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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-17/2 - 3q-15/2 + 7q-13/2 - 13q-11/2 + 17q-9/2 - 19q-7/2 + 19q-5/2 - 17q-3/2 + 12q-1/2 - 8q1/2 + 3q3/2 - q5/2

A2 (sl(3)) Invariant:

- q-26 + q-22 - 2q-20 + 4q-18 - q-16 - q-14 + 2q-12 - 4q-10 + 4q-8 - 2q-6 + 2q-4 + 3q-2 - 1 + 4q2 + q8

HOMFLY-PT Polynomial:

- a-1z-1 - 2a-1z - a-1z3 + az-1 + 3az + 5az3 + 2az5 - 2a3z - 3a3z3 - 3a3z5 - a3z7 + 3a5z + 5a5z3 + 2a5z5 - 2a7z - a7z3

Kauffman Polynomial:

- a-1z-1 + 4a-1z - 6a-1z3 + 4a-1z5 - a-1z7 + 1 + 2z2 - 10z4 + 10z6 - 3z8 - az-1 + 6az - 12az3 + 2az5 + 9az7 - 4az9 + 6a2z2 - 26a2z4 + 29a2z6 - 5a2z8 - 2a2z10 + 4a3z - 11a3z3 - 2a3z5 + 24a3z7 - 11a3z9 + 12a4z2 - 39a4z4 + 46a4z6 - 13a4z8 - 2a4z10 + 6a5z - 23a5z3 + 20a5z5 + 4a5z7 - 7a5z9 + 5a6z2 - 17a6z4 + 21a6z6 - 11a6z8 + 4a7z - 16a7z3 + 17a7z5 - 10a7z7 - 2a8z2 + 5a8z4 - 6a8z6 + 2a9z3 - 3a9z5 + a10z2 - 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                     2   j = 2                   6 1   j = 0                 6 2     j = -2               11 6       j = -4             10 8         j = -6           9 9           j = -8         8 10             j = -10       5 9               j = -12     2 8                 j = -14   1 5                   j = -16   2                     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, 354]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 354]]
Out[4]=	PD[X[12, 1, 13, 2], X[2, 13, 3, 14], X[8, 3, 9, 4], X[16, 11, 17, 12], > X[14, 8, 15, 7], X[6, 16, 7, 15], X[22, 17, 11, 18], X[4, 20, 5, 19], > X[18, 6, 19, 5], X[20, 9, 21, 10], X[10, 21, 1, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -8, 9, -6, 5, -3, 10, -11}, > {4, -1, 2, -5, 6, -4, 7, -9, 8, -10, 11, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 3 7 13 17 19 19 17 12 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 > 8 Sqrt[q] + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -22 2 4 -16 -14 2 4 4 2 2 3 -1 - q + q - --- + --- - q - q + --- - --- + -- - -- + -- + -- + 20 18 12 10 8 6 4 2 q q q q q q q q 2 8 > 4 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 354]][a, z]
Out[8]=	3 1 a 2 z 3 5 7 z 3 3 3 -(---) + - - --- + 3 a z - 2 a z + 3 a z - 2 a z - -- + 5 a z - 3 a z + a z z a a 5 3 7 3 5 3 5 5 5 3 7 > 5 a z - a z + 2 a z - 3 a z + 2 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 354]][a, z]
Out[9]=	1 a 4 z 3 5 7 2 2 2 1 - --- - - + --- + 6 a z + 4 a z + 6 a z + 4 a z + 2 z + 6 a z + a z z a 3 4 2 6 2 8 2 10 2 6 z 3 3 3 > 12 a z + 5 a z - 2 a z + a z - ---- - 12 a z - 11 a z - a 5 3 7 3 9 3 4 2 4 4 4 6 4 > 23 a z - 16 a z + 2 a z - 10 z - 26 a z - 39 a z - 17 a z + 5 8 4 10 4 4 z 5 3 5 5 5 7 5 > 5 a z - a z + ---- + 2 a z - 2 a z + 20 a z + 17 a z - a 7 9 5 6 2 6 4 6 6 6 8 6 z 7 > 3 a z + 10 z + 29 a z + 46 a z + 21 a z - 6 a z - -- + 9 a z + a 3 7 5 7 7 7 8 2 8 4 8 6 8 > 24 a z + 4 a z - 10 a z - 3 z - 5 a z - 13 a z - 11 a z - 9 3 9 5 9 2 10 4 10 > 4 a z - 11 a z - 7 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	8 11 1 2 1 5 2 8 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 9 8 10 9 9 10 6 t 2 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 6 t + --- + 2 t + 6 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 4 3 6 4 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a354

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