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DrawMorseLink

PD Presentation:

X12,1,13,2 X18,9,19,10 X14,6,15,5 X6,12,7,11 X22,15,11,16 X20,8,21,7 X8394 X16,21,17,22 X4,18,5,17 X10,13,1,14 X2,19,3,20

Gauss Code:

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

Jones Polynomial:

q-15/2 - 4q-13/2 + 10q-11/2 - 18q-9/2 + 25q-7/2 - 30q-5/2 + 30q-3/2 - 27q-1/2 + 20q1/2 - 13q3/2 + 5q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-24 + 3q-20 - 3q-18 + 4q-14 - 6q-12 + 4q-10 - q-8 + 5q-4 - 4q-2 + 8 - 3q2 + 5q6 - 3q8 + q10

HOMFLY-PT Polynomial:

- a-1z-1 - 2a-1z - a-1z3 - a-1z5 + az-1 + 4az + 6az3 + 3az5 + az7 - 3a3z - 5a3z3 - 3a3z5 + 2a5z + 3a5z3 - a7z

Kauffman Polynomial:

- a-3z5 + 2a-2z4 - 5a-2z6 - a-1z-1 + 3a-1z - 7a-1z3 + 16a-1z5 - 13a-1z7 + 1 + 2z2 - 12z4 + 27z6 - 18z8 - az-1 + 4az - 13az3 + 19az5 + 6az7 - 13az9 + 4a2z2 - 33a2z4 + 59a2z6 - 24a2z8 - 4a2z10 - 3a3z3 - 9a3z5 + 36a3z7 - 22a3z9 + 7a4z2 - 33a4z4 + 44a4z6 - 14a4z8 - 4a4z10 - 2a5z3 - 3a5z5 + 13a5z7 - 9a5z9 + 4a6z2 - 12a6z4 + 16a6z6 - 8a6z8 + a7z - 5a7z3 + 8a7z5 - 4a7z7 - a8z2 + 2a8z4 - 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                   9 1   j = 2                 11 4     j = 0               16 9       j = -2             16 13         j = -4           14 14           j = -6         11 16             j = -8       7 14               j = -10     3 11                 j = -12   1 7                   j = -14   3                     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, 351]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 351]]
Out[4]=	PD[X[12, 1, 13, 2], X[18, 9, 19, 10], X[14, 6, 15, 5], X[6, 12, 7, 11], > X[22, 15, 11, 16], X[20, 8, 21, 7], X[8, 3, 9, 4], X[16, 21, 17, 22], > X[4, 18, 5, 17], X[10, 13, 1, 14], X[2, 19, 3, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 7, -9, 3, -4, 6, -7, 2, -10}, > {4, -1, 10, -3, 5, -8, 9, -2, 11, -6, 8, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 4 10 18 25 30 30 27 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 > 13 q + 5 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 3 3 4 6 4 -8 5 4 2 6 8 8 - q + --- - --- + --- - --- + --- - q + -- - -- - 3 q + 5 q - 3 q + 20 18 14 12 10 4 2 q q q q q q q 10 > q
In[8]:=	HOMFLYPT[Link[11, Alternating, 351]][a, z]
Out[8]=	3 1 a 2 z 3 5 7 z 3 3 3 -(---) + - - --- + 4 a z - 3 a z + 2 a z - a z - -- + 6 a z - 5 a z + a z z a a 5 5 3 z 5 3 5 7 > 3 a z - -- + 3 a z - 3 a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 351]][a, z]
Out[9]=	1 a 3 z 7 2 2 2 4 2 6 2 8 2 1 - --- - - + --- + 4 a z + a z + 2 z + 4 a z + 7 a z + 4 a z - a z - a z z a 3 4 7 z 3 3 3 5 3 7 3 4 2 z 2 4 > ---- - 13 a z - 3 a z - 2 a z - 5 a z - 12 z + ---- - 33 a z - a 2 a 5 5 4 4 6 4 8 4 z 16 z 5 3 5 5 5 > 33 a z - 12 a z + 2 a z - -- + ----- + 19 a z - 9 a z - 3 a z + 3 a a 6 7 7 5 6 5 z 2 6 4 6 6 6 8 6 13 z > 8 a z + 27 z - ---- + 59 a z + 44 a z + 16 a z - a z - ----- + 2 a a 7 3 7 5 7 7 7 8 2 8 4 8 > 6 a z + 36 a z + 13 a z - 4 a z - 18 z - 24 a z - 14 a z - 6 8 9 3 9 5 9 2 10 4 10 > 8 a z - 13 a z - 22 a z - 9 a z - 4 a z - 4 a z
In[10]:=	Kh[L][q, t]
Out[10]=	13 1 3 1 7 3 11 7 14 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 11 16 14 14 16 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 9 t + 11 q t + 4 q t + 9 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 L11a351

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