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DrawMorseLink

PD Presentation:

X6172 X12,7,13,8 X4,13,1,14 X5,16,6,17 X8493 X9,21,10,20 X19,11,20,10 X17,22,18,15 X21,18,22,19 X15,14,16,5 X2,12,3,11

Gauss Code:

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

Jones Polynomial:

- 2q-6 + 5q-5 - 8q-4 + 11q-3 - 10q-2 + 11q-1 - 8 + 6q - 2q2 + q3

A2 (sl(3)) Invariant:

- q-20 - 2q-18 + 2q-16 - q-14 + q-12 + 3q-10 + 5q-6 + q-4 + 5q-2 + 4 + 2q2 + 5q4 + q6 + q8 + q10

HOMFLY-PT Polynomial:

a-2z-2 + 2a-2 + a-2z2 - 2z-2 - 5 - 5z2 - 2z4 + a2z-2 + 3a2 + 4a2z2 + 3a2z4 + a2z6 + a4 - a4z4 - a6

Kauffman Polynomial:

a-2z-2 - 4a-2 + 6a-2z2 - 4a-2z4 + a-2z6 - 2a-1z-1 + 3a-1z + 2a-1z3 - 5a-1z5 + 2a-1z7 + 2z-2 - 9 + 19z2 - 14z4 + 2z8 - 2az-1 + 7az - 2az3 - 11az5 + 4az7 + az9 + a2z-2 - 6a2 + 11a2z2 - 11a2z4 - 4a2z6 + 5a2z8 + 3a3z - 2a3z3 - 8a3z5 + 5a3z7 + a3z9 + 2a4 - 5a4z2 + 3a4z4 - 2a4z6 + 3a4z8 - 3a5z + 5a5z3 - 2a5z5 + 3a5z7 + 2a6 - 3a6z2 + 4a6z4 + a6z6 - 2a7z + 3a7z3

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                   1 j = 5                 1   j = 3               5 1   j = 1             3 1     j = -1           8 5       j = -3         6 7         j = -5       5 4           j = -7     3 6             j = -9   2 5               j = -11   3                 j = -13 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 353]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 353]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 7, 13, 8], X[4, 13, 1, 14], X[5, 16, 6, 17], > X[8, 4, 9, 3], X[9, 21, 10, 20], X[19, 11, 20, 10], X[17, 22, 18, 15], > X[21, 18, 22, 19], X[15, 14, 16, 5], X[2, 12, 3, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {-10, 4, -8, 9, -7, 6, -9, 8}, > {-4, -1, 2, -5, -6, 7, 11, -2, 3, 10}]
In[6]:=	Jones[L][q]
Out[6]=	2 5 8 11 10 11 2 3 -8 - -- + -- - -- + -- - -- + -- + 6 q - 2 q + q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 2 2 -14 -12 3 5 -4 5 2 4 6 4 - q - --- + --- - q + q + --- + -- + q + -- + 2 q + 5 q + q + 18 16 10 6 2 q q q q q 8 10 > q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 353]][a, z]
Out[8]=	2 2 2 2 4 6 2 1 a 2 z 2 2 4 -5 + -- + 3 a + a - a - -- + ----- + -- - 5 z + -- + 4 a z - 2 z + 2 2 2 2 2 2 a z a z z a 2 4 4 4 2 6 > 3 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 353]][a, z]
Out[9]=	2 4 2 4 6 2 1 a 2 2 a 3 z -9 - -- - 6 a + 2 a + 2 a + -- + ----- + -- - --- - --- + --- + 7 a z + 2 2 2 2 2 a z z a a z a z z 2 3 5 7 2 6 z 2 2 4 2 6 2 > 3 a z - 3 a z - 2 a z + 19 z + ---- + 11 a z - 5 a z - 3 a z + 2 a 3 4 2 z 3 3 3 5 3 7 3 4 4 z 2 4 > ---- - 2 a z - 2 a z + 5 a z + 3 a z - 14 z - ---- - 11 a z + a 2 a 5 6 4 4 6 4 5 z 5 3 5 5 5 z 2 6 > 3 a z + 4 a z - ---- - 11 a z - 8 a z - 2 a z + -- - 4 a z - a 2 a 7 4 6 6 6 2 z 7 3 7 5 7 8 2 8 > 2 a z + a z + ---- + 4 a z + 5 a z + 3 a z + 2 z + 5 a z + a 4 8 9 3 9 > 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 8 2 3 2 5 3 6 5 4 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 3 q 13 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q q t q t q t q t q t q t q t q t 6 5 t 2 3 2 3 3 5 3 7 4 > ---- + --- + 3 q t + q t + 5 q t + q t + q t + q t 3 q q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n353

L11n352 L11n354