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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X15,18,16,11 X9,21,10,20 X13,19,14,22 X21,15,22,14 X19,5,20,10 X17,8,18,9 X7,16,8,17 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-24 + 2q-20 + 3q-16 + 3q-14 + 4q-12 + 9q-10 + 10q-8 + 14q-6 + 11q-4 + 9q-2 + 8 + 3q2 + 3q4 + 2q6 + q8

HOMFLY-PT Polynomial:

- a-1z-3 - 3a-1z-1 - 4a-1z - a-1z3 + 3az-3 + 8az-1 + 11az + 7az3 + az5 - 3a3z-3 - 7a3z-1 - 9a3z - 4a3z3 - a3z5 + a5z-3 + 2a5z-1 + 3a5z + 2a5z3 - a7z

Kauffman Polynomial:

a-1z-3 - 5a-1z-1 + 11a-1z - 14a-1z3 + 7a-1z5 - a-1z7 - 3z-2 + 10 - 12z2 + 4z4 + 3az-3 - 12az-1 + 23az - 18az3 + 4az5 - 6a2z-2 + 19a2 - 15a2z2 - 4a2z4 + 5a2z6 - a2z8 + 3a3z-3 - 12a3z-1 + 21a3z - 17a3z3 + 2a3z5 + 3a3z7 - a3z9 - 3a4z-2 + 10a4 - 3a4z2 - 16a4z4 + 16a4z6 - 4a4z8 + a5z-3 - 5a5z-1 + 12a5z - 22a5z3 + 16a5z5 - a5z7 - a5z9 - 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                       1 j = 2                 1 1     j = 0               4         j = -2             2 4 1       j = -4           6 2           j = -6         3 6 1           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]=	4
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 456]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 456]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[15, 18, 16, 11], X[9, 21, 10, 20], > X[13, 19, 14, 22], X[21, 15, 22, 14], X[19, 5, 20, 10], X[17, 8, 18, 9], > X[7, 16, 8, 17], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {-7, 4, -6, 5}, {10, -1, -9, 8, -4, 7}, > {11, -2, -5, 6, -3, 9, -8, 3}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 4 6 5 7 3 4 5/2 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 3 4 9 10 14 11 9 2 4 8 - q + --- + --- + --- + --- + --- + -- + -- + -- + -- + 3 q + 3 q + 20 16 14 12 10 8 6 4 2 q q q q q q q q q 6 8 > 2 q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 456]][a, z]
Out[8]=	3 5 3 5 1 3 a 3 a a 3 8 a 7 a 2 a 4 z 3 -(----) + --- - ---- + -- - --- + --- - ---- + ---- - --- + 11 a z - 9 a z + 3 3 3 3 a z z z z a a z z z z 3 5 7 z 3 3 3 5 3 5 3 5 > 3 a z - a z - -- + 7 a z - 4 a z + 2 a z + a z - a z a
In[9]:=	Kauffman[Link[11, NonAlternating, 456]][a, z]
Out[9]=	3 5 2 4 2 4 1 3 a 3 a a 3 6 a 3 a 5 12 a 10 + 19 a + 10 a + ---- + --- + ---- + -- - -- - ---- - ---- - --- - ---- - 3 3 3 3 2 2 2 a z z a z z z z z z z 3 5 12 a 5 a 11 z 3 5 7 2 > ----- - ---- + ---- + 23 a z + 21 a z + 12 a z + 3 a z - 12 z - z z a 3 2 2 4 2 6 2 8 2 14 z 3 3 3 > 15 a z - 3 a z - a z - a z - ----- - 18 a z - 17 a z - a 5 5 3 7 3 4 2 4 4 4 6 4 8 4 7 z > 22 a z - 9 a z + 4 z - 4 a z - 16 a z - 5 a z + 3 a z + ---- + a 5 3 5 5 5 7 5 2 6 4 6 6 6 > 4 a z + 2 a z + 16 a z + 11 a z + 5 a z + 16 a z + 10 a z - 7 8 6 z 3 7 5 7 7 7 2 8 4 8 6 8 > a z - -- + 3 a z - a z - 3 a z - a z - 4 a z - 3 a z - a 3 9 5 9 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 2 1 2 2 4 3 3 4 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 3 1 6 6 1 2 2 t 2 2 2 > ----- + ----- + ----- + ----- + ---- + ---- + ---- + -- + q t + q t + 6 3 8 2 6 2 4 2 6 4 2 2 q t q t q t q t q t q t q t q 4 4 6 4 > q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n456

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