3q-11/2 + 4q-9/2 - 6q-7/2 + 4q-5/2 - 5q-3/2 + 3q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-30 - q-28 - q-26 - q-24 - q-22 - 3q-20 + q-18 + q-16 + 3q-14 + 5q-12 + 2q-10 + 4q-8 + q-4 - 1 + q2

HOMFLY-PT Polynomial:

- az - az3 - 2a3z-1 - a3z + 2a3z3 + a3z5 + 2a5z-1 - a5z3 + a7z-1 + 3a7z + a7z3 - a9z-1 - a9z

Kauffman Polynomial:

- az + 2az3 - az5 - a2z2 + 7a2z4 - 3a2z6 - 2a3z-1 + 3a3z - 2a3z3 + 7a3z5 - 3a3z7 + 2a4 - 2a4z2 + 4a4z4 - a4z8 - 2a5z-1 + 5a5z - 12a5z3 + 10a5z5 - 3a5z7 - 4a6 + 18a6z2 - 28a6z4 + 13a6z6 - 2a6z8 + a7z-1 + a7z - a7z3 - 11a7z5 + 7a7z7 - a7z9 - 9a8 + 32a8z2 - 40a8z4 + 17a8z6 - 2a8z8 + a9z-1 + 7a9z3 - 13a9z5 + 7a9z7 - a9z9 - 4a10 + 13a10z2 - 15a10z4 + 7a10z6 - a10z8

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 = 2                       1 j = 0                     2   j = -2                   3 1   j = -4                 3 4     j = -6             1 4 1       j = -8             1 3         j = -10         1 4 4           j = -12       1   1             j = -14       1 3               j = -16   1 1                   j = -18                         j = -20 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 28]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 28]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[5, 12, 6, 13], > X[8, 4, 9, 3], X[13, 22, 14, 5], X[21, 14, 22, 15], X[11, 18, 12, 19], > X[9, 20, 10, 21], X[19, 10, 20, 11], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {-4, -1, 2, -5, -9, 10, -8, 4, -6, 7, 11, -2, 3, 8, > -10, 9, -7, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) -(17/2) -(15/2) -(13/2) 3 4 6 4 5 q - q + q + q - ----- + ---- - ---- + ---- - ---- + 11/2 9/2 7/2 5/2 3/2 q q q q q 3 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 -26 -24 -22 3 -18 -16 3 5 2 -1 - q - q - q - q - q - --- + q + q + --- + --- + --- + 20 14 12 10 q q q q 4 -4 2 > -- + q + q 8 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 28]][a, z]
Out[8]=	3 5 7 9 -2 a 2 a a a 3 7 9 3 3 3 5 3 ----- + ---- + -- - -- - a z - a z + 3 a z - a z - a z + 2 a z - a z + z z z z 7 3 3 5 > a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 28]][a, z]
Out[9]=	3 5 7 9 4 6 8 10 2 a 2 a a a 3 5 2 a - 4 a - 9 a - 4 a - ---- - ---- + -- + -- - a z + 3 a z + 5 a z + z z z z 7 2 2 4 2 6 2 8 2 10 2 3 > a z - a z - 2 a z + 18 a z + 32 a z + 13 a z + 2 a z - 3 3 5 3 7 3 9 3 2 4 4 4 6 4 > 2 a z - 12 a z - a z + 7 a z + 7 a z + 4 a z - 28 a z - 8 4 10 4 5 3 5 5 5 7 5 9 5 > 40 a z - 15 a z - a z + 7 a z + 10 a z - 11 a z - 13 a z - 2 6 6 6 8 6 10 6 3 7 5 7 7 7 > 3 a z + 13 a z + 17 a z + 7 a z - 3 a z - 3 a z + 7 a z + 9 7 4 8 6 8 8 8 10 8 7 9 9 9 > 7 a z - a z - 2 a z - 2 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 3 1 1 1 1 1 3 1 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 9 16 8 16 7 14 6 12 6 14 5 10 5 q q q t q t q t q t q t q t q t 1 4 4 1 1 3 4 1 3 > ------ + ------ + ------ + ----- + ----- + ----- + ----- + ---- + ---- + 12 4 10 4 10 3 8 3 6 3 8 2 6 2 6 4 q t q t q t q t q t q t q t q t q t t 2 2 > 2 t + -- + q t 2 q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n28

L11n27 L11n29