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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X11,18,12,19 X7,16,8,17 X15,8,16,9 X13,21,14,20 X19,22,20,15 X21,13,22,12 X17,14,18,5 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-10 + 2q-9 - 2q-8 + 3q-7 - q-6 + 2q-5 + q-2 - q-1 + 1

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

2a2 + 4a2z2 + a2z4 + a4z-2 - a4 - 5a4z2 - 5a4z4 - a4z6 - 2a6z-2 - 2a6 + a6z2 + a8z-2 + 2a8 + 2a8z2 - a10

Kauffman Polynomial:

- 2a2 + 6a2z2 - 5a2z4 + a2z6 - a3z + 5a3z3 - 5a3z5 + a3z7 - a4z-2 + 2a4 + 4a4z2 - 6a4z4 + a4z6 + 2a5z-1 - 10a5z + 14a5z3 - 8a5z5 + a5z7 - 2a6z-2 + 12a6 - 26a6z2 + 28a6z4 - 14a6z6 + 2a6z8 + 2a7z-1 - 14a7z + 19a7z3 - 4a7z5 - 4a7z7 + a7z9 - a8z-2 + 12a8 - 35a8z2 + 46a8z4 - 25a8z6 + 4a8z8 - 7a9z + 16a9z3 - 6a9z5 - 3a9z7 + a9z9 + 3a10 - 11a10z2 + 17a10z4 - 11a10z6 + 2a10z8 - 2a11z + 6a11z3 - 5a11z5 + a11z7

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                         j = -3                 1 2 1   j = -5               1 1 1     j = -7             2 3 1       j = -9           2 2 2         j = -11         2 4 1           j = -13       2 1 1             j = -15     1 3 1               j = -17   1 1                   j = -19   1                     j = -21 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, NonAlternating, 354]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 354]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[11, 18, 12, 19], X[7, 16, 8, 17], > X[15, 8, 16, 9], X[13, 21, 14, 20], X[19, 22, 20, 15], X[21, 13, 22, 12], > X[17, 14, 18, 5], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {-5, 4, -9, 3, -7, 6, -8, 7}, > {10, -1, -4, 5, 11, -2, -3, 8, -6, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-10 2 2 3 -6 2 -2 1 1 - q + -- - -- + -- - q + -- + q - - 9 8 7 5 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -30 -28 3 4 4 5 3 4 -12 -10 1 - q - q + q + --- + --- + --- + --- + --- + --- + q + q + 24 22 20 18 16 14 q q q q q q -8 -2 > q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 354]][a, z]
Out[8]=	4 6 8 2 4 6 8 10 a 2 a a 2 2 4 2 6 2 2 a - a - 2 a + 2 a - a + -- - ---- + -- + 4 a z - 5 a z + a z + 2 2 2 z z z 8 2 2 4 4 4 4 6 > 2 a z + a z - 5 a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 354]][a, z]
Out[9]=	4 6 8 5 7 2 4 6 8 10 a 2 a a 2 a 2 a 3 -2 a + 2 a + 12 a + 12 a + 3 a - -- - ---- - -- + ---- + ---- - a z - 2 2 2 z z z z z 5 7 9 11 2 2 4 2 6 2 > 10 a z - 14 a z - 7 a z - 2 a z + 6 a z + 4 a z - 26 a z - 8 2 10 2 3 3 5 3 7 3 9 3 > 35 a z - 11 a z + 5 a z + 14 a z + 19 a z + 16 a z + 11 3 2 4 4 4 6 4 8 4 10 4 3 5 > 6 a z - 5 a z - 6 a z + 28 a z + 46 a z + 17 a z - 5 a z - 5 5 7 5 9 5 11 5 2 6 4 6 6 6 > 8 a z - 4 a z - 6 a z - 5 a z + a z + a z - 14 a z - 8 6 10 6 3 7 5 7 7 7 9 7 11 7 > 25 a z - 11 a z + a z + a z - 4 a z - 3 a z + a z + 6 8 8 8 10 8 7 9 9 9 > 2 a z + 4 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-5 2 1 1 1 1 1 3 2 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 21 9 19 8 17 8 17 7 15 7 15 6 13 6 q q t q t q t q t q t q t q t 1 1 2 1 4 2 1 2 > ------ + ------ + ------ + ------ + ------ + ----- + ------ + ----- + 15 5 13 5 11 5 13 4 11 4 9 4 11 3 9 3 q t q t q t q t q t q t q t q t 2 2 3 1 1 1 1 t 2 > ----- + ----- + ----- + ----- + ---- + ---- + ---- + -- + q t 7 3 9 2 7 2 5 2 7 5 3 3 q t q t q t q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n354

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