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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X18,12,19,11 X12,6,13,5 X4,19,5,20 X16,7,17,8 X20,13,21,14 X22,15,7,16 X14,21,15,22 X6,18,1,17

Gauss Code:

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

Jones Polynomial:

- q-21/2 + 3q-19/2 - 5q-17/2 + 8q-15/2 - 10q-13/2 + 11q-11/2 - 11q-9/2 + 9q-7/2 - 8q-5/2 + 4q-3/2 - 3q-1/2 + q1/2

A2 (sl(3)) Invariant:

q-30 - q-28 + q-26 - 2q-24 - q-22 - q-20 - 2q-18 + 3q-16 - q-14 + 4q-12 + 2q-10 + 3q-8 + 3q-6 + q-2 - 1

HOMFLY-PT Polynomial:

- 2a3z-1 - a3z + 6a3z3 + 5a3z5 + a3z7 + 3a5z-1 - 2a5z - 16a5z3 - 17a5z5 - 7a5z7 - a5z9 - a7z-1 + 2a7z + 7a7z3 + 5a7z5 + a7z7

Kauffman Polynomial:

2a2z2 - 7a2z4 + 5a2z6 - a2z8 - 2a3z-1 + 19a3z3 - 31a3z5 + 17a3z7 - 3a3z9 + 3a4 + 3a4z2 - 11a4z4 - a4z6 + 7a4z8 - 2a4z10 - 3a5z-1 - a5z + 29a5z3 - 56a5z5 + 38a5z7 - 8a5z9 + 3a6 + a6z2 - 20a6z4 + 17a6z6 + a6z8 - 2a6z10 - a7z-1 - a7z3 - 4a7z5 + 13a7z7 - 5a7z9 + a8 - 3a8z2 - 4a8z4 + 16a8z6 - 7a8z8 + a9z - 7a9z3 + 16a9z5 - 8a9z7 - 2a10z2 + 9a10z4 - 7a10z6 + 3a11z3 - 5a11z5 + a12z2 - 3a12z4 - a13z3

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 2                       1 j = 0                     2   j = -2                   2 1   j = -4                 6 2     j = -6               4 3       j = -8             7 5         j = -10           5 5           j = -12         5 6             j = -14       3 5               j = -16     2 5                 j = -18   1 3                   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, 147]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 147]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[18, 12, 19, 11], > X[12, 6, 13, 5], X[4, 19, 5, 20], X[16, 7, 17, 8], X[20, 13, 21, 14], > X[22, 15, 7, 16], X[14, 21, 15, 22], X[6, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -6, 5, -11}, > {7, -1, 2, -3, 4, -5, 8, -10, 9, -7, 11, -4, 6, -8, 10, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 3 5 8 10 11 11 9 8 4 -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]=	-30 -28 -26 2 -22 -20 2 3 -14 4 2 -1 + q - q + q - --- - q - q - --- + --- - q + --- + --- + 24 18 16 12 10 q q q q q 3 3 -2 > -- + -- + q 8 6 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 147]][a, z]
Out[8]=	3 5 7 -2 a 3 a a 3 5 7 3 3 5 3 7 3 ----- + ---- - -- - a z - 2 a z + 2 a z + 6 a z - 16 a z + 7 a z + z z z 3 5 5 5 7 5 3 7 5 7 7 7 5 9 > 5 a z - 17 a z + 5 a z + a z - 7 a z + a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 147]][a, z]
Out[9]=	3 5 7 4 6 8 2 a 3 a a 5 9 2 2 4 2 6 2 3 a + 3 a + a - ---- - ---- - -- - a z + a z + 2 a z + 3 a z + a z - z z z 8 2 10 2 12 2 3 3 5 3 7 3 9 3 > 3 a z - 2 a z + a z + 19 a z + 29 a z - a z - 7 a z + 11 3 13 3 2 4 4 4 6 4 8 4 10 4 > 3 a z - a z - 7 a z - 11 a z - 20 a z - 4 a z + 9 a z - 12 4 3 5 5 5 7 5 9 5 11 5 2 6 > 3 a z - 31 a z - 56 a z - 4 a z + 16 a z - 5 a z + 5 a z - 4 6 6 6 8 6 10 6 3 7 5 7 7 7 > a z + 17 a z + 16 a z - 7 a z + 17 a z + 38 a z + 13 a z - 9 7 2 8 4 8 6 8 8 8 3 9 5 9 7 9 > 8 a z - a z + 7 a z + a z - 7 a z - 3 a z - 8 a z - 5 a z - 4 10 6 10 > 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 6 1 2 1 3 2 5 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 22 8 20 7 18 7 18 6 16 6 16 5 14 5 q q q t q t q t q t q t q t q t 5 5 6 5 5 7 5 4 2 t > ------ + ------ + ------ + ------ + ------ + ----- + ---- + ---- + --- + 14 4 12 4 12 3 10 3 10 2 8 2 8 6 4 q t q t q t q t q t q t q t q t q 2 2 t 2 t 2 3 > --- + 2 t + -- + q t 2 2 q q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a147

L11a146 L11a148