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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X6718 X18,15,19,16 X16,6,17,5 X4,18,5,17 X20,11,21,12 X22,13,7,14 X12,21,13,22 X14,19,15,20

Gauss Code:

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

Jones Polynomial:

q-23/2 - 3q-21/2 + 6q-19/2 - 9q-17/2 + 11q-15/2 - 12q-13/2 + 11q-11/2 - 9q-9/2 + 6q-7/2 - 4q-5/2 + q-3/2 - q-1/2

A2 (sl(3)) Invariant:

- q-34 + q-32 - q-30 + q-26 - 2q-24 + 2q-22 - 2q-20 + q-18 + q-16 - q-14 + 3q-12 + 3q-8 + 2q-6 + q-4 + q-2

HOMFLY-PT Polynomial:

- 2a3z-1 - 7a3z - 5a3z3 - a3z5 + 3a5z-1 + 8a5z + 9a5z3 + 5a5z5 + a5z7 - a7z-1 + 4a7z3 + 4a7z5 + a7z7 - 2a9z - 3a9z3 - a9z5

Kauffman Polynomial:

- 2a3z-1 + 9a3z - 12a3z3 + 6a3z5 - a3z7 + 3a4 - 7a4z2 + a4z4 + 3a4z6 - a4z8 - 3a5z-1 + 11a5z - 16a5z3 + 8a5z5 + a5z7 - a5z9 + 3a6 - 15a6z2 + 19a6z4 - 8a6z6 + 3a6z8 - a6z10 - a7z-1 - 2a7z + 15a7z3 - 18a7z5 + 13a7z7 - 4a7z9 + a8 - 2a8z2 + 2a8z4 + 3a8z6 - a8z8 - a8z10 - 3a9z + 8a9z3 - 7a9z5 + 5a9z7 - 3a9z9 + 3a10z2 - 9a10z4 + 9a10z6 - 5a10z8 + a11z - 8a11z3 + 10a11z5 - 6a11z7 - 2a12z2 + 6a12z4 - 5a12z6 + 3a13z3 - 3a13z5 + a14z2 - a14z4

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 0                       1 j = -2                         j = -4                   4 1   j = -6                 3 1     j = -8               6 3       j = -10             6 4         j = -12           6 5           j = -14         5 6             j = -16       4 6               j = -18     2 5                 j = -20   1 4                   j = -22   2                     j = -24 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, 205]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 205]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[6, 7, 1, 8], > X[18, 15, 19, 16], X[16, 6, 17, 5], X[4, 18, 5, 17], X[20, 11, 21, 12], > X[22, 13, 7, 14], X[12, 21, 13, 22], X[14, 19, 15, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -7, 6, -4}, > {4, -1, 2, -3, 8, -10, 9, -11, 5, -6, 7, -5, 11, -8, 10, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) 3 6 9 11 12 11 9 6 4 q - ----- + ----- - ----- + ----- - ----- + ----- - ---- + ---- - ---- + 21/2 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 q q q q q q q q q -(3/2) 1 > q - ------- Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 -32 -30 -26 2 2 2 -18 -16 -14 3 3 -q + q - q + q - --- + --- - --- + q + q - q + --- + -- + 24 22 20 12 8 q q q q q 2 -4 -2 > -- + q + q 6 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 205]][a, z]
Out[8]=	3 5 7 -2 a 3 a a 3 5 9 3 3 5 3 7 3 ----- + ---- - -- - 7 a z + 8 a z - 2 a z - 5 a z + 9 a z + 4 a z - z z z 9 3 3 5 5 5 7 5 9 5 5 7 7 7 > 3 a z - a z + 5 a z + 4 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 205]][a, z]
Out[9]=	3 5 7 4 6 8 2 a 3 a a 3 5 7 9 3 a + 3 a + a - ---- - ---- - -- + 9 a z + 11 a z - 2 a z - 3 a z + z z z 11 4 2 6 2 8 2 10 2 12 2 14 2 > a z - 7 a z - 15 a z - 2 a z + 3 a z - 2 a z + a z - 3 3 5 3 7 3 9 3 11 3 13 3 4 4 > 12 a z - 16 a z + 15 a z + 8 a z - 8 a z + 3 a z + a z + 6 4 8 4 10 4 12 4 14 4 3 5 5 5 > 19 a z + 2 a z - 9 a z + 6 a z - a z + 6 a z + 8 a z - 7 5 9 5 11 5 13 5 4 6 6 6 8 6 > 18 a z - 7 a z + 10 a z - 3 a z + 3 a z - 8 a z + 3 a z + 10 6 12 6 3 7 5 7 7 7 9 7 11 7 > 9 a z - 5 a z - a z + a z + 13 a z + 5 a z - 6 a z - 4 8 6 8 8 8 10 8 5 9 7 9 9 9 6 10 > a z + 3 a z - a z - 5 a z - a z - 4 a z - 3 a z - a z - 8 10 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 4 1 2 1 4 2 5 4 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 24 9 22 8 20 8 20 7 18 7 18 6 16 6 q q t q t q t q t q t q t q t 6 5 6 6 5 6 4 6 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 16 5 14 5 14 4 12 4 12 3 10 3 10 2 8 2 q t q t q t q t q t q t q t q t 3 3 t 2 > ---- + ---- + -- + t 8 6 4 q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a205

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