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DrawMorseLink

PD Presentation:

X6172 X12,7,13,8 X4,13,1,14 X9,18,10,19 X3849 X5,14,6,15 X15,22,16,5 X17,20,18,21 X21,16,22,17 X19,10,20,11 X11,2,12,3

Gauss Code:

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

Jones Polynomial:

2q-23/2 - 3q-21/2 + 4q-19/2 - 6q-17/2 + 6q-15/2 - 6q-13/2 + 4q-11/2 - 3q-9/2 + q-7/2 - q-5/2

A2 (sl(3)) Invariant:

- q-40 - q-38 - 2q-36 - q-34 + q-32 + 2q-28 + q-26 + q-24 + 2q-22 + 2q-18 + q-16 + q-14 + 2q-12 + q-8

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a5z-1 - 4a5z + 4a5z3 - a5z5 - a6 + a6z2 + 2a6z4 - a6z6 + a7z-1 - 3a7z + 3a7z3 + a7z5 - a7z7 + a8z2 - 2a8z4 + 2a8z6 - a8z8 + a9z-1 - 4a9z + 7a9z3 - 9a9z5 + 4a9z7 - a9z9 - 4a10 + 16a10z2 - 25a10z4 + 13a10z6 - 3a10z8 + 2a11z-1 - 8a11z + 11a11z3 - 10a11z5 + 4a11z7 - a11z9 - 7a12 + 24a12z2 - 24a12z4 + 10a12z6 - 2a12z8 + a13z-1 - 3a13z + 3a13z3 + a13z5 - a13z7 - 3a14 + 8a14z2 - 3a14z4

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                 1 1 j = -8               2     j = -10             2 1     j = -12           4 2       j = -14         3 3         j = -16       3 3           j = -18     1 3             j = -20   2 3               j = -22   1                 j = -24 2

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, 42]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 42]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 7, 13, 8], X[4, 13, 1, 14], X[9, 18, 10, 19], > X[3, 8, 4, 9], X[5, 14, 6, 15], X[15, 22, 16, 5], X[17, 20, 18, 21], > X[21, 16, 22, 17], X[19, 10, 20, 11], X[11, 2, 12, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -5, -3}, {-6, -1, 2, 5, -4, 10, -11, -2, 3, 6, -7, 9, -8, 4, > -10, 8, -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 4 6 6 6 4 3 -(7/2) -(5/2) ----- - ----- + ----- - ----- + ----- - ----- + ----- - ---- + q - q 23/2 21/2 19/2 17/2 15/2 13/2 11/2 9/2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-40 -38 2 -34 -32 2 -26 -24 2 2 -16 -q - q - --- - q + q + --- + q + q + --- + --- + q + 36 28 22 18 q q q q -14 2 -8 > q + --- + q 12 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 42]][a, z]
Out[8]=	5 7 9 11 13 a a a 2 a a 5 7 9 11 5 3 -(--) + -- - -- + ----- - --- - 4 a z + a z - 2 a z + 3 a z - 4 a z - z z z z z 7 3 9 3 11 3 5 5 7 5 9 5 > 2 a z - 3 a z + a z - a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 42]][a, z]
Out[9]=	5 7 9 11 13 6 10 12 14 a a a 2 a a 5 7 -a - 4 a - 7 a - 3 a + -- + -- + -- + ----- + --- - 4 a z - 3 a z - z z z z z 9 11 13 6 2 8 2 10 2 12 2 > 4 a z - 8 a z - 3 a z + a z + a z + 16 a z + 24 a z + 14 2 5 3 7 3 9 3 11 3 13 3 6 4 > 8 a z + 4 a z + 3 a z + 7 a z + 11 a z + 3 a z + 2 a z - 8 4 10 4 12 4 14 4 5 5 7 5 9 5 > 2 a z - 25 a z - 24 a z - 3 a z - a z + a z - 9 a z - 11 5 13 5 6 6 8 6 10 6 12 6 7 7 > 10 a z + a z - a z + 2 a z + 13 a z + 10 a z - a z + 9 7 11 7 13 7 8 8 10 8 12 8 9 9 11 9 > 4 a z + 4 a z - a z - a z - 3 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 2 1 2 3 1 3 3 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 24 9 22 8 20 8 20 7 18 7 18 6 16 6 q t q t q t q t q t q t q t 3 3 3 4 2 2 1 2 1 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ---- 16 5 14 5 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 q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n42

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