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DrawMorseLink

PD Presentation:

X8192 X16,7,17,8 X10,4,11,3 X2,15,3,16 X14,10,15,9 X18,11,19,12 X12,5,13,6 X6,21,1,22 X20,14,21,13 X22,17,7,18 X4,20,5,19

Gauss Code:

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

Jones Polynomial:

q-15/2 - 5q-13/2 + 12q-11/2 - 20q-9/2 + 27q-7/2 - 32q-5/2 + 31q-3/2 - 28q-1/2 + 20q1/2 - 12q3/2 + 5q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-22 + 3q-20 - 3q-18 + 4q-14 - 6q-12 + 6q-10 - q-8 + 2q-6 + 5q-4 - 4q-2 + 7 - 4q2 + 3q6 - 3q8 + q10

HOMFLY-PT Polynomial:

- a-1z3 - a-1z5 - az-1 - az + az3 + 2az5 + az7 + a3z-1 + a3z3 + 2a3z5 + a3z7 - a5z3 - a5z5

Kauffman Polynomial:

- a-3z5 + 3a-2z4 - 5a-2z6 - 5a-1z3 + 14a-1z5 - 12a-1z7 + 2z2 - 11z4 + 24z6 - 17z8 + az-1 - az - 6az3 + 9az5 + 11az7 - 14az9 - a2 + 6a2z2 - 36a2z4 + 59a2z6 - 22a2z8 - 5a2z10 + a3z-1 - a3z - 18a3z5 + 47a3z7 - 26a3z9 + 6a4z2 - 36a4z4 + 53a4z6 - 16a4z8 - 5a4z10 - 2a5z3 - 4a5z5 + 19a5z7 - 12a5z9 + 2a6z2 - 13a6z4 + 22a6z6 - 11a6z8 - 3a7z3 + 8a7z5 - 5a7z7 + a8z4 - a8z6

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 = 8                       1 j = 6                     4   j = 4                   8 1   j = 2                 12 4     j = 0               16 8       j = -2             16 13         j = -4           16 15           j = -6         12 17             j = -8       8 15               j = -10     4 12                 j = -12   1 8                   j = -14   4                     j = -16 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 150]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 150]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 7, 17, 8], X[10, 4, 11, 3], X[2, 15, 3, 16], > X[14, 10, 15, 9], X[18, 11, 19, 12], X[12, 5, 13, 6], X[6, 21, 1, 22], > X[20, 14, 21, 13], X[22, 17, 7, 18], X[4, 20, 5, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -11, 7, -8}, > {2, -1, 5, -3, 6, -7, 9, -5, 4, -2, 10, -6, 11, -9, 8, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 5 12 20 27 32 31 28 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 20 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 12 q + 5 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 3 3 4 6 6 -8 2 5 4 2 6 7 - q + --- - --- + --- - --- + --- - q + -- + -- - -- - 4 q + 3 q - 20 18 14 12 10 6 4 2 q q q q q q q q 8 10 > 3 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 150]][a, z]
Out[8]=	3 3 5 a a z 3 3 3 5 3 z 5 3 5 5 5 -(-) + -- - a z - -- + a z + a z - a z - -- + 2 a z + 2 a z - a z + z z a a 7 3 7 > a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 150]][a, z]
Out[9]=	3 3 2 a a 3 2 2 2 4 2 6 2 5 z -a + - + -- - a z - a z + 2 z + 6 a z + 6 a z + 2 a z - ---- - z z a 4 3 5 3 7 3 4 3 z 2 4 4 4 > 6 a z - 2 a z - 3 a z - 11 z + ---- - 36 a z - 36 a z - 2 a 5 5 6 4 8 4 z 14 z 5 3 5 5 5 7 5 > 13 a z + a z - -- + ----- + 9 a z - 18 a z - 4 a z + 8 a z + 3 a a 6 7 6 5 z 2 6 4 6 6 6 8 6 12 z 7 > 24 z - ---- + 59 a z + 53 a z + 22 a z - a z - ----- + 11 a z + 2 a a 3 7 5 7 7 7 8 2 8 4 8 6 8 > 47 a z + 19 a z - 5 a z - 17 z - 22 a z - 16 a z - 11 a z - 9 3 9 5 9 2 10 4 10 > 14 a z - 26 a z - 12 a z - 5 a z - 5 a z
In[10]:=	Kh[L][q, t]
Out[10]=	13 1 4 1 8 4 12 8 15 16 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 12 17 16 15 16 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 8 t + 12 q t + 4 q t + 8 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a150

L11a149 L11a151