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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X16,7,17,8 X8,15,5,16 X11,19,12,18 X17,9,18,22 X21,13,22,12 X13,21,14,20 X19,15,20,14 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-6 - 2q-5 + 6q-4 - 6q-3 + 9q-2 - 8q-1 + 9 - 6q + 3q2 - 2q3

A2 (sl(3)) Invariant:

q-20 + 2q-18 + q-16 + 5q-14 + 6q-12 + 5q-10 + 8q-8 + 4q-6 + 4q-4 + 2q-2 - 1 - 5q4 - 2q6 - q8 - 2q10

HOMFLY-PT Polynomial:

- a-2z-2 - 3a-2 - 2a-2z2 + 3z-2 + 7 + 5z2 + 2z4 - 2a2z-2 - a2 + 3a2z2 + 2a2z4 - a4z-2 - 4a4 - 3a4z2 + a6z-2 + a6

Kauffman Polynomial:

2a-3z-1 - 7a-3z + 3a-3z3 - a-2z-2 + a-2 - a-2z2 + a-2z6 + 8a-1z-1 - 27a-1z + 35a-1z3 - 17a-1z5 + 4a-1z7 - 3z-2 + 5 - 9z2 + 23z4 - 15z6 + 4z8 + 10az-1 - 34az + 50az3 - 29az5 + 5az7 + az9 - 2a2z-2 + 4a2 - 8a2z2 + 19a2z4 - 19a2z6 + 6a2z8 + 2a3z-1 - 10a3z + 18a3z3 - 17a3z5 + 3a3z7 + a3z9 + a4z-2 - 3a4 + 6a4z2 - 8a4z4 - 2a4z6 + 2a4z8 - 2a5z-1 + 4a5z - 5a5z5 + 2a5z7 + a6z-2 - 4a6 + 6a6z2 - 4a6z4 + a6z6

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 7                   2 j = 5                 1   j = 3               5 2   j = 1             4 1     j = -1           5 6       j = -3         4 4 1       j = -5       2 5           j = -7     4 4             j = -9   1 5               j = -11   1                 j = -13 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, 273]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 273]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 7, 17, 8], X[8, 15, 5, 16], > X[11, 19, 12, 18], X[17, 9, 18, 22], X[21, 13, 22, 12], X[13, 21, 14, 20], > X[19, 15, 20, 14], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -4}, > {11, -2, -5, 7, -8, 9, 4, -3, -6, 5, -9, 8, -7, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-6 2 6 6 9 8 2 3 9 + q - -- + -- - -- + -- - - - 6 q + 3 q - 2 q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 2 -16 5 6 5 8 4 4 2 4 6 -1 + q + --- + q + --- + --- + --- + -- + -- + -- + -- - 5 q - 2 q - 18 14 12 10 8 6 4 2 q q q q q q q q 8 10 > q - 2 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 273]][a, z]
Out[8]=	2 4 6 2 3 2 4 6 3 1 2 a a a 2 2 z 2 2 7 - -- - a - 4 a + a + -- - ----- - ---- - -- + -- + 5 z - ---- + 3 a z - 2 2 2 2 2 2 2 2 a z a z z z z a 4 2 4 2 4 > 3 a z + 2 z + 2 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 273]][a, z]
Out[9]=	2 4 6 -2 2 4 6 3 1 2 a a a 2 8 5 + a + 4 a - 3 a - 4 a - -- - ----- - ---- + -- + -- + ---- + --- + 2 2 2 2 2 2 3 a z z a z z z z a z 3 5 2 10 a 2 a 2 a 7 z 27 z 3 5 2 z > ---- + ---- - ---- - --- - ---- - 34 a z - 10 a z + 4 a z - 9 z - -- - z z z 3 a 2 a a 3 3 2 2 4 2 6 2 3 z 35 z 3 3 3 4 > 8 a z + 6 a z + 6 a z + ---- + ----- + 50 a z + 18 a z + 23 z + 3 a a 5 2 4 4 4 6 4 17 z 5 3 5 5 5 > 19 a z - 8 a z - 4 a z - ----- - 29 a z - 17 a z - 5 a z - a 6 7 6 z 2 6 4 6 6 6 4 z 7 3 7 > 15 z + -- - 19 a z - 2 a z + a z + ---- + 5 a z + 3 a z + 2 a a 5 7 8 2 8 4 8 9 3 9 > 2 a z + 4 z + 6 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-3 6 1 1 1 5 4 4 2 q + - + 4 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 q t q t q t q t q t q t q t 5 4 4 5 3 3 2 5 2 7 3 > ----- + ----- + ---- + --- + q t + 5 q t + 2 q t + q t + 2 q t 5 2 3 2 3 q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n273

L11n272 L11n274