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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X16,9,17,10 X8,15,9,16 X4,17,1,18 X11,22,12,19 X10,4,11,3 X5,21,6,20 X21,5,22,18 X19,12,20,13 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

- q-8 + q-7 - q-6 + q-5 - q-4 + q-3 + 2q-2 + 2 - q + q2

A2 (sl(3)) Invariant:

- q-24 - q-22 - 2q-20 - q-18 - q-16 - q-14 + q-12 + 2q-10 + 6q-8 + 6q-6 + 6q-4 + 5q-2 + 3 + 3q2 + q4 + q6

HOMFLY-PT Polynomial:

2z-2 + 5 + 4z2 + z4 - 5a2z-2 - 14a2 - 16a2z2 - 7a2z4 - a2z6 + 4a4z-2 + 13a4 + 16a4z2 + 7a4z4 + a4z6 - a6z-2 - 4a6 - 4a6z2 - a6z4

Kauffman Polynomial:

- 2z-2 + 9 - 16z2 + 16z4 - 7z6 + z8 + 5az-1 - 12az + 4az3 + 8az5 - 6az7 + az9 - 5a2z-2 + 23a2 - 53a2z2 + 52a2z4 - 22a2z6 + 3a2z8 + 9a3z-1 - 31a3z + 34a3z3 - 7a3z5 - 4a3z7 + a3z9 - 4a4z-2 + 22a4 - 49a4z2 + 50a4z4 - 22a4z6 + 3a4z8 + 5a5z-1 - 25a5z + 40a5z3 - 21a5z5 + 3a5z7 - a6z-2 + 7a6 - 10a6z2 + 10a6z4 - 6a6z6 + a6z8 + a7z-1 - 4a7z + 6a7z3 - 5a7z5 + a7z7 + 2a8z2 - 4a8z4 + a8z6 + 2a9z - 4a9z3 + a9z5

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 5                       1 j = 3                         j = 1                   2 1   j = -1               3 1       j = -3             1 5 2       j = -5           3 2 2         j = -7         1 2 1           j = -9       2 4 2             j = -11       1 1               j = -13   1 2 1                 j = -15                         j = -17 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, 404]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 404]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[16, 9, 17, 10], X[8, 15, 9, 16], > X[4, 17, 1, 18], X[11, 22, 12, 19], X[10, 4, 11, 3], X[5, 21, 6, 20], > X[21, 5, 22, 18], X[19, 12, 20, 13], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 7, -5}, {-10, 8, -9, 6}, > {-8, -1, 2, -4, 3, -7, -6, 10, 11, -2, 4, -3, 5, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-8 -7 -6 -5 -4 -3 2 2 2 - q + q - q + q - q + q + -- - q + q 2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 2 -18 -16 -14 -12 2 6 6 6 5 3 - q - q - --- - q - q - q + q + --- + -- + -- + -- + -- + 20 10 8 6 4 2 q q q q q q 2 4 6 > 3 q + q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 404]][a, z]
Out[8]=	2 4 6 2 4 6 2 5 a 4 a a 2 2 2 4 2 5 - 14 a + 13 a - 4 a + -- - ---- + ---- - -- + 4 z - 16 a z + 16 a z - 2 2 2 2 z z z z 6 2 4 2 4 4 4 6 4 2 6 4 6 > 4 a z + z - 7 a z + 7 a z - a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 404]][a, z]
Out[9]=	2 4 6 3 5 7 2 4 6 2 5 a 4 a a 5 a 9 a 5 a a 9 + 23 a + 22 a + 7 a - -- - ---- - ---- - -- + --- + ---- + ---- + -- - 2 2 2 2 z z z z z z z z 3 5 7 9 2 2 2 > 12 a z - 31 a z - 25 a z - 4 a z + 2 a z - 16 z - 53 a z - 4 2 6 2 8 2 3 3 3 5 3 7 3 > 49 a z - 10 a z + 2 a z + 4 a z + 34 a z + 40 a z + 6 a z - 9 3 4 2 4 4 4 6 4 8 4 5 > 4 a z + 16 z + 52 a z + 50 a z + 10 a z - 4 a z + 8 a z - 3 5 5 5 7 5 9 5 6 2 6 4 6 > 7 a z - 21 a z - 5 a z + a z - 7 z - 22 a z - 22 a z - 6 6 8 6 7 3 7 5 7 7 7 8 2 8 > 6 a z + a z - 6 a z - 4 a z + 3 a z + a z + z + 3 a z + 4 8 6 8 9 3 9 > 3 a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 5 3 1 1 2 1 1 2 1 -- + -- + - + ------ + ------ + ------ + ------ + ------ + ----- + ------ + 5 3 q 17 7 13 6 13 5 13 4 11 4 9 4 11 3 q q q t q t q t q t q t q t q t 4 1 2 2 3 1 2 1 2 t t > ----- + ----- + ----- + ----- + ----- + ---- + ---- + ---- + --- + - + 9 3 7 3 9 2 7 2 5 2 7 5 3 3 q q t q t q t q t q t q t q t q t q 2 3 5 4 > 2 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n404

L11n403 L11n405