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DrawMorseLink

PD Presentation:

X6172 X14,3,15,4 X9,22,10,5 X7,19,8,18 X17,9,18,8 X19,13,20,12 X11,21,12,20 X15,10,16,11 X21,16,22,17 X2536 X4,13,1,14

Gauss Code:

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

Jones Polynomial:

q-15/2 - 3q-13/2 + 4q-11/2 - 6q-9/2 + 5q-7/2 - 5q-5/2 + 4q-3/2 - 2q-1/2 + q3/2 - q5/2

A2 (sl(3)) Invariant:

- q-24 + 2q-20 + 3q-16 + 2q-14 + q-12 + 2q-10 - q-8 + q-6 - 2q-4 - q-2 + 1 - q2 + q4 + q6 + q8

HOMFLY-PT Polynomial:

- a-1z-1 - 3a-1z - a-1z3 + 3az-1 + 7az + 6az3 + az5 - 4a3z-1 - 8a3z - 4a3z3 - a3z5 + 2a5z-1 + 3a5z + 2a5z3 - a7z

Kauffman Polynomial:

- a-1z-1 + 5a-1z - 9a-1z3 + 6a-1z5 - a-1z7 - 1 + 7z2 - 12z4 + 7z6 - z8 - 3az-1 + 16az - 25az3 + 11az5 - az7 - 3a2 + 15a2z2 - 27a2z4 + 14a2z6 - 2a2z8 - 4a3z-1 + 23a3z - 34a3z3 + 12a3z5 + 2a3z7 - a3z9 - 2a4 + 10a4z2 - 23a4z4 + 18a4z6 - 4a4z8 - 2a5z-1 + 15a5z - 27a5z3 + 18a5z5 - a5z7 - a5z9 - a6 + a6z2 - 5a6z4 + 10a6z6 - 3a6z8 + 3a7z - 9a7z3 + 11a7z5 - 3a7z7 - a8z2 + 3a8z4 - 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 = 6                       1 j = 4                         j = 2                 1 1 1   j = 0               3 1       j = -2             3 2 1       j = -4           3 3 1         j = -6         3 3             j = -8       3 3 1             j = -10     2 4                 j = -12   1 2                   j = -14   2                     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, NonAlternating, 125]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 125]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[9, 22, 10, 5], X[7, 19, 8, 18], > X[17, 9, 18, 8], X[19, 13, 20, 12], X[11, 21, 12, 20], X[15, 10, 16, 11], > X[21, 16, 22, 17], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -4, 5, -3, 8, -7, 6, 11, -2, -8, 9, -5, 4, > -6, 7, -9, 3}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 4 6 5 5 4 2 3/2 5/2 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + q - q 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 2 3 2 -12 2 -8 -6 2 -2 2 4 6 1 - q + --- + --- + --- + q + --- - q + q - -- - q - q + q + q + 20 16 14 10 4 q q q q q 8 > q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 125]][a, z]
Out[8]=	3 5 3 1 3 a 4 a 2 a 3 z 3 5 7 z -(---) + --- - ---- + ---- - --- + 7 a z - 8 a z + 3 a z - a z - -- + a z z z z a a 3 3 3 5 3 5 3 5 > 6 a z - 4 a z + 2 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 125]][a, z]
Out[9]=	3 5 2 4 6 1 3 a 4 a 2 a 5 z 3 -1 - 3 a - 2 a - a - --- - --- - ---- - ---- + --- + 16 a z + 23 a z + a z z z z a 3 5 7 2 2 2 4 2 6 2 8 2 9 z > 15 a z + 3 a z + 7 z + 15 a z + 10 a z + a z - a z - ---- - a 3 3 3 5 3 7 3 4 2 4 4 4 > 25 a z - 34 a z - 27 a z - 9 a z - 12 z - 27 a z - 23 a z - 5 6 4 8 4 6 z 5 3 5 5 5 7 5 > 5 a z + 3 a z + ---- + 11 a z + 12 a z + 18 a z + 11 a z + a 7 6 2 6 4 6 6 6 8 6 z 7 3 7 > 7 z + 14 a z + 18 a z + 10 a z - a z - -- - a z + 2 a z - a 5 7 7 7 8 2 8 4 8 6 8 3 9 5 9 > a z - 3 a z - z - 2 a z - 4 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 2 1 2 1 2 2 4 3 3 + q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 q q t q t q t q t q t q t q t 3 3 1 3 3 3 3 t 2 > ----- + ----- + ----- + ----- + ----- + ---- + ---- + t + -- + q t + 8 3 6 3 8 2 6 2 4 2 4 2 2 q t q t q t q t q t q t q t q 2 2 2 3 6 4 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n125

L11n124 L11n126