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DrawMorseLink

PD Presentation:

X8192 X10,3,11,4 X14,6,15,5 X16,11,17,12 X20,13,21,14 X22,17,7,18 X18,21,19,22 X12,19,13,20 X4,16,5,15 X2738 X6,9,1,10

Gauss Code:

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

Jones Polynomial:

q-21/2 - 3q-19/2 + 7q-17/2 - 11q-15/2 + 15q-13/2 - 18q-11/2 + 17q-9/2 - 16q-7/2 + 11q-5/2 - 7q-3/2 + 3q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-32 + q-30 - q-28 - 4q-26 + 2q-24 - 2q-22 + 4q-18 + 5q-14 + 2q-10 + 3q-8 - 3q-6 + 3q-4 - 1 + q2

HOMFLY-PT Polynomial:

- az - az3 - a3z-1 - 2a3z + a3z5 + a5z + 3a5z3 + 2a5z5 + 2a7z-1 + 2a7z + a7z3 + a7z5 - a9z-1 - a9z - a9z3

Kauffman Polynomial:

- az + 2az3 - az5 - 2a2z2 + 5a2z4 - 3a2z6 - a3z-1 + 4a3z - 6a3z3 + 8a3z5 - 5a3z7 + a4 - 3a4z2 + 2a4z4 + 4a4z6 - 5a4z8 + 4a5z - 15a5z3 + 17a5z5 - 5a5z7 - 3a5z9 - 3a6 + 7a6z2 - 15a6z4 + 19a6z6 - 9a6z8 - a6z10 + 2a7z-1 - 5a7z + 5a7z3 - 5a7z5 + 10a7z7 - 7a7z9 - 5a8 + 18a8z2 - 29a8z4 + 27a8z6 - 9a8z8 - a8z10 + a9z-1 - 4a9z + 7a9z3 - 5a9z5 + 7a9z7 - 4a9z9 - 2a10 + 8a10z2 - 14a10z4 + 14a10z6 - 5a10z8 - 5a11z3 + 8a11z5 - 3a11z7 - 2a12z2 + 3a12z4 - a12z6

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 = 2                       1 j = 0                     2   j = -2                   5 1   j = -4                 7 3     j = -6               9 4       j = -8             9 8         j = -10           9 8           j = -12         6 9             j = -14       5 9               j = -16     2 6                 j = -18   1 5                   j = -20   2                     j = -22 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, 151]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 151]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[14, 6, 15, 5], X[16, 11, 17, 12], > X[20, 13, 21, 14], X[22, 17, 7, 18], X[18, 21, 19, 22], X[12, 19, 13, 20], > X[4, 16, 5, 15], X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -9, 3, -11}, > {10, -1, 11, -2, 4, -8, 5, -3, 9, -4, 6, -7, 8, -5, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 3 7 11 15 18 17 16 11 7 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q 3 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -30 -28 4 2 2 4 5 2 3 3 3 2 -1 - q + q - q - --- + --- - --- + --- + --- + --- + -- - -- + -- + q 26 24 22 18 14 10 8 6 4 q q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 151]][a, z]
Out[8]=	3 7 9 a 2 a a 3 5 7 9 3 5 3 -(--) + ---- - -- - a z - 2 a z + a z + 2 a z - a z - a z + 3 a z + z z z 7 3 9 3 3 5 5 5 7 5 > a z - a z + a z + 2 a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 151]][a, z]
Out[9]=	3 7 9 4 6 8 10 a 2 a a 3 5 7 a - 3 a - 5 a - 2 a - -- + ---- + -- - a z + 4 a z + 4 a z - 5 a z - z z z 9 2 2 4 2 6 2 8 2 10 2 12 2 > 4 a z - 2 a z - 3 a z + 7 a z + 18 a z + 8 a z - 2 a z + 3 3 3 5 3 7 3 9 3 11 3 2 4 > 2 a z - 6 a z - 15 a z + 5 a z + 7 a z - 5 a z + 5 a z + 4 4 6 4 8 4 10 4 12 4 5 3 5 > 2 a z - 15 a z - 29 a z - 14 a z + 3 a z - a z + 8 a z + 5 5 7 5 9 5 11 5 2 6 4 6 6 6 > 17 a z - 5 a z - 5 a z + 8 a z - 3 a z + 4 a z + 19 a z + 8 6 10 6 12 6 3 7 5 7 7 7 9 7 > 27 a z + 14 a z - a z - 5 a z - 5 a z + 10 a z + 7 a z - 11 7 4 8 6 8 8 8 10 8 5 9 7 9 > 3 a z - 5 a z - 9 a z - 9 a z - 5 a z - 3 a z - 7 a z - 9 9 6 10 8 10 > 4 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 1 2 1 5 2 6 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q q t q t q t q t q t q t q t 9 6 9 9 8 9 8 9 4 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q t q t q t q t q t q t q t q t q t 7 t 2 2 > ---- + 2 t + -- + q t 4 2 q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a151

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