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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X3849 X13,22,14,5 X21,14,22,15 X11,18,12,19 X9,20,10,21 X19,10,20,11 X15,2,16,3

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-23/2 - q-21/2 + 4q-19/2 - 6q-17/2 + 5q-15/2 - 7q-13/2 + 5q-11/2 - 4q-9/2 + 2q-7/2 - q-5/2

A2 (sl(3)) Invariant:

- q-40 - q-38 - q-36 - 2q-34 - q-32 - 2q-30 + q-28 + 3q-26 + 3q-24 + 5q-22 + q-20 + 2q-18 + q-16 + q-12 - q-10 + q-8

HOMFLY-PT Polynomial:

- a5z - 3a5z3 - a5z5 - 2a7z-1 - 5a7z - 7a7z3 - 2a7z5 + 2a9z-1 - 3a9z3 - a9z5 + a11z-1 + 4a11z + a11z3 - a13z-1

Kauffman Polynomial:

- a5z + 3a5z3 - a5z5 - a6z2 + 5a6z4 - 2a6z6 - 2a7z-1 + 7a7z - 11a7z3 + 10a7z5 - 3a7z7 + 2a8 - 4a8z2 - 2a8z4 + 5a8z6 - 2a8z8 - 2a9z-1 + 10a9z - 11a9z3 + a9z5 + 2a9z7 - a9z9 - 4a10 + 14a10z2 - 24a10z4 + 13a10z6 - 3a10z8 + a11z-1 + 3a11z + 4a11z3 - 11a11z5 + 5a11z7 - a11z9 - 9a12 + 21a12z2 - 18a12z4 + 6a12z6 - a12z8 + a13z-1 + a13z + a13z3 - a13z5 - 4a14 + 4a14z2 - a14z4

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -4                   1 j = -6                 2 1 j = -8               2     j = -10             3 2     j = -12           4 2       j = -14         2 4         j = -16       4 3           j = -18       2             j = -20   1 4               j = -22                     j = -24 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, 26]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 26]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[5, 12, 6, 13], > X[3, 8, 4, 9], 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[15, 2, 16, 3]]
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]=	-(23/2) -(21/2) 4 6 5 7 5 4 2 q - q + ----- - ----- + ----- - ----- + ----- - ---- + ---- - 19/2 17/2 15/2 13/2 11/2 9/2 7/2 q q q q q q q -(5/2) > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-40 -38 -36 2 -32 2 -28 3 3 5 -20 2 -q - q - q - --- - q - --- + q + --- + --- + --- + q + --- + 34 30 26 24 22 18 q q q q q q -16 -12 -10 -8 > q + q - q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 26]][a, z]
Out[8]=	7 9 11 13 -2 a 2 a a a 5 7 11 5 3 7 3 ----- + ---- + --- - --- - a z - 5 a z + 4 a z - 3 a z - 7 a z - z z z z 9 3 11 3 5 5 7 5 9 5 > 3 a z + a z - a z - 2 a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 26]][a, z]
Out[9]=	7 9 11 13 8 10 12 14 2 a 2 a a a 5 7 2 a - 4 a - 9 a - 4 a - ---- - ---- + --- + --- - a z + 7 a z + z z z z 9 11 13 6 2 8 2 10 2 12 2 > 10 a z + 3 a z + a z - a z - 4 a z + 14 a z + 21 a z + 14 2 5 3 7 3 9 3 11 3 13 3 6 4 > 4 a z + 3 a z - 11 a z - 11 a z + 4 a z + a z + 5 a z - 8 4 10 4 12 4 14 4 5 5 7 5 9 5 > 2 a z - 24 a z - 18 a z - a z - a z + 10 a z + a z - 11 5 13 5 6 6 8 6 10 6 12 6 7 7 > 11 a z - a z - 2 a z + 5 a z + 13 a z + 6 a z - 3 a z + 9 7 11 7 8 8 10 8 12 8 9 9 11 9 > 2 a z + 5 a z - 2 a z - 3 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 1 4 2 4 3 2 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 24 9 20 8 20 7 18 6 16 6 16 5 14 5 q t q t q t q t q t q t q t 4 4 2 3 2 2 2 > ------ + ------ + ------ + ------ + ------ + ----- + ---- 14 4 12 4 12 3 10 3 10 2 8 2 6 q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n26

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