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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X8,9,1,10 X20,15,21,16 X16,8,17,7 X18,6,19,5 X6,18,7,17 X4,20,5,19 X22,13,9,14 X14,21,15,22

Gauss Code:

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

Jones Polynomial:

q-17/2 - 2q-15/2 + 4q-13/2 - 7q-11/2 + 8q-9/2 - 9q-7/2 + 8q-5/2 - 7q-3/2 + 5q-1/2 - 3q1/2 + q3/2 - q5/2

A2 (sl(3)) Invariant:

- q-26 - q-20 + 2q-18 + q-14 + q-12 - q-10 + 2q-8 - q-6 + q-4 + q-2 + 2q2 + q4 + q6 + q8

HOMFLY-PT Polynomial:

- a-1z-1 - 3a-1z - a-1z3 + az-1 + 2az + 3az3 + az5 + 3a3z + 6a3z3 + 2a3z5 + 2a5z3 + a5z5 - 2a7z - a7z3

Kauffman Polynomial:

- a-1z-1 + 6a-1z - 11a-1z3 + 6a-1z5 - a-1z7 + 1 - 2z2 - 3z4 + 4z6 - z8 - az-1 + 4az - 5az3 + 3az7 - az9 - 10a2z2 + 22a2z4 - 14a2z6 + 5a2z8 - a2z10 - 6a3z + 27a3z3 - 26a3z5 + 13a3z7 - 3a3z9 - 3a4z2 + 11a4z4 - 8a4z6 + 3a4z8 - a4z10 + 3a5z3 - 7a5z5 + 5a5z7 - 2a5z9 - 8a6z4 + 7a6z6 - 3a6z8 + 4a7z - 15a7z3 + 11a7z5 - 4a7z7 - 3a8z2 + 5a8z4 - 3a8z6 + 3a9z3 - 2a9z5 + 2a10z2 - a10z4

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 = 6                       1 j = 4                         j = 2                   3 1   j = 0                 2       j = -2               5 3       j = -4             5 4         j = -6           4 3           j = -8         4 5             j = -10       3 4               j = -12     1 4                 j = -14   1 3                   j = -16   1                     j = -18 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, 319]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 319]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[8, 9, 1, 10], > X[20, 15, 21, 16], X[16, 8, 17, 7], X[18, 6, 19, 5], X[6, 18, 7, 17], > X[4, 20, 5, 19], X[22, 13, 9, 14], X[14, 21, 15, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -9, 7, -8, 6, -4}, > {4, -1, 2, -3, 10, -11, 5, -6, 8, -7, 9, -5, 11, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 2 4 7 8 9 8 7 5 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q 3/2 5/2 > 3 Sqrt[q] + q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -20 2 -14 -12 -10 2 -6 -4 -2 2 4 -q - q + --- + q + q - q + -- - q + q + q + 2 q + q + 18 8 q q 6 8 > q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 319]][a, z]
Out[8]=	3 1 a 3 z 3 7 z 3 3 3 5 3 -(---) + - - --- + 2 a z + 3 a z - 2 a z - -- + 3 a z + 6 a z + 2 a z - a z z a a 7 3 5 3 5 5 5 > a z + a z + 2 a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 319]][a, z]
Out[9]=	1 a 6 z 3 7 2 2 2 4 2 1 - --- - - + --- + 4 a z - 6 a z + 4 a z - 2 z - 10 a z - 3 a z - a z z a 3 8 2 10 2 11 z 3 3 3 5 3 7 3 > 3 a z + 2 a z - ----- - 5 a z + 27 a z + 3 a z - 15 a z + a 5 9 3 4 2 4 4 4 6 4 8 4 10 4 6 z > 3 a z - 3 z + 22 a z + 11 a z - 8 a z + 5 a z - a z + ---- - a 3 5 5 5 7 5 9 5 6 2 6 4 6 > 26 a z - 7 a z + 11 a z - 2 a z + 4 z - 14 a z - 8 a z + 7 6 6 8 6 z 7 3 7 5 7 7 7 8 > 7 a z - 3 a z - -- + 3 a z + 13 a z + 5 a z - 4 a z - z + a 2 8 4 8 6 8 9 3 9 5 9 2 10 4 10 > 5 a z + 3 a z - 3 a z - a z - 3 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 5 1 1 1 3 1 4 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 10 4 q q q t q t q t q t q t q t q t 4 4 5 4 3 5 3 t 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 2 t + --- + 3 q t + 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q 2 3 6 4 > q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a319

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